Existence and uniqueness solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms

Abstract

In this article, we study the existence and uniqueness solution for a hyperbolic relaxation of the Caginalp phase-field system with singular nonlinear terms with homogenous Dirichlet boundary conditions, in a bounded smooth domain.

Keywords

Hyperbolic relaxation Caginalp phase-field system logarithmic potential Dirichlet boundary conditions

1. Introduction

In this article, we are interested in the study of the following hyperbolic relaxation of the Caginalp phase-field system which is based on a variant of the Maxwell–Cattaneo law for thermal conduction rather than Fourier’s law, in a bounded smooth domain $Ω \subset R^{n}$ ( $1 ⩽ n ⩽ 3$ ). $\begin{array}{l} (1) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u + f (u) = α, \\ (2) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{rcl} (3) & u |_{\partial Ω} = α |_{\partial Ω} = 0 \end{array}$ and initial conditions $\begin{array}{rcl} (4) & u |_{t = 0} = u (0), \frac{\partial u}{\partial t} |_{t = 0} = u_{1}, α |_{t = 0} = α (0), \frac{\partial α}{\partial t} |_{t = 0} = α_{1}, \end{array}$ where $ϵ > 0$ , is a relaxation parameter, $u = u (t, x)$ the order parameter, $α = α (t, x)$ the relative temperature, with $α (t, x) = \int_{0}^{t} θ (x, τ) d τ + α (0)$ , Δ is the Laplacian, f is a given singular potential function.

From a physical point of view, equation (1) can be justified for $ϵ = 0$ , we can rewrite this equation in the form (see [3]) $\begin{array}{rcl} \frac{\partial u}{\partial t} = - \frac{δ Ψ}{δ u}, \end{array}$ where $\frac{δ Ψ}{δ u}$ denotes the variational derivative of the free energy $\begin{array}{rcl} Ψ (u, θ) = \int_{Ω} (\frac{1}{2} | \nabla u |^{2} + F (u) - u \int_{0}^{t} θ (x, τ) d τ) d x, \end{array}$ with F is a primitive of f. The quantity $\frac{δ Ψ}{δ u}$ can be seen as a generalized force which arises as a consequence of the tendency of the free energy to decay towards a minimum. Thus, the relation $\frac{\partial u}{\partial t} = - \frac{δ Ψ}{δ u}$ means that the response of u to the generalized force is instantaneous (see [3]). In certain situations (like rapid phase transformations), the response of u to the generalized force should be subject to a delay expressed by a time dependent relaxation kernel, i.e., $\begin{array}{rcl} \frac{\partial u}{\partial t} = - \int_{- \infty}^{t} k (t - s) \frac{δ Ψ}{δ u} (s) d s, \end{array}$ for a proper relaxation kernel k. A simple and classical choice of kernel reads $\begin{array}{rcl} k (t) = \frac{1}{ϵ} e^{- \frac{t}{ϵ}} . \end{array}$ When $ϵ ⟶ 0$ , then, $k (t) ⟶ δ_{0}$ (the Dirac mass at 0) and we recover (1) for $ϵ = 0$ . Now,when $ϵ > 0$ , we find, by differentiation with respect to t, (1).

Similar parabolic studies have already been done with $ϵ = 0$ in many books with different types of boundary conditions and regular or singular potentials, see for example [1,2,4–9,11].

The main difficulty in this article is to prove the existence of the solution of the problem (1)–(2) when $ϵ > 0$ which can not be obtained by classical methods including the Faedo Galerkin method. For this, the proof follows a perturbation technique used by Grasseli, Miranville, Pata and Zelik (see [3]).

Thus, our aim in this article is to establish the existence and uniqueness of the solution of the problem (1)–(2) as follows:

In Section 1, we establish uniform estimates; then in Section 2, we study the existence of the solution of the auxiliary problem (with $ϵ = 0$ ) obtained by modifying the equation (1) via the uniform estimates and the strict separation; and then, we show that the solution of the problem (1)–(2) coincides with that of the auxiliary problem ( $ϵ = 0$ ), when ϵ is small enough; however, we can use, in Section 3, the existence of the solution of the problem (1)–(2) with $ϵ > 0$ ; and in Section 4, we study the uniqueness of the solution of the problem (1)–(2) with $ϵ > 0$ . Finally, in Section 5, we establish the uniform estimates with more regularity.

f satisfies the following assumptions: $\begin{array}{l} (5) & f \in C^{3} (] - 1, 1 [), lim_{s \to \pm 1} f (s) = \pm \infty, lim_{s \to \pm 1} f^{'} (s) = + \infty, f (0) = 0, \\ (6) & - c_{0} ⩽ F (s) ⩽ f (s) s + c_{0}, c_{0} ⩾ 0, for all s \in] - 1, 1 [, where F (s) = \int_{0}^{s} f (τ) d τ \end{array}$ and $\begin{matrix} (7) & f^{'} (s) ⩾ - c_{1}, c_{1} ⩾ 0, for all s \in] - 1, 1 [. \end{matrix}$ We recall that a typical example of function f is: $\begin{matrix} f (s) = k_{1} ln (\frac{1 + s}{1 - s}) - k_{0} s, for all s \in] - 1, 1 [, with 0 < k_{1} < k_{0} . \end{matrix}$

1.1. Notations

It is worth emphasizing that equations (1) and (2) have a sense only if $\begin{array}{rcl} - 1 < u (t, x) < 1, \end{array}$ for almost all $(t, x) \in [0, T] \times Ω$ . That is the reason why it is natural to introduce the following quantity: $\begin{matrix} D [u (t)] = {(1 - {‖ u (t) ‖}_{L^{\infty}})}^{- 1} . \end{matrix}$ We also introduce, for every $k = 1, 2$ , the standard energy norm for the equation (1): $\begin{matrix} {‖ ζ_{u} (t) ‖}_{ε^{k} (ϵ)}^{2} = {‖ u (t) ‖}_{H^{k + 1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{k}}^{2} + {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{k - 1}}^{2} . \end{matrix}$ Thus, the space $ε^{k} (ϵ)$ coincides with $[H^{k + 1} (Ω) \times H^{k} (Ω)] \cap {ζ |_{\partial Ω} = 0}$ if $ϵ > 0$ and with $[H^{k + 1} (Ω) \times H^{k - 1} (Ω)] \cap {ζ |_{\partial Ω} = 0}$ if $ϵ = 0$ , whenever the traces make sense. We denote by $‖ \cdot ‖$ and $(\cdot, \cdot)$ (or $‖ \cdot ‖_{Φ}$ and ${(\cdot, \cdot)}_{Φ}$ ) the norm and the scalar product in $L^{2} (Ω)$ (in Φ).

2. Auxiliary problem

This auxiliary problem is a problem corresponding to $ϵ = 0$ of the problem (1)–(2), obtained by modifying the equation (1) as follows $\begin{array}{l} (8) & \frac{\partial u_{0}}{\partial t} - Δ u_{0} + f (u_{0}) + L u_{0} = α_{0} + L u, \\ (9) & \frac{\partial^{2} α_{0}}{\partial t^{2}} + \frac{\partial α_{0}}{\partial t} - Δ \frac{\partial α_{0}}{\partial t} - Δ α_{0} = - u_{0} - \frac{\partial u_{0}}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{rcl} (10) & u_{0} |_{\partial Ω} = α_{0} |_{\partial Ω} = 0 \end{array}$ and initial conditions $\begin{array}{rcl} (11) & u_{0} |_{t = 0} = u (0), α_{0} |_{t = 0} = α (0), \frac{\partial α_{0}}{\partial t} |_{t = 0} = α_{1}, \end{array}$ where $L > 0$ , and f checks the conditions of the starting problem.

3. Uniform estimates and existence of solution

Theorem 3.1.
Let $ϵ ⩽ ϵ_{0}$ and f nonlinearity satisfying ( 5 ). Then there is a decreasing function $R : (0, ϵ_{0}] ⟶ R_{+}$ such that $\begin{array}{rcl} (12) & lim_{ϵ \to 0} R (ϵ) = + \infty \end{array}$ and for every initial data $(ζ_{u} (0), α (0), α_{1})$ satisfying $\begin{array}{rcl} (13) & D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2} ⩽ R (ϵ), \end{array}$ the system ( 1 )–( 2 ) has a unique global solution $(u, α)$ such that $u, α \in L^{\infty} (0, T; H^{2} (Ω) \cap H_{0}^{1} (Ω))$ sufficiently regular which is separated from the singular points, i.e. $\begin{array}{rcl} (14) & ‖ u ‖_{L^{\infty} ((0, T) \times Ω)} < 1, \end{array}$ satisfies the following estimate $\begin{array}{l} D [u (t)] + {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ \\ (15) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

We begin by proving the following classical lemma giving uniform estimates of $‖ u ‖_{H^{1}}^{2}$ , $‖ α ‖^{2}$ and $‖ \frac{\partial α}{\partial t} ‖^{2}$ independent of ϵ.
Lemma 3.1.
Let $ϵ ⩽ ϵ_{0}$ and f nonlinearity satisfying ( 5 ). Then, the solution $(ζ_{u} (t), α (t))$ satisfies the following estimate $\begin{array}{l} {‖ u (t) ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ α (t) ‖}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}^{2} + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}^{2}) e^{- β (t - τ)} d τ \\ (16) & ⩽ Q ({‖ u (0) ‖}_{H^{1}}^{2} + ϵ_{0} ‖ u_{1} ‖^{2} + {‖ α (0) ‖}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.
Proof.
Multiplying (1) by u and integreting over Ω, we get $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (‖ u ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, u)) + ‖ \nabla u ‖^{2} + \int_{Ω} f (u) u d x = ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + (α, u), \end{array}$ which implies, using (6) $\begin{array}{rcl} (17) & \frac{d}{d t} (‖ u ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, u)) + ‖ \nabla u ‖^{2} ⩽ 2 ϵ_{0} {‖ \frac{\partial u}{\partial t} ‖}^{2} + c_{p}^{2} ‖ \nabla α ‖^{2} + C, \end{array}$ where $c_{p}$ is Poincare’s constant.

Multiplying (1) by $\frac{\partial u}{\partial t}$ and integreting over Ω, we have $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial u}{\partial t} ‖}^{2} = (α, \frac{\partial u}{\partial t}), \end{array}$ we get using the inequalities of Hölder and Young $\begin{array}{rcl} (18) & \frac{d}{d t} (ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial u}{\partial t} ‖}^{2} & ⩽ & c_{p} ‖ \nabla α ‖^{2} . \end{array}$

Multiplying (2) by α and integreting over Ω, and using the inequalities of Hölder and Young, we find $\begin{array}{rcl} (19) & \frac{d}{d t} (‖ α ‖^{2} + ‖ \nabla α ‖^{2} + 2 (\frac{\partial α}{\partial t}, α)) + ‖ \nabla α ‖^{2} ⩽ 2 {‖ \frac{\partial α}{\partial t} ‖}^{2} + 2 c_{p}^{2} ‖ \nabla u ‖^{2} + 2 c_{p} {‖ \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$

Multiplying (2) by $\frac{\partial α}{\partial t}$ and integreting over Ω, we have $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} ({‖ \frac{\partial α}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2}) + {‖ \frac{\partial α}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} = - (u, \frac{\partial α}{\partial t}) - (\frac{\partial u}{\partial t}, \frac{\partial α}{\partial t}), \end{array}$ after applying the inequalities of Hölder and Young, we get $\begin{array}{rcl} \frac{d}{d t} ({‖ \frac{\partial α}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2}) + {‖ \frac{\partial α}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} & ⩽ & 2 c_{p} ‖ \nabla u ‖^{2} + 2 {‖ \frac{\partial u}{\partial t} ‖}^{2}, \end{array}$ in particular $\begin{array}{rcl} (20) & \frac{d}{d t} ({‖ \frac{\partial α}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2}) + {‖ \frac{\partial α}{\partial t} ‖}^{2} & ⩽ & 2 c_{p} ‖ \nabla u ‖^{2} + 2 {‖ \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{1}$ (17), $γ_{2}$ (18), $γ_{3}$ (19) and $γ_{4}$ (20), with $γ_{i} > 0$ such that $\begin{array}{l} γ_{1} - 2 c_{p}^{2} γ_{3} - 2 c_{p} γ_{4} > 0 \\ γ_{2} - 2 ϵ_{0} γ_{1} - 2 c_{p} γ_{3} - 2 γ_{4} > 0 \\ γ_{3} - c_{p}^{2} γ_{1} - c_{p} γ_{2} > 0 \\ γ_{4} - 2 γ_{3} > 0, \end{array}$ we have $\begin{array}{rcl} (21) & \frac{d}{d t} E_{3} + C_{1} ‖ \nabla u ‖^{2} + C_{2} {‖ \frac{\partial u}{\partial t} ‖}^{2} + C_{3} ‖ \nabla α ‖^{2} + C_{4} {‖ \frac{\partial α}{\partial t} ‖}^{2} ⩽ C, C_{i}, C > 0, \end{array}$ which implies $\begin{array}{rcl} (22) & \frac{d}{d t} E_{3} + C_{1} ‖ \nabla u ‖^{2} + C_{2} {‖ \frac{\partial u}{\partial t} ‖}^{2} + \tilde{C_{3}} ‖ α ‖^{2} + C_{4} {‖ \frac{\partial α}{\partial t} ‖}^{2} ⩽ C, C_{i}, \tilde{C_{3}}, C > 0, \end{array}$ where $\begin{array}{rcl} E_{3} & = & γ_{1} (‖ u ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, u)) + γ_{2} (ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ \nabla u ‖^{2} + 2 \int_{Ω} F (u) d x) \\ + γ_{3} (‖ α ‖^{2} + ‖ \nabla α ‖^{2} + 2 (\frac{\partial α}{\partial t}, α)) + γ_{4} ({‖ \frac{\partial α}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2}) . \end{array}$

We know that $\begin{array}{rcl} (23) & γ_{2} ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + γ_{1} (‖ u ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, u)) ⩾ ϵ (γ_{2} - 2 γ_{1}) {‖ \frac{\partial u}{\partial t} ‖}^{2} + \frac{γ_{1}}{2} ‖ u ‖^{2}, \end{array}$ and $\begin{array}{rcl} (24) & γ_{4} {‖ \frac{\partial α}{\partial t} ‖}^{2} + γ_{3} (‖ α ‖^{2} + 2 (\frac{\partial α}{\partial t}, α)) ⩾ (γ_{4} - 2 γ_{3}) {‖ \frac{\partial α}{\partial t} ‖}^{2} + \frac{γ_{3}}{2} ‖ α ‖^{2} . \end{array}$ Choosing $γ_{1}$ and $γ_{2}$ such that $\begin{array}{rcl} γ_{2} - 2 γ_{1} > 0, \end{array}$ there is then a positive constant C independent of ϵ, and according (6), such that $\begin{array}{l} C^{- 1} ({‖ u (t) ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ α (t) ‖}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}^{2} - 2 c_{0} | Ω |) \\ ⩽ E_{3} (t) \\ (25) & ⩽ C ({‖ u (t) ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ α (t) ‖}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}^{2} + \int_{Ω} F (u) d x) . \end{array}$ We obtain using (25) an estimate of the form $\begin{array}{rcl} (26) & \frac{d}{d t} E_{3} + β E_{3} + C_{1} {‖ \frac{\partial u}{\partial t} ‖}^{2} + C_{2} {‖ \frac{\partial α}{\partial t} ‖}^{2} ⩽ C, \end{array}$ where β, C and $C_{i}$ are positive constants independent of ϵ.

Applying Gronwall’s lemma to the relation (26) and taking into account (25), we find, for all $t \in [0, T]$ $\begin{array}{l} {‖ u (t) ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ α (t) ‖}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}^{2} + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}^{2}) e^{- β (t - τ)} d τ \\ (27) & ⩽ Q ({‖ u (0) ‖}_{H^{1}}^{2} + ϵ_{0} ‖ u_{1} ‖^{2} + {‖ α (0) ‖}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ. □

The following lemma gives uniform estimates of $‖ α ‖_{H^{2}}^{2}$ , $‖ α ‖_{H^{1}}^{2}$ and $‖ \frac{\partial α}{\partial t} ‖_{H^{1}}^{2}$ independent of ϵ.
Lemma 3.2.
Let $ϵ ⩽ ϵ_{0}$ and f nonlinearity satisfying ( 5 ). Then, the solution $(u, α)$ satisfies the following estimate $\begin{array}{l} {‖ α (t) ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} + \int_{0}^{t} ({‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ α (τ) ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (28) & ⩽ Q ({‖ u (0) ‖}_{H^{1}}^{2} + ϵ_{0} ‖ u_{1} ‖^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.
Proof.
Multiplying (2) by $- Δ \frac{\partial α}{\partial t}$ and integrating Ω, and then using Hölder Young’s inequalities, we get $\begin{array}{rcl} (29) & \frac{d}{d t} ({‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + ‖ Δ α ‖^{2}) + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial α}{\partial t} ‖}^{2} ⩽ ‖ \nabla u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$ Multiplying (2) by $- Δ α$ and integrating over Ω, then using the inequalities of Hölder and Young, we obtain $\begin{array}{rcl} (30) & \frac{d}{d t} (‖ \nabla α ‖^{2} + ‖ Δ α ‖^{2} + 2 (\nabla \frac{\partial α}{\partial t}, \nabla α)) + ‖ Δ α ‖^{2} ⩽ 2 {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + 2 {‖ \frac{\partial u}{\partial t} ‖}^{2} + 2 c_{p} ‖ \nabla u ‖^{2} . \end{array}$

Adding (21), $γ_{5}$ (29) and $γ_{6}$ (30), with $γ_{i} > 0$ such that $\begin{array}{l} C_{1} - γ_{5} - 2 c_{p} γ_{6} > 0 \\ C_{2} - γ_{5} - 2 γ_{6} > 0 \\ γ_{5} - 2 γ_{6} > 0, \end{array}$ we have $\begin{array}{l} \frac{d}{d t} E_{4} + C_{1} ‖ \nabla u ‖^{2} + C_{2} {‖ \frac{\partial u}{\partial t} ‖}^{2} + C_{3} ‖ \nabla α ‖^{2} + C_{4} ‖ Δ α ‖^{2} + C_{5} {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + C_{6} {‖ Δ \frac{\partial α}{\partial t} ‖}^{2} ⩽ C, \\ C_{i}, C > 0, \end{array}$ where $\begin{array}{rcl} E_{4} = E_{3} + γ_{5} ({‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + ‖ Δ α ‖^{2}) + γ_{6} (‖ \nabla α ‖^{2} + ‖ Δ α ‖^{2} + 2 (\nabla \frac{\partial α}{\partial t}, \nabla α)) . \end{array}$

We know that $\begin{array}{rcl} (31) & γ_{5} {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + γ_{6} (‖ \nabla α ‖^{2} + 2 (\nabla \frac{\partial α}{\partial t}, \nabla α)) ⩾ (γ_{5} - 2 γ_{6}) {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + \frac{γ_{6}}{2} ‖ \nabla α ‖^{2} . \end{array}$ There is then a positive constant C independent of ϵ, and using (6), such that $\begin{array}{l} C^{- 1} ({‖ u (t) ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ α (t) ‖}_{H^{1}}^{2} + {‖ α (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} - 2 c_{0} | Ω |) \\ ⩽ E_{4} (t) \\ (32) & ⩽ C ({‖ u (t) ‖}_{H^{1}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} + {‖ α (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} + \int_{Ω} F (u) d x) . \end{array}$ We obtain using (32) a estimate of the form $\begin{array}{rcl} (33) & \frac{d}{d t} E_{4} + β E_{4} + C_{1} {‖ \frac{\partial α}{\partial t} ‖}_{H^{1}}^{2} + C_{2} ‖ α ‖_{H^{2}}^{2} ⩽ C, \end{array}$ where C and β are positive constants independent of ϵ.

Applying Gronwall’s lemma to the relation (33) and taking into account (32), we find, for all $t \in [0, T]$ $\begin{array}{l} {‖ α (t) ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} + \int_{0}^{t} ({‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ α (τ) ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (34) & ⩽ Q ({‖ u (0) ‖}_{H^{1}}^{2} + ϵ_{0} ‖ u_{1} ‖^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ. □

We know that if the problem (1)–(4) possesses a solution such that $‖ u ‖_{L^{\infty}} < 1$ , then this solution has the regularity given by the lemma 3.1.

Let’s compare the solution $(u, α)$ of problem (1)–(4) with the solution $(u_{0}, α_{0})$ of auxiliary problem (8)–(11).

We start with the following lemma
Lemma 3.3.
Assume f nonlinearity satisfying ( 5 ) and u such that $‖ u (t) ‖_{L^{\infty}} < 1$ . Then, the problem ( 8 )–( 11 ) possesses at least a solution $(u_{0} (t), α_{0} (t))$ satisfying the following estimate $\begin{array}{l} D [u_{0} (t)] + {‖ u_{0} (t) ‖}_{H^{2}}^{2} + {‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (35) & ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

The proof begins with the following lemma. For this, assume the existence of the solution of the auxiliary problem (8)–(11).
Lemma 3.4.
Assume f nonlinearity satisfying ( 5 ) and $u (t)$ such that $‖ u (t) ‖_{L^{\infty}} ⩽ 1$ and if the problem ( 8 )–( 11 ) possesses a solution $(u_{0} (t), α_{0} (t))$ , then this satisfies the following estimate $\begin{array}{l} {‖ u_{0} (t) ‖}_{H^{1}}^{2} + {‖ α_{0} (t) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial α_{0} (τ)}{\partial t} ‖}^{2} + {‖ α_{0} (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial u_{0} (τ)}{\partial t} ‖}^{2}) e^{- β (t - τ)} d τ \\ (36) & ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{1}}^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.
Proof.
Multiplying (8) by $u_{0}$ and integrating over Ω, we obtain $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (‖ u_{0} ‖^{2}) + ‖ \nabla u_{0} ‖^{2} + L ‖ u_{0} ‖^{2} + \int_{Ω} f (u_{0}) u_{0} d x = (α_{0}, u_{0}) + (L u, u_{0}), \end{array}$ which implies using inequalities of Hölder, Young and the relation (6) $\begin{array}{rcl} (37) & \frac{d}{d t} ‖ u_{0} ‖^{2} + 2 ‖ \nabla u_{0} ‖^{2} + 2 (L - c_{p}) ‖ u_{0} ‖^{2} ⩽ ‖ \nabla α_{0} ‖^{2} + C, C, L - c_{p} > 0 . \end{array}$ Multiplying (8) by $\frac{\partial u_{0}}{\partial t}$ and integrating over Ω, we have $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} (L ‖ u_{0} ‖^{2} + ‖ \nabla u_{0} ‖^{2} + 2 \int_{Ω} F (u_{0}) d x) + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} = (α_{0}, \frac{\partial u_{0}}{\partial t}) + (L u, \frac{\partial u_{0}}{\partial t}), \end{array}$ we get using inequalities of Hölder and Young $\begin{array}{rcl} (38) & \frac{d}{d t} (L ‖ u_{0} ‖^{2} + ‖ \nabla u_{0} ‖^{2} + 2 \int_{Ω} F (u_{0}) d x) + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} ⩽ 2 c_{p} ‖ \nabla α_{0} ‖^{2} + C . \end{array}$

Multiplying now (9) by $α_{0}$ and integrating over Ω, then applying inequalities of Hölder and Young, we have $\begin{array}{rcl} (39) & \frac{d}{d t} (‖ α_{0} ‖^{2} + ‖ \nabla α_{0} ‖^{2} + 2 (\frac{\partial α_{0}}{\partial t}, α_{0})) + ‖ \nabla α_{0} ‖^{2} ⩽ 2 {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + 2 c_{p}^{2} ‖ \nabla u_{0} ‖^{2} + 2 c_{p} {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} . \end{array}$

Multiplying (9) by $\frac{\partial α_{0}}{\partial t}$ and integrating over Ω, we have $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} ({‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + ‖ \nabla α_{0} ‖^{2}) + {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} = - (u_{0}, \frac{\partial α_{0}}{\partial t}) - (\frac{\partial u_{0}}{\partial t}, \frac{\partial α_{0}}{\partial t}), \end{array}$ applying inequalities Hölder and Young we get $\begin{array}{rcl} \frac{d}{d t} ({‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + ‖ \nabla α_{0} ‖^{2}) + {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} & ⩽ & 2 c_{p} ‖ \nabla u_{0} ‖^{2} + 2 {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}, \end{array}$ in particular $\begin{array}{rcl} (40) & \frac{d}{d t} ({‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + ‖ \nabla α_{0} ‖^{2}) + {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} & ⩽ & 2 c_{p} ‖ \nabla u_{0} ‖^{2} + 2 {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{7}$ (37), $γ_{8}$ (38), $γ_{9}$ (39) and $γ_{10}$ (40), with $γ_{i} > 0$ such that $\begin{array}{l} 2 γ_{7} - 2 c_{p}^{2} γ_{9} - 2 c_{p} γ_{10} > 0 \\ γ_{8} - 2 γ_{9} - 2 γ_{10} > 0 \\ γ_{9} - γ_{7} - 2 c_{p} γ_{8} > 0 \\ γ_{10} - 2 γ_{9} > 0, \end{array}$ we have $\begin{array}{rcl} \frac{d}{d t} E_{5} + C_{1} ‖ \nabla u_{0} ‖^{2} + C_{2} ‖ u_{0} ‖^{2} + C_{3} ‖ \nabla α_{0} ‖^{2} + C_{4} {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} + C_{5} {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} ⩽ C, \end{array}$ where $\begin{array}{rcl} E_{5} & = & γ_{7} ‖ u_{0} ‖^{2} + γ_{8} (L ‖ u_{0} ‖^{2} + ‖ \nabla u_{0} ‖^{2} + 2 \int_{Ω} F (u_{0}) d x) \\ + γ_{9} (‖ α_{0} ‖^{2} + ‖ \nabla α_{0} ‖^{2} + 2 (\frac{\partial α_{0}}{\partial t}, α_{0})) + γ_{10} ({‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + ‖ \nabla α_{0} ‖^{2}) . \end{array}$

We know that $\begin{array}{rcl} (41) & γ_{10} {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + γ_{9} (‖ α_{0} ‖^{2} + 2 (\frac{\partial α_{0}}{\partial t}, α_{0})) ⩾ (γ_{10} - 2 γ_{9}) {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + \frac{γ_{9}}{2} ‖ α_{0} ‖^{2} . \end{array}$ There is then a positive constant C independent of ϵ, and using (6), such that $\begin{array}{l} C^{- 1} ({‖ u_{0} (t) ‖}_{H^{1}}^{2} + {‖ α_{0} (t) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}^{2} - 2 c_{0} | Ω |) \\ ⩽ E_{5} (t) \\ (42) & ⩽ C ({‖ u_{0} (t) ‖}_{H^{1}}^{2} + {‖ α_{0} (t) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}^{2} + \int_{Ω} F (u_{0}) d x) . \end{array}$ We obtain using (42) a estimate of the form $\begin{array}{rcl} (43) & \frac{d}{d t} E_{5} + β E_{5} + C_{1} {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + C_{2} ‖ α_{0} ‖_{H^{1}}^{2} + C_{3} {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} ⩽ C . \end{array}$ Gronwall’s lemma and (42) we have, for all $t \in [0, T]$ $\begin{array}{l} {‖ u_{0} (t) ‖}_{H^{1}}^{2} + {‖ α_{0} (t) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial α_{0} (τ)}{\partial t} ‖}^{2} + {‖ α_{0} (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial u_{0} (τ)}{\partial t} ‖}^{2}) e^{- β (t - τ)} d τ \\ (44) & ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{1}}^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ. □

Our now aim is to obtain uniform estimates of $‖ u_{0} (t) ‖_{H^{2}}^{2}$ , $‖ α_{0} (t) ‖_{H^{2}}^{2}$ , $‖ \frac{\partial α_{0} (t)}{\partial t} ‖_{H^{1}}^{2}$ and $D [u_{0} (t)]$ . then, strict separation and the existence of the solution of auxiliary problem (8)–(11). For this, we proof the following lemma
Lemma 3.5.
Assume the hypothesis of Lemma 3.4 such that $‖ u_{0} ‖_{L^{\infty} ((0, T) \times Ω)} < 1$ . Then, the solution $(u_{0} (t), α_{0} (t))$ satisfies the following estimate $\begin{array}{l} D [u_{0} (t)] + {‖ u_{0} (t) ‖}_{H^{2}}^{2} + {‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (45) & ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.
Proof.
We differentiate with respect to t the equation (8), we obtain $\begin{array}{l} (46) & \frac{\partial^{2} u_{0}}{\partial t^{2}} - Δ \frac{\partial u_{0}}{\partial t} + f^{'} (u_{0}) \frac{\partial u_{0}}{\partial t} + L \frac{\partial u_{0}}{\partial t} = \frac{\partial α_{0}}{\partial t} + L \frac{\partial u}{\partial t}, \\ (47) & \frac{\partial u_{0}}{\partial t} |_{t = 0} = Δ u (0) - f (u (0)) + α (0) . \end{array}$

We now multiply (46) by $\frac{\partial u_{0}}{\partial t}$ , then integrate over Ω, and which, using the inequalities of Hölder, Young and the property (7) $\begin{array}{rcl} (48) & \frac{d}{d t} {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial u_{0}}{\partial t} ‖}^{2} + β {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} ⩽ C ({‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2}), β > 0 . \end{array}$ Gronwall’s lemma implies $\begin{array}{l} {‖ \frac{\partial u_{0} (t)}{\partial t} ‖}^{2} + \int_{0}^{t} {‖ \nabla \frac{\partial u_{0} (τ)}{\partial t} ‖}^{2} e^{- β (t - τ)} d τ \\ (49) & ⩽ {‖ \frac{\partial u_{0} (0)}{\partial t} ‖}^{2} e^{- β t} + C \int_{0}^{t} ({‖ \frac{\partial α_{0} (τ)}{\partial t} ‖}^{2} + {‖ \frac{\partial u (τ)}{\partial t} ‖}^{2}) e^{- β (t - τ)} d τ . \end{array}$ We note that $\begin{array}{rcl} {‖ \frac{\partial u_{0} (0)}{\partial t} ‖}^{2} & = & {‖ Δ u (0) - f (u (0)) + α (0) ‖}^{2} . \end{array}$ We also note that, obviously, $\begin{array}{rcl} (50) & {‖ \frac{\partial u_{0} (0)}{\partial t} ‖}^{2} ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{1}}^{2}) . \end{array}$ Using (27), (44) and (50), the estimate (49) becomes $\begin{array}{l} {‖ \frac{\partial u_{0} (t)}{\partial t} ‖}^{2} + \int_{0}^{t} {‖ \nabla \frac{\partial u_{0} (τ)}{\partial t} ‖}^{2} e^{- β (t - τ)} d τ \\ (51) & ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C . \end{array}$ Thus, $\begin{array}{rcl} (52) & {‖ \frac{\partial u_{0} (t)}{\partial t} ‖}^{2} ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C . \end{array}$ Multiplying (8) by $- Δ u_{0}$ and integrating over Ω, we have $\begin{array}{rcl} ‖ Δ u_{0} ‖^{2} + (f^{'} (u_{0}) \nabla u_{0}, \nabla u_{0}) = (α_{0}, - Δ u_{0}) + (L u, - Δ u_{0}) + (\frac{\partial u_{0}}{\partial t}, Δ u_{0}) + (L u_{0}, Δ u_{0}), \end{array}$ which, using Hölder and Young inequalities and (7) $\begin{array}{rcl} ‖ Δ u_{0} ‖^{2} ⩽ C (‖ \nabla u_{0} ‖^{2} + ‖ α_{0} ‖^{2} + ‖ u ‖^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}), \end{array}$ therefore $\begin{array}{rcl} (53) & {‖ u_{0} (t) ‖}_{H^{2}}^{2} ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C . \end{array}$ Asking $\begin{array}{rcl} (54) & \frac{\partial^{2} α_{0}}{\partial t^{2}} + \frac{\partial α_{0}}{\partial t} - Δ \frac{\partial α_{0}}{\partial t} - Δ α_{0} = g_{u_{0}} (t) & = & - u_{0} - \frac{\partial u_{0}}{\partial t} . \end{array}$ We Multiply (54) by $- Δ α_{0}$ and integrate over Ω, we obtain applying Hölder and Young inequalities $\begin{array}{l} \frac{d}{d t} (‖ \nabla α_{0} ‖^{2} + ‖ Δ α_{0} ‖^{2} + 2 (\nabla \frac{\partial α_{0}}{\partial t}, \nabla α_{0})) + ‖ Δ α_{0} ‖^{2} \\ (55) & ⩽ 2 {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + 2 c_{p}^{2} ‖ \nabla u_{0} ‖^{2} + 2 {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} . \end{array}$ Multiplying (54) by $- Δ \frac{\partial α_{0}}{\partial t}$ and integrate over Ω, we have applying Hölder and Young inequalities $\begin{array}{rcl} (56) & \frac{d}{d t} ({‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + ‖ Δ α_{0} ‖^{2}) + 2 {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + {‖ Δ \frac{\partial α_{0}}{\partial t} ‖}^{2} ⩽ 2 c_{p}^{2} ‖ \nabla u_{0} ‖^{2} + 2 {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} . \end{array}$ Adding $γ_{11}$ (55) and $γ_{12}$ (56), with $γ_{i} > 0$ such that $\begin{array}{rcl} γ_{12} - γ_{11} > 0, \end{array}$ we have $\begin{array}{rcl} \frac{d}{d t} E_{6} + C_{1} ‖ Δ α_{0} ‖^{2} + C_{2} {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + C_{3} {‖ \frac{\partial α_{0}}{\partial t} ‖}^{2} ⩽ C (‖ \nabla u_{0} ‖^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}), C_{i}, C > 0, \end{array}$ where $\begin{array}{rcl} E_{6} = γ_{11} (‖ \nabla α_{0} ‖^{2} + ‖ Δ α_{0} ‖^{2} + 2 (\nabla \frac{\partial α_{0}}{\partial t}, \nabla α_{0})) + γ_{12} ({‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + ‖ Δ α_{0} ‖^{2}) . \end{array}$

We know that $\begin{array}{rcl} (57) & γ_{12} {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + γ_{11} (‖ \nabla α_{0} ‖^{2} + 2 (\nabla \frac{\partial α_{0}}{\partial t}, \nabla α_{0})) ⩾ (γ_{12} - 2 γ_{11}) {‖ \nabla \frac{\partial α_{0}}{\partial t} ‖}^{2} + \frac{γ_{11}}{2} ‖ \nabla α_{0} ‖^{2} . \end{array}$ Shoosing $γ_{11}$ and $γ_{12}$ such that $\begin{array}{rcl} γ_{12} - 2 γ_{11} > 0, \end{array}$ there exist then the positive constant C independent of ϵ such that $\begin{array}{rcl} (58) & C^{- 1} ({‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2}) ⩽ E_{6} (t) ⩽ C ({‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2}) . \end{array}$ We obtain a inequality of the form $\begin{array}{rcl} (59) & \frac{d}{d t} E_{6} + β E_{6} + C_{1} {‖ \frac{\partial α_{0}}{\partial t} ‖}_{H^{2}}^{2} ⩽ C (‖ \nabla u_{0} ‖^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2} + ‖ \nabla α_{0} ‖^{2}) . \end{array}$ Gronwall’s lemma and the estimates (44) and (52), then using (58), we end to $\begin{array}{l} {‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2} + \int_{0}^{t} ({‖ \frac{\partial α_{0} (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ ⩽ Q ({‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ Therefore $\begin{array}{rcl} (60) & {‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2} ⩽ Q ({‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C . \end{array}$ Adding the estimates (53) and (60), we obtain $\begin{array}{l} {‖ u_{0} (t) ‖}_{H^{2}}^{2} + {‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (61) & ⩽ Q ({‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

In order to do so, we rewrite the equation of (1) as follows: $\begin{array}{rcl} (62) & \frac{\partial u_{0}}{\partial t} - Δ u_{0} + f (u_{0}) = h_{α_{0}, u_{0}, u} (t) = α_{0} + L (u - u_{0}), \end{array}$ and $H^{2} (Ω) \subset C (\overline{Ω})$ , then we have $\begin{array}{rcl} (63) & {‖ h_{α_{0}, u_{0}, u} (t) ‖}_{L^{\infty}}^{2} ⩽ Q ({‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C . \end{array}$ We consider the functions $y_{\pm} (t)$ solve the following ODEs: $\begin{array}{rcl} (64) & y_{\pm}^{'} + f (y_{\pm}) = h_{\pm} (t) = \pm {‖ h_{α_{0}, u_{0}, u} (t) ‖}_{L^{\infty}}^{2}, \end{array}$ where $y_{\pm} (0) = \pm ‖ u_{0} ‖_{L^{\infty}}$ and $h_{\pm} (t) = \pm ‖ α_{0} (t) + L (u - u_{0}) (t) ‖_{L^{\infty}}^{2}$ .

Moreover, the solutions of (64) satisfy the following inequalities (see [3] and [12]): $\begin{array}{l} (65) & \begin{array}{l} {‖ y_{\pm} (t) ‖}_{L^{\infty}} ⩽ 1 - δ (D [u (0)] + {‖ h_{\pm} (t) ‖}_{L^{\infty} ([0, 1])}), 0 ⩽ t ⩽ 1, \\ {‖ y_{\pm} (t + 1) ‖}_{L^{\infty}} ⩽ 1 - δ ({‖ h_{\pm} (t) ‖}_{L^{\infty} ([t, t + 1])}), t ⩾ 0, δ > 0 . \end{array} \end{array}$ Applying to the comparison principle to, we have $\begin{array}{rcl} (66) & y_{-} (t) ⩽ u (t, x) ⩽ y_{+} (t), (t, x) \in (0, T) \times Ω . \end{array}$ And using the estimates (63), (65) and (66), we deduce that $\begin{array}{rcl} (67) & D [u_{0} (t)] ⩽ Q ({‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ therefore $\begin{array}{l} D [u_{0} (t)] + {‖ u_{0} (t) ‖}_{H^{2}}^{2} + {‖ α_{0} (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α_{0} (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (68) & ⩽ Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ.

Finally, we have the following strict separation $\begin{array}{rcl} (69) & ‖ u_{0} ‖_{L^{\infty} ((0, T) \times Ω)} ⩽ 1 - δ, \end{array}$ where δ positive constant.

We show the existence of solution of the auxiliary problem (8)–(11).

To show the existence of solution of the auxiliary problem (8)–(11), we first regularize the potential f by a potential $f_{δ}$ of class $C^{1}$ defined by: $\begin{array}{rcl} (70) & f_{δ} (s) = \{\begin{matrix} f (δ) + f^{'} (δ) (s - δ) & si s > δ, \\ f (s) & si - δ ⩽ s ⩽ δ, \\ f (- δ) + f^{'} (- δ) (s + δ) & si s < - δ, \end{matrix} \end{array}$ where δ is the constant appearing above that we choose close enough to 1, such that $\begin{array}{rcl} (71) & f (δ) ⩾ 0, f (- δ) ⩽ 0, f^{'} (δ) ⩾ 0 and f^{'} (- δ) ⩾ 0 . \end{array}$ We consider the same auxiliary problem (8)–(11) and we replace $(u_{0}, α_{0})$ with $(u_{0}^{δ}, α_{0}^{δ})$ , which gives $\begin{array}{l} (72) & \frac{\partial u_{0}^{δ}}{\partial t} - Δ u_{0}^{δ} + f_{δ} (u_{0}^{δ}) + L u_{0}^{δ} = α_{0}^{δ} + L u, \\ (73) & \frac{\partial^{2} α_{0}^{δ}}{\partial t^{2}} + \frac{\partial α_{0}^{δ}}{\partial t} - Δ \frac{\partial α_{0}^{δ}}{\partial t} - Δ α_{0}^{δ} = - u_{0}^{δ} - \frac{\partial u_{0}^{δ}}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{rcl} (74) & u_{0}^{δ} |_{\partial Ω} = α_{0}^{δ} |_{\partial Ω} = 0 \end{array}$ and initial conditions $\begin{array}{rcl} (75) & u_{0}^{δ} |_{t = 0} = u (0), α_{0}^{δ} |_{t = 0} = α (0), \frac{\partial α_{0}^{δ}}{\partial t} |_{t = 0} = α_{1}, \end{array}$ with $u \in L^{\infty} (0, T; H^{1} (Ω))$ .

We know that the problem (72)–(75) possesses a regular solution $(u_{0}^{δ}, α_{0}^{δ})$ (see [2]).

Moreover, we know that the auxiliary problem (8)–(11) possesses at least one solution if $f_{δ}$ and $F_{δ}$ satisfy the same properties as f and F. We have:
For $s > δ$ , we have $\begin{array}{rcl} f_{δ}^{'} (s) = f^{'} (δ) ⩾ - c_{1} and F_{δ} (s) ⩾ - c_{0} . \end{array}$

For $- δ ⩽ s ⩽ δ$ , we have $\begin{array}{rcl} f_{δ}^{'} (s) = f^{'} (δ) ⩾ - c_{1} and F_{δ} (s) ⩾ - c_{0} . \end{array}$

For $s < - δ$ , we have $\begin{array}{rcl} f_{δ}^{'} (s) = f^{'} (δ) ⩾ - c_{1} and F_{δ} (s) ⩾ - c_{0} . \end{array}$
Therefore the estimates made above are true.

In particular, $\begin{array}{rcl} {‖ u_{0}^{δ} (t) ‖}_{L^{\infty} ((0, T) \times Ω)} ⩽ 1 - δ, \end{array}$ consequently $\begin{array}{rcl} f_{δ} (u_{0}^{δ}) = f (u_{0}^{δ}) over [- δ, δ], \end{array}$ and the couple $(u_{0}^{δ}, α_{0}^{δ})$ is a solution of our auxiliary problem (8)–(11). □

For comparison, we state the following lemma
Lemma 3.6.
Assume $(u, α)$ solution of problem ( 1 )–( 2 ) satisfying $‖ u ‖_{L^{\infty}} ⩽ 1$ . Then, there exists a sufficiently large constant L independent ϵ such that $\begin{array}{l} {‖ u (t) - u_{0} (t) ‖}^{2} + \int_{0}^{t} ({‖ α (τ) - α_{0} (τ) ‖}^{2}) e^{- β (t - τ)} d τ \\ (76) & ⩽ ϵ (Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + ϵ ‖ u_{1} ‖^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C), \end{array}$ where $(u_{0}, α_{0})$ is a solution of the auxiliary problem ( 8 )–( 11 ), with C and β positive constants and the monotonic function Q are independent of ϵ.
Proof.
We set $\overline{u} (t) = u (t) - u_{0} (t)$ and $\overline{α} (t) = α (t) - α_{0} (t)$ . These functions satisfy the following equations: $\begin{array}{l} (77) & \frac{\partial \overline{u}}{\partial t} - Δ \overline{u} + f (u) - f (u_{0}) + L \overline{u} = \overline{α} - ϵ \frac{\partial^{2} u}{\partial t^{2}}, \\ (78) & \frac{\partial \overline{α}}{\partial t} - Δ \overline{α} = - \overline{u} - \frac{\partial \overline{u}}{\partial t} - \frac{\partial^{2} \overline{α}}{\partial t^{2}} + Δ \frac{\partial \overline{α}}{\partial t} . \end{array}$

Multiplying (77) by $\overline{u}$ ant integrate over Ω, we have $\begin{array}{rcl} \frac{1}{2} \frac{d}{d t} ‖ \overline{u} ‖^{2} + ‖ \nabla \overline{u} ‖^{2} + (f (u) - f (u_{0}), \overline{u}) + L ‖ \overline{u} ‖^{2} = (\overline{α}, \overline{u}) - ϵ (\frac{\partial^{2} u}{\partial t^{2}}, \overline{u}), \end{array}$ Applying (7) and Hölder and Young inequalities, we obtain $\begin{array}{l} \frac{d}{d t} (‖ \overline{u} ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, \overline{u})) + β (‖ \overline{u} ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, \overline{u})) + 2 (L - c_{1}) ‖ \overline{u} ‖^{2} \\ (79) & ⩽ c_{p} ‖ \overline{α} ‖^{2} + ϵ C^{″} ({‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}) . \end{array}$

We multiply (78) by $\overline{α}$ and integrate over Ω, we have using Hölder, Young and Poincare inequalities $\begin{array}{l} \frac{d}{d t} (‖ \overline{α} ‖^{2} + ‖ \nabla \overline{α} ‖^{2} + 2 (\frac{\partial \overline{α}}{\partial t}, \overline{α})) + C ‖ \overline{α} ‖^{2} \\ (80) & ⩽ 2 {‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + 2 c_{p} ‖ \overline{u} ‖^{2} + 2 c_{p} ({‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}) . \end{array}$

We now multiply (78) by $\frac{\partial \overline{α}}{\partial t}$ and integrate over Ω. We find using the Hölder and Young inequalities $\begin{array}{rcl} (81) & \frac{d}{d t} ({‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + ‖ \nabla \overline{α} ‖^{2}) + \frac{1}{4} {‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial \overline{α}}{\partial t} ‖}^{2} ⩽ ‖ \overline{u} ‖^{2} + ({‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}) . \end{array}$

Adding $γ_{13}$ (79), $γ_{14}$ (80) and $γ_{15}$ (81), with $γ_{i} > 0$ such that $\begin{array}{l} 2 γ_{13} (L - c_{1}) - 2 c_{p} γ_{14} - γ_{15} > 0 \\ C γ_{14} - c_{p} γ_{13} > 0 \\ \frac{γ_{15}}{4} - 2 γ_{14} > 0, \end{array}$ we have $\begin{array}{l} \frac{d}{d t} E_{7} + C_{1} (‖ \overline{u} ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, \overline{u})) + C_{2} ‖ \overline{u} ‖^{2} + C_{3} ‖ \overline{α} ‖^{2} + C_{4} {‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + C_{5} {‖ \nabla \frac{\partial \overline{α}}{\partial t} ‖}^{2} \\ ⩽ C_{6} ({‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}), C_{i} > 0, \end{array}$ where $\begin{array}{rcl} E_{7} = γ_{13} (‖ \overline{u} ‖^{2} + 2 ϵ (\frac{\partial u}{\partial t}, \overline{u})) + γ_{14} (‖ \overline{α} ‖^{2} + ‖ \nabla \overline{α} ‖^{2} + 2 (\frac{\partial \overline{α}}{\partial t}, \overline{α})) + γ_{15} ({‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + ‖ \nabla \overline{α} ‖^{2}) . \end{array}$

We know that $\begin{array}{rcl} (82) & γ_{15} {‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + γ_{14} (‖ \overline{α} ‖^{2} + 2 (\frac{\partial \overline{α}}{\partial t}, \overline{α})) ⩾ (γ_{15} - 2 γ_{14}) {‖ \frac{\partial \overline{α}}{\partial t} ‖}^{2} + \frac{γ_{14}}{2} ‖ \overline{α} ‖^{2} . \end{array}$ Shoosing $γ_{14}$ and $γ_{15}$ such that $\begin{array}{rcl} γ_{15} - 2 γ_{14} > 0, \end{array}$ we obtain a inequality of the form $\begin{array}{rcl} (83) & \frac{d}{d t} E_{7} + β E_{7} + C_{1} ‖ \overline{α} ‖^{2} ⩽ C ({‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial u_{0}}{\partial t} ‖}^{2}) . \end{array}$ Applying Gronwall’s lemma, we find, for all $t \in [0, T]$ $\begin{array}{l} {‖ u (t) - u_{0} (t) ‖}^{2} + \int_{0}^{t} ({‖ α (τ) - α_{0} (τ) ‖}^{2}) e^{- β (t - τ)} d τ \\ (84) & ⩽ ϵ (Q (D [u (0)] + {‖ u (0) ‖}_{H^{2}}^{2} + ϵ ‖ u_{1} ‖^{2} + {‖ α (0) ‖}_{H^{1}}^{2} + ‖ α_{1} ‖^{2}) e^{- β t} + C), \end{array}$ where C and β are positive constants and Q a monotonic function are independent of ϵ. □
Remark 3.1.
The solution $(u, α)$ of problem ( 1 )–( 4 ) is getting closer to the solution $(u_{0}, α_{0})$ auxiliary problem ( 8 )–( 11 ), when ϵ goes to zero.

Finally, $\begin{array}{rcl} (85) & ‖ u ‖_{L^{\infty} ((0, T) \times Ω)} ⩽ 1 - δ, δ > 0 . \end{array}$
Proof.
From the Theorem 3.1, now determine the estimates of $‖ u (t) ‖_{H^{2}}^{2}$ and $D [u (t)]$ to establish the estimate (15).

Multiplying (1) by $- Δ \frac{\partial u}{\partial t}$ then integrating over Ω, we have, applying Hölder and Young inequalities $\begin{array}{rcl} (86) & \frac{d}{d t} (ϵ {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + ‖ Δ u ‖^{2}) + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ 2 ({‖ \nabla f (u) ‖}^{2} + ‖ \nabla α ‖^{2}) . \end{array}$

We Multiply equation (1) by $- Δ u$ and integrate over Ω, we find, using the inequalities Hölder, Young and (7) $\begin{array}{rcl} (87) & \frac{d}{d t} (‖ \nabla u ‖^{2} + 2 ϵ (\nabla \frac{\partial u}{\partial t}, \nabla u)) + ‖ Δ u ‖^{2} ⩽ C_{2} ‖ \nabla u ‖^{2} + 2 ϵ_{0} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + c_{p} ‖ \nabla α ‖^{2} . \end{array}$

We add (86) and $γ_{16}$ (87), with $γ_{16} > 0$ such that $\begin{array}{rcl} 1 - 2 γ_{16} > 0, \end{array}$ we have $\begin{array}{rcl} (88) & \frac{d}{d t} E_{8} + C_{1} ‖ Δ u ‖^{2} + C_{2} {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} ⩽ C ({‖ \nabla f (u) ‖}^{2} + ‖ \nabla u ‖^{2} + ‖ \nabla α ‖^{2}), \end{array}$ where $\begin{array}{rcl} E_{8} = (ϵ {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + ‖ Δ u ‖^{2}) + γ_{16} (‖ \nabla u ‖^{2} + 2 ϵ (\nabla \frac{\partial u}{\partial t}, \nabla u)) . \end{array}$

We know that $\begin{array}{rcl} (89) & ϵ {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + γ_{16} (‖ \nabla u ‖^{2} + 2 ϵ (\nabla \frac{\partial u}{\partial t}, \nabla u)) ⩾ ϵ (1 - 2 γ_{16}) {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2} + \frac{γ_{16}}{2} ‖ \nabla u ‖^{2} . \end{array}$ Shoosing $γ_{16}$ such that $\begin{array}{rcl} 1 - 2 γ_{16} > 0, \end{array}$ then there exist a positive constant C such that $\begin{array}{rcl} (90) & C^{- 1} ({‖ u (t) ‖}_{H^{2}}^{2} + ϵ {‖ \nabla \frac{\partial u (t)}{\partial t} ‖}^{2}) ⩽ E_{8} (t) ⩽ C ({‖ u (t) ‖}_{H^{2}}^{2} + ϵ {‖ \nabla \frac{\partial u (t)}{\partial t} ‖}^{2}), \end{array}$ we obtain a estimate of the form $\begin{array}{rcl} (91) & \frac{d}{d t} E_{8} + ρ E_{8} + \frac{ρ}{2} (‖ Δ u ‖^{2} + {‖ \nabla \frac{\partial u}{\partial t} ‖}^{2}) ⩽ C ({‖ \nabla f (u) ‖}^{2} + ‖ \nabla u ‖^{2} + ‖ \nabla α ‖^{2}) . \end{array}$ We estimate the term $‖ f (u_{0} (t)) - f (u (t)) ‖_{H^{1}}^{2}$ to deduce $‖ f (u (t)) ‖_{H^{1}}^{2}$ (see [3]), we have $\begin{array}{l} {‖ f (u_{0} (t)) - f (u (t)) ‖}_{H^{1}}^{2} \\ = {‖ \nabla (f (u_{0} (t)) - f (u (t))) ‖}^{2} \\ = {‖ \nabla (l (t) (u_{0} (t) - u (t))) ‖}^{2} \\ ⩽ M_{f} (\frac{1}{1 - ‖ u_{0} (t) ‖_{L^{\infty}} - ‖ u (t) - u_{0} (t) ‖_{L^{\infty}}}) (1 + {‖ u_{0} (t) ‖}_{H^{2}}^{2} + {‖ u (t) ‖}_{H^{2}}^{2}) \\ (92) & \times {‖ u (t) - u_{0} (t) ‖}_{H^{1}}^{2}, \end{array}$ where $\begin{array}{rcl} l (t) = \int_{0}^{1} f^{'} (s u_{0} (t) + (1 - s) u (t)) d s, \end{array}$ and $M_{f}$ is a growing function dependent on f and such that ${lim}_{z \to \infty} M_{f} (z) = \infty$ .

Using the interpolation inequality (see [3]), we have:

if $(1 - θ) m_{1} + m_{2} θ = 1$ , $θ = \frac{1}{2}$ , $m_{1} = 0$ and $m_{2} = 2$ $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{H^{1}} & ⩽ & C {‖ u (t) - u_{0} (t) ‖}^{\frac{1}{2}} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{\frac{1}{2}} . \end{array}$ Among’s inequality given:

for $n = 1$ , $θ = \frac{1}{4}$ , $m_{1} = 0$ and $m_{2} = 2$ $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{L^{\infty}} & ⩽ & C {‖ u (t) - u_{0} (t) ‖}^{\frac{3}{4}} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{\frac{1}{4}}, \end{array}$ for $n = 2$ , $θ = \frac{1}{2}$ , $m_{1} = 0$ and $m_{2} = 2$ $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{L^{\infty}} & ⩽ & C {‖ u (t) - u_{0} (t) ‖}^{\frac{1}{2}} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{\frac{1}{2}} . \end{array}$ For $n = 3$ , $θ = \frac{3}{4}$ , $m_{1} = 0$ and $m_{2} = 2$ $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{L^{\infty}} & ⩽ & C {‖ u (t) - u_{0} (t) ‖}^{\frac{1}{4}} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{\frac{3}{4}} . \end{array}$ Then, for $1 ⩽ n ⩽ 3$ $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{L^{\infty}} & ⩽ & C {‖ u (t) - u_{0} (t) ‖}^{\frac{1}{4}} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{\frac{3}{4}} \\ (93) & ⩽ & C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{\frac{3}{4}}, \end{array}$ with C and β positive constants and Q a monotonic function are independent of ϵ.

Using (90) $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{H^{2}}^{2} & ⩽ & {‖ u (t) + u_{0} (t) ‖}_{H^{2}}^{2} \\ ⩽ & C ({‖ u (t) ‖}_{H^{2}}^{2} + {‖ u_{0} (t) ‖}_{H^{2}}^{2}) \\ ⩽ & C (E_{8} (t) + {‖ u_{0} (t) ‖}_{H^{2}}^{2}), \end{array}$ then applying the lemma 3.5, $‖ u_{0} (t) ‖_{H^{2}}^{2}$ is bounded using $D [u (0)]$ , $‖ u (0) ‖_{H^{2}}^{2}$ , $‖ α (0) ‖_{H^{2}}^{2}$ and $‖ \frac{\partial α (0)}{\partial t} ‖_{H^{1}}^{2}$ , and on the other hand, $\begin{array}{rcl} {‖ u (t) - u_{0} (t) ‖}_{L^{\infty}} ⩾ {‖ u (t) ‖}_{L^{\infty}} - {‖ u_{0} (t) ‖}_{L^{\infty}}, \end{array}$ therefore $\begin{array}{rcl} 1 - {‖ u (t) ‖}_{L^{\infty}} & ⩾ & 1 - {‖ u_{0} (t) ‖}_{L^{\infty}} - {‖ u (t) - u_{0} (t) ‖}_{L^{\infty}} \\ ⩾ & 1 - {‖ u_{0} (t) ‖}_{L^{\infty}} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{8} (t) + 1)}^{\frac{3}{8}} \\ (94) & ⩾ & {(Q e^{- β t} + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{8} (t) + 1)}^{\frac{3}{8}} . \end{array}$ Consequently, $\begin{array}{rcl} {‖ f (u (t)) ‖}_{H^{1}}^{2} & ⩽ & C ϵ (Q e^{- β t} + C) {(E_{8} (t) + 1)}^{\frac{3}{2}} \\ \times M_{f} (\frac{1}{{(Q e^{- β t} + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{8} (t) + 1)}^{\frac{3}{8}}}) \\ (95) & + (Q e^{- β t} + C), \end{array}$ where $\begin{array}{rcl} (96) & Q = Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ with $\begin{array}{rcl} (97) & {(Q e^{- β t} + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{8} (t) + 1)}^{\frac{3}{8}} > 0 . \end{array}$ The estimate (91) becomes $\begin{array}{rcl} \frac{d}{d t} E_{8} + ρ E_{8} & ⩽ & C ϵ (Q e^{- β t} + C) {(E_{8} (t) + 1)}^{\frac{3}{2}} \\ \times M_{f} (\frac{1}{{(Q e^{- β t} + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{8} (t) + 1)}^{\frac{3}{8}}}) \\ (98) & + (Q e^{- β t} + C) . \end{array}$

We state the following lemma giving the estimate of $E_{8} (t)$ and assuming without loss of generality, that $β < ρ$ Lemma 3.7.
Assume the initial conditions $(ζ_{u} (0), α (0), α_{1})$ satisfies the estimate $\begin{array}{rcl} (99) & D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2} ⩽ R (ϵ), \end{array}$ where $R = R (ϵ)$ (see [ 3 ]) solves the equation $\begin{array}{rcl} C & = & Q (R) ϵ {(1 + 2 {(ρ - β)}^{- 1} Q (R) + 3 β^{- 1} C)}^{\frac{3}{2}} \\ (100) & \times M_{f} (\frac{1}{{(Q (R) + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q (R) + C)}^{\frac{1}{8}} {(1 + 2 {(ρ - β)}^{- 1} Q (R) + 3 β^{- 1} C)}^{\frac{3}{8}}}) . \end{array}$ Then, we have the following estimate $\begin{array}{rcl} (101) & E_{8} (t) ⩽ E_{0} (t), \end{array}$ where $\begin{array}{rcl} E_{0} (t) = 2 {(ρ - β)}^{- 1} Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + 3 β^{- 1} C . \end{array}$
Proof.
Using the estimate (100) and the fact that the R decreases function, $E_{0} (t)$ satisfies the following estimate $\begin{array}{rcl} \frac{d}{d t} E_{0} + ρ E_{0} & ⩾ & C ϵ (Q e^{- β t} + C) {(E_{0} (t) + 1)}^{\frac{3}{2}} \\ \times M_{f} (\frac{1}{{(Q e^{- β t} + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{0} (t) + 1)}^{\frac{3}{8}}}) \\ (102) & + (Q e^{- β t} + C) . \end{array}$ Using the estimate (99), and since the function Q is monotonic, we can assume without loss of generality that $\begin{array}{rcl} (103) & E_{8} (0) ⩽ E_{0} (0) . \end{array}$ Consequently, using the comparison principle for first order differential inequalities, we have $\begin{array}{rcl} (104) & E_{8} (t) ⩽ E_{0} (t) . \end{array}$ □

Which implies $\begin{array}{l} {‖ u (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{1}}^{2} \\ (105) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and Q a monotonic function are independent of ϵ.

Using (100), we have $\begin{array}{rcl} (106) & lim_{ϵ \to 0} R (ϵ) = + \infty . \end{array}$

Moreover, it follows from (100) and (101) the $\begin{array}{rcl} {(Q e^{- β t} + C)}^{- 1} - C ϵ^{\frac{1}{8}} {(Q e^{- β t} + C)}^{\frac{1}{8}} {(E_{0} (0) + 1)}^{\frac{3}{8}} > 0 . \end{array}$ Then, inserting the estimate (101) in the second member of (94), we verify that the solution $u (t)$ is indeed separate from the singular points, if $ϵ > 0$ is small enough and the following estimate $\begin{array}{rcl} (107) & D [u (t)] ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C \end{array}$ is valid for C and β positive constants and the monotonic function Q are independent of ϵ.

Thus $\begin{array}{rcl} (108) & {‖ u (t) ‖}_{L^{\infty} (Ω)} < 1 . \end{array}$

Finally, $\begin{array}{l} D [u (t)] + {‖ u (t) ‖}_{H^{2}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{1}}^{2} + {‖ α (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ \\ (109) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and the monotonic function Q are independent of ϵ.

We now estimate $‖ \frac{\partial u (t)}{\partial t} ‖^{2}$ , for that we consider the following hyperbolic equation $\begin{array}{l} (110) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u = h (t) = - f (u) + α, \\ (111) & ζ_{u} (t) |_{\partial Ω} = h (t) |_{\partial Ω} = 0 . \end{array}$ Applying Proposition A.2 (see [10]) to the equation (110) gives $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ C ({‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ h (0) ‖}^{2}) e^{- β t} \\ (112) & + C \int_{0}^{t} ({‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2}) e^{- β (t - τ)} d τ . \end{array}$ Let’s determine a estimate of $\begin{array}{rcl} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array}$ We have $\begin{array}{l} (113) & \begin{array}{l} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} \\ = {‖ α (τ) - f (u (τ)) ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} - f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖}_{H^{- 1}}^{2} \\ ⩽ {‖ α (τ) ‖}_{H^{1}}^{2} + {‖ f (u (τ)) ‖}_{H^{1}}^{2} + {‖ f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖}_{H^{- 1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{- 1}}^{2} . \end{array} \end{array}$ Let’s determine the estimates of $‖ α (τ) ‖_{H^{1}}$ , $‖ f (u (τ)) ‖_{H^{1}}$ , $‖ f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖_{H^{- 1}}$ and $‖ \frac{\partial α (τ)}{\partial t} ‖_{H^{- 1}}$ .

Using the uniform estimates de $‖ u ‖_{H^{2}}^{2}$ , $‖ α ‖_{H^{2}}^{2}$ and $‖ \frac{\partial α}{\partial t} ‖_{H^{1}}^{2}$ independent of ϵ, we have $\begin{array}{rcl} (114) & {‖ f (u (τ)) ‖}_{H^{1}} ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{\frac{- β τ}{2}} + C, \end{array}$ and $\begin{array}{rcl} (115) & {‖ α (τ) ‖}_{H^{1}} ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{\frac{- β τ}{2}} + C . \end{array}$ Furthermore $\begin{array}{rcl} {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{- 1}} & ⩽ & C {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}} \\ (116) & ⩽ & Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{\frac{- β τ}{2}} + C . \end{array}$ We have $\begin{array}{rcl} (117) & \forall w \in H_{0}^{1} (Ω), | (f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t}, w) | & ⩽ & C {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}} ‖ w ‖_{H^{1}}, \end{array}$ therefore $\begin{array}{rcl} (118) & {‖ f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖}_{H^{- 1}} ⩽ C {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}} . \end{array}$ The estimate becomes $\begin{array}{rcl} {‖ h (τ) ‖}_{H^{1}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}_{H^{- 1}}^{2} & ⩽ & Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β τ} \\ + C (1 + {‖ \frac{\partial (u (τ))}{\partial t} ‖}_{H^{1}}^{2}) . \end{array}$ Using the estimate above, the estimate (112) becomes $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} \\ + \int_{0}^{t} (Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β τ} + 1) e^{- β (t - τ)} d τ \\ + C \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, for all t \in [0, T], \end{array}$ which implies, for all $t \in [0, T]$ , $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} e^{- β (t - τ)} d τ \\ ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C . \end{array}$ Therefore $\begin{array}{rcl} (119) & {‖ \frac{\partial u (t)}{\partial t} ‖}^{2} ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C . \end{array}$ Combining estimates (109) and (119), we obtain $\begin{array}{l} D [u (t)] + {‖ ζ_{u} (t) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (t) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{1}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{1}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{1}}^{2}) e^{- β (t - τ)} d τ \\ (120) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{1} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{2}}^{2} + ‖ α_{1} ‖_{H^{1}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and the monotonic function Q are independent of ϵ. □

This shows the estimate (15) and the estimates necessary for the existence of the solution of the problem (1)–(4).

To show the existence of the solution of the problem (1)–(4), we regularize, as in the case of the auxiliary problem, the potential f by a regular potential $f_{δ}$ of $C^{1}$ class defined by: $\begin{array}{rcl} (121) & f_{δ} (s) = \{\begin{matrix} f (δ) + f^{'} (δ) (s - δ) & si s > δ, \\ f (s) & si - δ ⩽ s ⩽ δ, \\ f (- δ) + f^{'} (- δ) (s + δ) & si s < - δ, \end{matrix} \end{array}$ where δ is the constant we choose close enough to 1, such that $\begin{array}{rcl} (122) & f (δ) ⩾ 0, f (- δ) ⩽ 0, f^{'} (δ) ⩾ 0 and f^{'} (- δ) ⩾ 0 . \end{array}$ We consider the same problem (1)–(4) and we replace $(u^{δ}, α^{δ})$ instead of $(u, α)$ by $\begin{array}{l} (123) & ϵ \frac{\partial^{2} u^{δ}}{\partial t^{2}} + \frac{\partial u^{δ}}{\partial t} - Δ u^{δ} + f_{δ} (u^{δ}) = α^{δ}, \\ (124) & \frac{\partial^{2} α^{δ}}{\partial t^{2}} + \frac{\partial α^{δ}}{\partial t} - Δ \frac{\partial α^{δ}}{\partial t} - Δ α^{δ} = - u^{δ} - \frac{\partial u^{δ}}{\partial t}, \end{array}$ with homogenous Dirichlet conditions $\begin{array}{rcl} (125) & u^{δ} |_{\partial Ω} = α^{δ} |_{\partial Ω} = 0 \end{array}$ and initial conditions $\begin{array}{rcl} (126) & u^{δ} |_{t = 0} = u (0), \frac{\partial u^{δ}}{\partial t} |_{t = 0} = u_{1}, α^{δ} |_{t = 0} = α (0), \frac{\partial α^{δ}}{\partial t} |_{t = 0} = α_{1} . \end{array}$

We know that the problem (123)–(126) possesses a regular global solution (see [10]). But as $f_{δ}$ and f coincide over $[- δ, δ]$ , we deduce a local solution then maximal for the initial singular problem. This solution verifies all the above estimates (we take δ small enough that $f_{δ}$ and $F_{δ}$ check again the same conditions as f and F). Consequently the solution is bounded (in infinity norm) and is therefore global.
4. Uniqueness of the solution

Theorem 4.1.
Assume the hypothesis of Theorem 3.1 verified, then the system ( 1 )–( 2 ) possesses a uniqueness solution $(u, α)$ such that $u, α \in L^{\infty} (0, T; H^{2} (Ω) \cap H_{0}^{1} (Ω))$ , $\frac{\partial u}{\partial t} \in L^{\infty} (0, T; H_{0}^{1} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω))$ , $\frac{\partial α}{\partial t} \in L^{\infty} (0, T; H_{0}^{1} (Ω)) \cap L^{2} (0, T; H_{0}^{1} (Ω))$ , $\frac{\partial^{2} α}{\partial t^{2}} \in L^{2} (0, T; L^{2} (Ω))$ and $\frac{\partial^{2} u}{\partial t^{2}} \in L^{2} (0, T; L^{2} (Ω))$ , for all $T > 0$ .
Proof.
Let $(u^{(1)}, α^{(1)})$ and $(u^{(2)}, α^{(2)})$ two solutions of the system (1)–(2) with initial conditions $(u_{0}^{(1)}, u_{1}^{(1)}, α_{0}^{(1)}, α_{1}^{(1)})$ and $(u_{0}^{(2)}, u_{1}^{(2)}, α_{0}^{(2)}, α_{1}^{(2)})$ ; respectively.

Let $u = u^{(1)} - u^{(2)}$ and $α = α^{(1)} - α^{(2)}$ . Then $(u, α)$ satisfies $\begin{array}{l} (127) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u + l . u = α, \\ (128) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \end{array}$ where $l (t, x) = \int_{0}^{1} f^{'} (s u^{(1)} (t, x) + (1 - s) u^{(2)} (t, x)) d s$ . Following Theorem 3.1, $u^{(1)}$ and $u^{(2)}$ belong to $L^{\infty} (0, T; H^{2} (Ω))$ and are such that $- 1 < u^{(1)} (t, x) < 1$ and $- 1 < u^{(2)} (t, x) < 1$ a.e., for all $(t, x) \in [0, T] \times Ω$ , then $s u^{(1)} + (1 - s) u^{(2)} \in L^{\infty} (0, T; H^{2} (Ω))$ , for all $s \in (0, 1)$ and $- 1 < s u^{(1)} (t, x) + (1 - s) u^{(2)} (t, x) < 1$ a.e., for all $(s, t, x) \in (0, 1) \times [0, T] \times Ω$ . Since $f^{'}$ is continuous and ${lim}_{s \to \pm 1} f^{'} (s) = + \infty$ , then there exists $c \in (0, 1)$ independent of t and x such that $f^{'} (c) > 0$ and $| s u^{(1)} (t, x) + (1 - s) u^{(2)} (t, x) | ⩽ c$ . Moreover $f^{'}$ convex and even, therefore $\begin{matrix} | l (t, x) | ⩽ \int_{0}^{1} | f^{'} (s u^{(1)} (t, x) + (1 - s) u^{(2)} (t, x)) | d s ⩽ f^{'} (c), for all (t, x) \in [0, T] \times Ω . \end{matrix}$

Multiplying (127) by $\frac{\partial u}{\partial t}$ and (128) by $\frac{\partial α}{\partial t}$ and integrating over Ω, we obtain $\begin{array}{l} (129) & \frac{d}{d t} (ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ \nabla u ‖^{2}) + {‖ \frac{\partial u}{\partial t} ‖}^{2} ⩽ C ‖ u ‖^{2} + 2 ‖ α ‖^{2}, C > 0 . \\ (130) & \frac{d}{d t} ({‖ \frac{\partial α}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2}) + {‖ \frac{\partial α}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} ⩽ 2 ‖ u ‖^{2} + 2 {‖ \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$ Adding (129) and (130), we have $\begin{array}{rcl} (131) & \frac{d}{d t} E_{3} + {‖ \frac{\partial α}{\partial t} ‖}^{2} + 2 {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} ⩽ C ‖ u ‖^{2} + 2 ‖ α ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2}, C > 0, \end{array}$ we obtain a differential inequality of the form, $\begin{array}{rcl} (132) & \frac{d}{d t} E_{3} ⩽ c^{'} E_{3}, c^{'} > 0, \end{array}$ where $\begin{array}{rcl} E_{3} = ‖ \nabla u ‖^{2} + ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}, \end{array}$ satisfies $\begin{array}{rcl} (133) & E_{3} ⩾ c (‖ \nabla u ‖^{2} + ‖ \nabla α ‖^{2} + ϵ {‖ \frac{\partial u}{\partial t} ‖}^{2}), c > 0 . \end{array}$ Gronwall’s lemma, together with (133), then yields the continuous dependence of the solution relative to the initial conditions, hence the uniqueness of the solution. □
5. Uniform estimates with more regularity

Theorem 5.1.

Let $ϵ ⩽ 1$ and $(u, α)$ a solution of problem ( 1 )–( 4 ) such that $(ζ_{u} (0), α (0), α_{1}) \in Φ_{2}$ . Then, the solution $(u, α)$ satisfies the following estimate: $\begin{array}{l} D [u (t)] + {‖ ζ_{u} (t) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2} + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (134) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and the monotonic function Q are independent of ϵ.

The following lemma gives uniform estimates of $‖ u ‖_{H^{3}}^{2}$ , $‖ α ‖_{H^{3}}^{2}$ and $‖ \frac{\partial α}{\partial t} ‖_{H^{2}}^{2}$ independant of ϵ.

Lemma 5.1.

Let $ϵ ⩽ 1$ and $(u, α)$ a solution of problem ( 1 )–( 4 ) such that $(ζ_{u} (0), α (0), α_{1}) \in Φ_{2}$ . Then, the solution $(u, α)$ satisfies the following estimate: $\begin{array}{l} D [u (t)] + {‖ u (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (135) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and the monotonic function Q are independent of ϵ.

Proof.

Multiplying (1) by $Δ^{2} \frac{\partial u}{\partial t}$ and integrating over Ω, we find $\begin{array}{rcl} (136) & \frac{d}{d t} (ϵ {‖ \frac{\partial u}{\partial t} ‖}_{H^{2}}^{2} + ‖ u ‖_{H^{3}}^{2}) + \frac{1}{2} {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} ⩽ c_{p} C_{1} ‖ \nabla Δ u ‖^{2} + c_{p} ‖ \nabla Δ α ‖^{2}, C_{1} > 0 . \end{array}$

Multiplying (2) by $Δ^{2} \frac{\partial α}{\partial t}$ and integrating over Ω, we have $\begin{array}{rcl} (137) & \frac{d}{d t} (‖ α ‖_{H^{3}}^{2} + {‖ \frac{\partial α}{\partial t} ‖}_{H^{2}}^{2}) + 2 {‖ \frac{\partial α}{\partial t} ‖}_{H^{3}}^{2} + {‖ \frac{\partial α}{\partial t} ‖}_{H^{2}}^{2} & ⩽ & 2 c_{p} ‖ \nabla Δ u ‖^{2} + 2 {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$

Multiplying (1) by $Δ^{2} u$ and integrating over Ω, we have using Hölder and Young inequalities $\begin{array}{rcl} \frac{d}{d t} (‖ Δ u ‖^{2} + 2 ϵ (Δ \frac{\partial u}{\partial t}, Δ u)) + ‖ u ‖_{H^{3}}^{2} ⩽ 2 {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} + c_{p} C_{2} ‖ \nabla Δ α ‖^{2} + C ‖ Δ u ‖^{2}, C > 0 . \\ (138) \end{array}$ Multiplying (2) by $Δ^{2} α$ and integrating over Ω, we find $\begin{array}{rcl} (139) & \frac{d}{d t} (‖ α ‖_{H^{3}}^{2} + ‖ Δ α ‖^{2} + 2 (Δ \frac{\partial α}{\partial t}, Δ α)) + ‖ α ‖_{H^{3}}^{2} ⩽ 2 {‖ Δ \frac{\partial α}{\partial t} ‖}^{2} + C_{2} {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} . \end{array}$

Adding (22), $γ_{6}$ (136), $γ_{7}$ (137), $γ_{8}$ (138) and $γ_{9}$ (139)), with $γ_{i} > 0$ such that $\begin{array}{l} \frac{γ_{6}}{2} - 2 γ_{7} - 2 γ_{8} - C_{2} γ_{9} > 0, \\ γ_{7} - 2 γ_{9} > 0, \\ γ_{9} + c_{p} C_{2} γ_{8} - c_{p} γ_{6} > 0, \\ γ_{8} - 2 c_{p} γ_{7} - c_{p} C_{1} γ_{6} > 0, \\ γ_{3} - C γ_{8} > 0, \end{array}$ we have $\begin{array}{l} \frac{d}{d t} E_{9} + C_{1} ‖ u ‖_{H^{3}}^{2} + C_{2} ‖ α ‖_{H^{3}}^{2} + C_{3} ‖ u ‖_{H^{2}}^{2} + C_{4} ‖ α ‖_{H^{2}}^{2} + C_{5} ‖ u ‖_{H^{1}}^{2} + C_{6} {‖ \frac{\partial u}{\partial t} ‖}_{H^{2}}^{2} \\ (140) & + C_{7} {‖ \frac{\partial α}{\partial t} ‖}_{H^{2}}^{2} + C_{8} {‖ \frac{\partial α}{\partial t} ‖}_{H^{3}}^{2} + C_{9} {‖ \frac{\partial u}{\partial t} ‖}_{H^{1}}^{2} + C_{10} {‖ \frac{\partial α}{\partial t} ‖}_{H^{1}}^{2} ⩽ C, \end{array}$ where $\begin{array}{rcl} E_{9} & = & E_{3} + γ_{6} (‖ u ‖_{H^{3}}^{2} + ϵ {‖ Δ \frac{\partial u}{\partial t} ‖}^{2}) + γ_{7} (‖ α ‖_{H^{3}}^{2} + ‖ Δ \frac{\partial α}{\partial t} ‖) \\ + γ_{8} (‖ Δ u ‖^{2} + 2 ϵ (Δ \frac{\partial u}{\partial t}, Δ u)) + γ_{9} (‖ α ‖_{H^{3}}^{2} + ‖ Δ α ‖^{2} + 2 (Δ \frac{\partial α}{\partial t}, Δ α)) . \end{array}$

We know that $\begin{array}{rcl} (141) & γ_{6} ϵ {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} + γ_{8} (‖ Δ u ‖^{2} + 2 ϵ (Δ \frac{\partial u}{\partial t}, \nabla u)) ⩾ ϵ (γ_{6} - 2 γ_{8}) {‖ Δ \frac{\partial u}{\partial t} ‖}^{2} + \frac{γ_{8}}{2} ‖ Δ u ‖^{2}, \end{array}$ and $\begin{array}{rcl} (142) & γ_{7} {‖ Δ \frac{\partial α}{\partial t} ‖}^{2} + γ_{9} (‖ Δ α ‖^{2} + 2 (Δ \frac{\partial α}{\partial t}, Δ α)) ⩾ (γ_{7} - 2 γ_{9}) {‖ Δ \frac{\partial α}{\partial t} ‖}^{2} + \frac{γ_{9}}{2} ‖ Δ α ‖^{2} . \end{array}$ Shoosing $γ_{6}$ and $γ_{8}$ such that $\begin{array}{rcl} γ_{6} - 2 γ_{8} > 0, \end{array}$ there is then a positive constant C independent of ϵ such that $\begin{array}{l} C^{- 1} ({‖ u (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2}) \\ ⩽ E_{9} (t) \\ (143) & ⩽ C ({‖ u (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2}) . \end{array}$ Using (143), we come to a estimate of form $\begin{array}{rcl} (144) & \frac{d}{d t} E_{9} + β E_{9} + C_{1} {‖ \frac{\partial u}{\partial t} ‖}_{H^{2}}^{2} + C_{2} {‖ \frac{\partial α}{\partial t} ‖}_{H^{2}}^{2} + C_{3} {‖ \frac{\partial α}{\partial t} ‖}_{H^{3}}^{2} ⩽ C, \end{array}$ where C and β positive constants are independent of ϵ.

Gronwall’s lemma appliying to the estimate (144) and using the estimate (143), we have $\begin{array}{l} {‖ u (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (145) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C . \end{array}$ Combining estimates (107) and (145), we find $\begin{array}{l} D [u (t)] + {‖ u (t) ‖}_{H^{3}}^{2} + ϵ {‖ \frac{\partial u (t)}{\partial t} ‖}_{H^{2}}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and the monotonic function Q are independent of ϵ. □

Proof.

We consider the following hyperbolic equation $\begin{array}{l} (146) & ϵ \frac{\partial^{2} u}{\partial t^{2}} + \frac{\partial u}{\partial t} - Δ u = h (t) = - f (u) + α, \\ (147) & ζ_{u} (t) |_{\partial Ω} = h (t) |_{\partial Ω} = 0 . \end{array}$ Applying Proposition A.1 (see [10]) to the equation (146) gives $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{2} (ϵ)}^{2} + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ ⩽ C ({‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ h (0) ‖}_{H^{1}}^{2}) e^{- β t} \\ (148) & + C \int_{0}^{t} ({‖ h (τ) ‖}_{H^{2}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}^{2}) e^{- β (t - τ)} d τ . \end{array}$ Let’s determine a estimate of $\begin{array}{rcl} {‖ h (τ) ‖}_{H^{2}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}^{2} . \end{array}$ We have $\begin{array}{l} (149) & \begin{array}{l} {‖ h (τ) ‖}_{H^{2}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}^{2} \\ = {‖ α (τ) - f (u (τ)) ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} - f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖}^{2} \\ ⩽ {‖ α (τ) ‖}_{H^{2}}^{2} + {‖ f (u (τ)) ‖}_{H^{2}}^{2} + {‖ f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}^{2} . \end{array} \end{array}$ Let’s determine the estimates of $‖ α (τ) ‖_{H^{2}}$ , $‖ f (u (τ)) ‖_{H^{2}}$ , $‖ f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖$ and $‖ \frac{\partial α (τ)}{\partial t} ‖^{2}$ .

Let us note that $\begin{array}{rcl} {‖ f (u (τ)) ‖}_{H^{2}} & = & ‖ Δ f (u (τ)) ‖ \\ = & ‖ f^{″} (u (τ)) {(\nabla u (τ))}^{2} + f^{'} (u (τ)) Δ u (τ) ‖ \\ (150) & ⩽ & C ({‖ u (τ) ‖}_{H^{1}} + {‖ u (τ) ‖}_{H^{2}}), \end{array}$ which implies $\begin{array}{l} (151) & {‖ f (u (τ)) ‖}_{H^{2}} ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{\frac{- β τ}{2}} + C, \\ (152) & ‖ α (τ)) ‖_{H^{2}} ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{\frac{- β τ}{2}} + C, \end{array}$ and $\begin{array}{rcl} ‖ \frac{\partial α (τ)}{\partial t} ‖ & ⩽ & C {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}} \\ (153) & ⩽ & Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{\frac{- β τ}{2}} + C . \end{array}$ Moreover, $\begin{array}{rcl} ‖ f^{'} (u (τ)) \frac{\partial u (τ)}{\partial t} ‖ & ⩽ & C ‖ \frac{\partial u (τ)}{\partial t} ‖ \\ (154) & ⩽ & C {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}} . \end{array}$ The estimate becomes $\begin{array}{rcl} {‖ h (τ) ‖}_{H^{2}}^{2} + {‖ \frac{\partial h (τ)}{\partial t} ‖}^{2} & ⩽ & Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β τ} \\ + C (1 + {‖ \frac{\partial (u (τ))}{\partial t} ‖}_{H^{2}}^{2}) . \end{array}$ Using the estimate above and (135), the estimate (148) becomes $\begin{array}{l} {‖ ζ_{u} (t) ‖}_{ε^{2} (ϵ)}^{2} + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} \\ + \int_{0}^{t} (Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β τ} + 1) e^{- β (t - τ)} d τ \\ + C \int_{0}^{t} {‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} e^{- β (t - τ)} d τ \\ (155) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C, for all t \in [0, T] . \end{array}$ Using (135) and (155), we obtain $\begin{array}{l} D [u (t)] + {‖ ζ_{u} (t) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (t) ‖}_{H^{3}}^{2} + {‖ \frac{\partial α (t)}{\partial t} ‖}_{H^{2}}^{2} \\ + \int_{0}^{t} ({‖ \frac{\partial u (τ)}{\partial t} ‖}_{H^{2}}^{2} + {‖ \frac{\partial α (τ)}{\partial t} ‖}_{H^{2}}^{2}) e^{- β (t - τ)} d τ \\ (156) & ⩽ Q (D [u (0)] + {‖ ζ_{u} (0) ‖}_{ε^{2} (ϵ)}^{2} + {‖ α (0) ‖}_{H^{3}}^{2} + ‖ α_{1} ‖_{H^{2}}^{2}) e^{- β t} + C, \end{array}$ where C and β positive constants and the monotonic function Q are independent of ϵ. □

References

Cherfils,

Gatti and

Miranville, Exitence and globalsolutions to the Caginalp phase-fields system with dynamic boundary conditions and singular potentials, Vol. 348, 2008, pp. 1029–1030.

B.L.

Doumbé Bongola, textitEtude de modèles de champ de phases de type Caginalp. Thèse soutenue à la Faculté des Sciences Fondamentales et Appliquées de Poitiers, le 30 mai 2013.

Grasselli,

Miranville,

Pata and

Zelik, Well-posedness and long time behavior of a parabolic-hyperbolic phasefield system with singular potentials, Math. Nachr. 280(13–14) (2007), 1475–1509. doi:10.1002/mana.200510560.

Miranville and

Quintanilla, Some generalizations of the Caginalp phase-field system, Vol. 88, 2009, pp. 877–894.

Miranville and

Quintanilla, A generalization of the Caginalp phase-field system based on the Maxwell–Cattaneo law, Nonlinear Anal. 71(5–6) (2009), 2278–2298. doi:10.1016/j.na.2009.01.061.

Miranville and

Quintanilla, Some generalizations of the Caginalp phase-field system based on the Cattaneo law, Vol. 71, 2009, pp. 2278–2290.

Miranville and

Quintanilla, A Caginalp phase-field system with nonlinear coupling, Nonlinear Anal. Real World Appl. 11 (2010), 2849–2861. doi:10.1016/j.nonrwa.2009.10.008.

Miranville and

Quintanilla, A Caginalp phase-field system based on type III heat conduction with two temperature.

Moukoko, Well-posedness and long time behavior of a hyperbolic Caginalp system, JAAC 4(2) (2014), 151–196.

10.

Moukoko, Etude de modèles hyperboliques de champ de phases de Caginalp, Thèse unique, Faculté des Sciences et Techniques, Université Marien NGOUABI 23 janvier (2015).

11.

Moukoko,

Moukamba and

F.D.R.

Langa, Global Attractor for Caginalp Hyperbolic Field-phase System with Singular Potentiel, Journal of Mathematics Research 7(3) (2015). doi:10.5539/jmr.v7n3p165.

12.

Wehbe, Etude asymptotique de modèles en transition de phase. Thèse Mathématiques et leurs interactions, 2014.