Long-time behavior of the higher-order anisotropic Caginalp phase-field systems based on the Cattaneo law

Abstract

The aim of this paper is to study higher-order Caginalp phase-field systems based on the Maxwell–Cattaneo law, instead of the classical Fourier law. More precisely, one obtains well-posedness results, as well as the existence of finite-dimensional attractors.

Keywords

Phase-field systems higher-order systems Anisotropy Maxwell–Cattaneo law well-posedness long-time behavior dissipativity global attractor exponential attractor

1. Introduction

The Caginalp phase-field model $\begin{array}{l} (1.1) & \frac{\partial u}{\partial t} - Δ u + f (u) = θ, \\ (1.2) & \frac{\partial θ}{\partial t} - Δ θ = - \frac{\partial u}{\partial t}, \end{array}$ proposed in [4], has been extensively studied (see, e.g., [3,5,6,8,9,11,15–20,22] and the references therein). Here u denotes the order parameter, θ the relative temperature (defined as $θ = \tilde{θ} - θ_{E}$ , where $\tilde{θ}$ is the absolute temperature and $θ_{E}$ the equilibrium melting temperature) and f the derivative of a double-well potential F (a typical choice of potential is $F (s) = \frac{1}{4} {(s^{2} - 1)}^{2}$ , hence the usual cubic nonlinear term $f (s) = s^{3} - s$ ). Furthermore, all physical constants have been set equal to one. This system models, e.g., melting-solidition phenomena in certain classes of materials.

These equations can be derived as follows: one introduces the (total Ginzburg-Landau) free energy $\begin{matrix} (1.3) & Ψ (u, θ) = \int_{Ω} (\frac{1}{2} | \nabla u |^{2} + F (u) - u θ - \frac{1}{2} θ^{2}) d x, \end{matrix}$ where Ω is the domain occupied by the system (one assumes here that it is a bounded and regular domain of $R^{3}$ , with boundary Γ). One then defines the enthalpy H as $\begin{matrix} (1.4) & H = - \partial_{θ} Ψ, \end{matrix}$ where ∂ denotes a variational derivative, which gives $\begin{matrix} (1.5) & H = u + θ . \end{matrix}$ The governing equations for u and θ are then given by (see [5]) $\begin{array}{l} (1.6) & \frac{\partial u}{\partial t} = - \partial_{u} Ψ, \\ (1.7) & \frac{\partial H}{\partial t} + div q = 0, \end{array}$ where q is the thermal flux vector. Assuming the classical Fourier law $\begin{matrix} (1.8) & q = - \nabla θ, \end{matrix}$ one finds (1.1) and (1.2).

Now, a drawback of the Fourier law is the so-called “paradox of heat conduction”, namely, it predicts that thermal signals propagate with infinite speed, which, in particular, violates causality (see [12]). One possible modification, in order to correct this unrealistic feature, is the Maxwell–Cattaneo law $\begin{matrix} (1.9) & (1 + \frac{\partial}{\partial t}) q = - \nabla θ . \end{matrix}$ In that case, it follows from (1.7) that $\begin{matrix} (1.10) & (1 + \frac{\partial}{\partial t}) \frac{\partial H}{\partial t} - Δ θ = 0, \end{matrix}$ hence the following equation for the temperature: $\begin{matrix} (1.11) & \frac{\partial^{2} θ}{\partial t^{2}} + \frac{\partial θ}{\partial t} - Δ θ = - \frac{\partial^{2} u}{\partial t^{2}} - \frac{\partial u}{\partial t} . \end{matrix}$ Integrating (1.11) between 0 and t (the terme $\frac{\partial^{2} u}{\partial t^{2}}$ in (1.11) would be very difficult to handle from a mathematical point of view) and setting $\begin{matrix} (1.12) & α = \int_{0}^{t} θ d s + α_{0} (θ = \frac{\partial α}{\partial t}), \end{matrix}$ where α is called thermal displacement variable (here, $α_{0}$ is a priori fixed arbitrarily; see also below), one finally obtains $\begin{matrix} (1.13) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ α = - \frac{\partial u}{\partial t} - u + φ, \end{matrix}$ where $\begin{array}{l} φ & = \frac{\partial θ}{\partial t} (0) + θ (0) - Δ α_{0} + \frac{\partial u}{\partial t} (0) + u (0) \\ (1.14) & = \frac{\partial^{2} α}{\partial t^{2}} (0) + \frac{\partial α}{\partial t} (0) - Δ α_{0} + \frac{\partial u}{\partial t} (0) + u (0) . \end{array}$ For simplification, one takes in (1.13): $φ = 0$ ; one notes that it follows from (1.14) that, if $θ (0)$ and $\frac{\partial θ}{\partial t} (0)$ are described, then, providing suitable assumptions on the initial data, one can always choose $α_{0}$ such that $φ = 0$ in (1.13), also one notes that $\frac{\partial u}{\partial t} (0)$ can be obtained from (1.1), hence (1.13) becomes $\begin{matrix} (1.15) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t} . \end{matrix}$

Caginalp G. and Esenturk E. proposed in [7] higher-order phase-field models in order to account for anisotropic interfaces (see also [23] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these authors proposed the following modified (total) free energy $\begin{matrix} (1.16) & Ψ_{HOGL} = \int_{Ω} (\frac{1}{2} \sum_{i = 1}^{k} \sum_{| β | = i} a_{β} {| D^{β} u |}^{2} + F (u) - u θ - \frac{1}{2} θ^{2}) d x, \end{matrix}$ where, for $β = (k_{1}, k_{2}, k_{3}) \in {(N \cup {0})}^{3}$ , $\begin{matrix} | β | = k_{1} + k_{2} + k_{3} \end{matrix}$ and, for $β \neq (0, 0, 0)$ , $\begin{matrix} D^{β} = \frac{\partial^{| β |}}{\partial x_{1}^{k_{1}} \partial x_{2}^{k_{2}} \partial x_{3}^{k_{3}}} \end{matrix}$ (one agrees that $D^{(0, 0, 0)} v = v$ ). Noting that $θ = \frac{\partial α}{\partial t}$ , this then yields the following evolution equation for the order parameter u: $\begin{matrix} (1.17) & \frac{\partial u}{\partial t} + \sum_{i = 1}^{k} {(- 1)}^{i} \sum_{| β | = i} a_{β} D^{2 β} u + f (u) = \frac{\partial α}{\partial t} . \end{matrix}$ In particular, for $k = 1$ (anisotropic Caginalp phase-field system), one has an equation of the form $\begin{matrix} (1.18) & \frac{\partial u}{\partial t} - \sum_{i = 1}^{3} a_{i} \frac{\partial^{2} u}{\partial x_{i}^{2}} + f (u) = \frac{\partial α}{\partial t} \end{matrix}$ and, for $k = 2$ (fourth-order anisotropic Caginalp phase-field system), one has an equation of the form $\begin{matrix} (1.19) & \frac{\partial u}{\partial t} + \sum_{i, j = 1}^{3} a_{i j} \frac{\partial^{4} u}{\partial x_{i}^{2} \partial x_{j}^{2}} - \sum_{i = 1}^{3} b_{i} \frac{\partial^{2} u}{\partial x_{i}^{2}} + f (u) = \frac{\partial α}{\partial t} . \end{matrix}$

Cherfils L., Miranville A. and Peng S. studied in [10] the corresponding higner-order isotropic equation (without the coupling with the temperature), namely the equation $\begin{matrix} \frac{\partial u}{\partial t} + P (- Δ) u + f (u) = 0, \end{matrix}$ where $\begin{matrix} P (s) = \sum_{i = 1}^{k} a_{i} s^{i}, a_{k} > 0, k ⩾ 1, \end{matrix}$ endowed with the Dirichlet/Navier boundary conditions $\begin{matrix} u = Δ u = \dots = Δ^{k - 1} u = 0 on Γ . \end{matrix}$

The aim of this paper is to study the model consisting of the higher-order anisotropic equation (1.17) and the temperature equation (1.15). In particular, one obtains well-posedness results, as well as the existence of finite-dimensional attractors.

2. Setting of the problem

One considers in this section the following initial and boundary value problem, for $k \in N$ : $\begin{array}{l} (2.1) & \frac{\partial u}{\partial t} + \sum_{i = 1}^{k} {(- 1)}^{i} \sum_{| β | = i} a_{β} D^{2 β} u + f (u) = \frac{\partial α}{\partial t}, \\ (2.2) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \\ (2.3) & D^{β} u = α = 0 on Γ, | β | ⩽ k - 1, \\ (2.4) & u |_{t = 0} = u_{0}, α |_{t = 0} = α_{0}, \frac{\partial α}{\partial t} |_{t = 0} = α_{1} . \end{array}$

One assumes that $\begin{matrix} (2.5) & a_{β} > 0 for | β | = k, \end{matrix}$ and one introduces the elliptic operator $A_{k}$ defined by $\begin{matrix} (2.6) & {⟨ A_{k} v, w ⟩}_{H^{- k} (Ω), H_{0}^{k} (Ω)} = \sum_{| β | = k} a_{β} ((D^{β} v, D^{β} w)), \end{matrix}$ where $H^{- k} (Ω)$ is the topological dual of $H_{0}^{k} (Ω)$ . Furthermore, $((\cdot, \cdot))$ denotes the usual $L^{2}$ -scalar product, with associated norm $‖ \cdot ‖$ ; more generally, one denotes by $‖ \cdot ‖_{X}$ the norm on the Banach space X. One can note that $\begin{matrix} (v, w) \in H_{0}^{k} {(Ω)}^{2} \mapsto \sum_{| β | = k} a_{β} ((D^{β} v, D^{β} w)) \end{matrix}$ is bilinear, symmetric, continuous and coercive, so that $\begin{matrix} A_{k} : H_{0}^{k} (Ω) \to H^{- k} (Ω) \end{matrix}$ is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order $2 k$ (see [1,2]) that $A_{k}$ is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain $\begin{matrix} D (A_{k}) = H^{2 k} (Ω) \cap H_{0}^{k} (Ω), \end{matrix}$ where, for $v \in D (A_{k})$ , $\begin{matrix} A_{k} v = {(- 1)}^{k} \sum_{| β | = k} a_{β} D^{2 β} v . \end{matrix}$ One further notes that $D (A_{k}^{\frac{1}{2}}) = H_{0}^{k} (Ω)$ and, for $(v, w) \in D {(A_{k}^{\frac{1}{2}})}^{2}$ , $\begin{matrix} ((A_{k}^{\frac{1}{2}} v, A_{k}^{\frac{1}{2}} w)) = \sum_{| β | = k} a_{β} ((D^{β} v, D^{β} w)) . \end{matrix}$ One finally notes that (see, e.g., [24]) $‖ A_{k} . ‖$ (resp., $‖ A_{k}^{\frac{1}{2}} . ‖$ ) is equivalent to the usual $H^{2 k}$ -norm (resp., $H^{k}$ -norm) on $D (A_{k})$ (resp., $D (A_{k}^{\frac{1}{2}})$ ).

Similarly, one can define the linear operator ${\overline{A}}_{k} = - Δ A_{k}$ $\begin{matrix} {\overline{A}}_{k} : H_{0}^{k + 1} (Ω) \to H^{- k - 1} (Ω) \end{matrix}$ which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain $\begin{matrix} D ({\overline{A}}_{k}) = H^{2 k + 2} (Ω) \cap H_{0}^{k + 1} (Ω), \end{matrix}$ where, for $v \in D ({\overline{A}}_{k})$ , $\begin{matrix} {\overline{A}}_{k} v = {(- 1)}^{k + 1} Δ \sum_{| β | = k} a_{β} D^{2 β} v . \end{matrix}$ Furthermore, $D ({\overline{A}}_{k}^{\frac{1}{2}}) = H_{0}^{k + 1} (Ω)$ and, for $(v, w) \in D {({\overline{A}}_{k}^{\frac{1}{2}})}^{2}$ , $\begin{matrix} (({\overline{A}}_{k}^{\frac{1}{2}} v, {\overline{A}}_{k}^{\frac{1}{2}} w)) = \sum_{| β | = k} a_{β} ((\nabla D^{β} v, \nabla D^{β} w)) . \end{matrix}$ Besides, $‖ {\overline{A}}_{k} . ‖$ (resp., $‖ {\overline{A}}_{k}^{\frac{1}{2}} . ‖$ ) is equivalent to the usual $H^{2 k + 2}$ -norm (resp., $H^{k + 1}$ -norm) on $D ({\overline{A}}_{k})$ (resp., $D ({\overline{A}}_{k}^{\frac{1}{2}})$ ).

Having this, one rewrites (2.1) as $\begin{matrix} (2.7) & \frac{\partial u}{\partial t} + A_{k} u + B_{k} u + f (u) = \frac{\partial α}{\partial t}, \end{matrix}$ where $B_{1} = 0$ and, for $k ⩾ 2$ , $\begin{matrix} B_{k} v = \sum_{i = 1}^{k - 1} {(- 1)}^{i} \sum_{| β | = i} a_{β} D^{2 β} v . \end{matrix}$

As far as the nonlinear term f is concerned, one assumes that $\begin{array}{l} (2.8) & f \in C^{1} (R), f (0) = 0, \\ (2.9) & f^{'} (s) ⩾ - c_{0}, c_{0} ⩾ 0, s \in R, \\ (2.10) & | f^{'} (s) | ⩽ c_{1} (| s |^{2} + 1), c_{1} ⩾ 0, s \in R, \\ (2.11) & f (s) s ⩾ c_{2} F (s) - c_{3} ⩾ - c_{4}, c_{2} > 0, c_{3}, c_{4} ⩾ 0, s \in R, \\ (2.12) & F (s) ⩾ c_{5} s^{4} - c_{6}, c_{5} > 0, c_{6} ⩾ 0, s \in R, \end{array}$ where $F (s) = \int_{0}^{s} f (τ) d τ$ . In particular, the usual cubic nonlinear term $f (s) = s^{3} - s$ satisfies these asumptions.

Throughout the paper, the same letters c, $c^{'}$ and $c^{″}$ denote (generally positive) constants which may vary from line to line. Similarly, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.

3. A priori estimates

The estimates derived in this section are formal, but they can easily be justified within a Galerkin approximation.

One multiplies (2.7) by $\frac{\partial u}{\partial t}$ , one has, integrating over Ω and by parts, $\begin{matrix} (3.1) & \frac{1}{2} \frac{d}{d t} ({‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + 2 \int_{Ω} F (u) d x) + {‖ \frac{\partial u}{\partial t} ‖}^{2} = ((\frac{\partial u}{\partial t}, \frac{\partial α}{\partial t})), \end{matrix}$ where $B_{1}^{\frac{1}{2}} [u] = 0$ and, for $k ⩾ 2$ , $\begin{matrix} (3.2) & B_{k}^{\frac{1}{2}} [u] = \sum_{i = 1}^{k - 1} \sum_{| β | = i} a_{β} {‖ D^{β} u ‖}^{2} \end{matrix}$ (one notes that $B_{k}^{\frac{1}{2}} [u]$ is not necessarily nonnegative). One can note that, owing to the interpolation inequality $\begin{array}{l} (3.3) & ‖ v ‖_{H^{i} (Ω)} ⩽ c (i) ‖ v ‖_{H^{m} (Ω)}^{\frac{i}{m}} ‖ v ‖^{1 - \frac{i}{m}}, \\ v \in H^{m} (Ω), i \in {1, \dots, m - 1}, m \in N, m ⩾ 2, \end{array}$ there holds $\begin{matrix} (3.4) & | B_{k}^{\frac{1}{2}} [u] | ⩽ \frac{1}{2} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + c ‖ u ‖^{2} . \end{matrix}$ This yields, employing (2.12), $\begin{array}{l} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + 2 \int_{Ω} F (u) d x ⩾ & \frac{1}{2} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} \\ + \int_{Ω} F (u) d x + c ‖ u ‖_{L^{4} (Ω)}^{4} - c^{'} ‖ u ‖^{2} - c^{″}, \end{array}$ whence $\begin{matrix} (3.5) & {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + 2 \int_{Ω} F (u) d x ⩾ c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x) - c^{'}, c > 0, \end{matrix}$ noting that, owing to Young’s inequality, $\begin{matrix} (3.6) & ‖ u ‖^{2} ⩽ ϵ ‖ u ‖_{L^{4} (Ω)}^{4} + c (ϵ), \forall ϵ > 0 . \end{matrix}$

One then multiplies (2.7) by u and one has, owing to (2.11) and the interpolation inequality (3.3), $\begin{matrix} (3.7) & \frac{1}{2} \frac{d}{d t} ‖ u ‖^{2} + c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x) ⩽ ((\frac{\partial α}{\partial t}, u)) + c^{'} . \end{matrix}$

Multiplying then (2.2) by $\frac{\partial α}{\partial t}$ , one finds $\begin{matrix} (3.8) & \frac{1}{2} \frac{d}{d t} (‖ \nabla α ‖^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) + {‖ \frac{\partial α}{\partial t} ‖}^{2} = - ((u + \frac{\partial u}{\partial t}, \frac{\partial α}{\partial t})) . \end{matrix}$

Summing (3.1), (3.7) and (3.8), one obtains a differential inequality $\begin{matrix} (3.9) & \frac{d E_{1}}{d t} + c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x + {‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) ⩽ c^{'}, c > 0, \end{matrix}$ where $\begin{matrix} E_{1} = {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] + ‖ u ‖^{2} + 2 \int_{Ω} F (u) d x + ‖ \nabla α ‖^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2} \end{matrix}$ satisfies, owing to (3.5), $\begin{matrix} (3.10) & E_{1} ⩾ c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) - c^{'}, c > 0 . \end{matrix}$ One notes indeed that $\begin{matrix} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + B_{k}^{\frac{1}{2}} [u] ⩽ c ‖ u ‖_{H^{k} (Ω)}^{2}, \end{matrix}$ which yields $\begin{array}{l} E_{1} & ⩽ c ‖ u ‖_{H^{k} (Ω)}^{2} + 2 \int_{Ω} F (u) d x + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2} \\ (3.11) & ⩽ c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) - c^{'}, c > 0, c^{'} ⩾ 0 . \end{array}$

One multiplies (2.2) by α, one finds $\begin{matrix} (3.12) & \frac{d}{d t} [‖ α ‖^{2} + 2 ((\frac{\partial α}{\partial t}, α))] + c ‖ \nabla α ‖^{2} ⩽ c^{'} (‖ u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing finally (3.9) and $δ_{1}$ times (3.12), where $δ_{1} > 0$ is chosen small enough so that $\begin{matrix} {‖ \frac{\partial α}{\partial t} ‖}^{2} + δ_{1} [‖ α ‖^{2} + 2 ((\frac{\partial α}{\partial t}, α))] ⩾ c ({‖ \frac{\partial α}{\partial t} ‖}^{2} + ‖ α ‖^{2}), c > 0, \end{matrix}$ one obtains, owing to the interpolation inequality (3.3), a differential inequality of the form $\begin{matrix} (3.13) & \frac{d E_{2}}{d t} + c (E_{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2}) ⩽ c^{'}, c > 0, \end{matrix}$ where $\begin{matrix} E_{2} = E_{1} + δ_{1} [‖ α ‖^{2} + 2 ((\frac{\partial α}{\partial t}, α))] \end{matrix}$ satisfies $\begin{matrix} (3.14) & E_{2} ⩾ c (‖ u ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u) d x + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) - c^{'}, c > 0 . \end{matrix}$

If follows from (3.13), (3.14) and Gronwall’s lemma that $\begin{array}{l} {‖ u (t) ‖}_{H^{k} (Ω)}^{2} + {‖ α (t) ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} (t) ‖}^{2} \\ (3.15) & ⩽ c e^{- c^{'} t} (‖ u_{0} ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u_{0}) d x + ‖ α_{0} ‖_{H^{1} (Ω)}^{2} + ‖ α_{1} ‖^{2}) + c^{″}, c^{'} > 0, t ⩾ 0 \end{array}$ and $\begin{array}{l} \int_{t}^{t + r} {‖ \frac{\partial u}{\partial t} ‖}^{2} d s \\ (3.16) & ⩽ c e^{- c^{'} t} (‖ u_{0} ‖_{H^{k} (Ω)}^{2} + \int_{Ω} F (u_{0}) d x + ‖ α_{0} ‖_{H^{1} (Ω)}^{2} + ‖ α_{1} ‖^{2}) + c^{″} (r), c^{'} > 0, t ⩾ 0, \end{array}$ for $r > 0$ , given.

One now distinguishes between the cases $k = 1$ and $k ⩾ 2$ .

3.1. First case: $k = 1$

Setting $A = A_{1} = - \sum_{i = 1}^{3} a_{i} \frac{\partial^{2}}{\partial x_{i}^{2}}$ , $a_{i} > 0$ , $i = 1, 2$ and 3, one then considers the initial and boundary value problem: $\begin{array}{l} (3.17) & \frac{\partial u}{\partial t} + A u + f (u) = \frac{\partial α}{\partial t}, \\ (3.18) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \\ (3.19) & u = α = 0 on Γ, \\ (3.20) & u |_{t = 0} = u_{0}, α |_{t = 0} = α_{0}, \frac{\partial α}{\partial t} |_{t = 0} = α_{1} . \end{array}$

Multipying (3.17) by $A u$ , one has $\begin{matrix} \frac{1}{2} \frac{d}{d t} {‖ A^{\frac{1}{2}} u ‖}^{2} + ‖ A u ‖^{2} + \sum_{i = 1}^{3} a_{i} \int_{Ω} f^{'} (u) {| \frac{\partial u}{\partial x_{i}} |}^{2} d x = ((\frac{\partial α}{\partial t}, A u)), \end{matrix}$ which yields, owing to (2.9), $\begin{matrix} (3.21) & \frac{d}{d t} {‖ A^{\frac{1}{2}} u ‖}^{2} + c ‖ u ‖_{H^{2} (Ω)}^{2} ⩽ c^{'} (‖ u ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

One now differentiates (3.17) with respect to time to find, owing to (3.18), $\begin{matrix} (3.22) & \frac{\partial}{\partial t} \frac{\partial u}{\partial t} + A \frac{\partial u}{\partial t} + f^{'} (u) \frac{\partial u}{\partial t} = Δ α - \frac{\partial α}{\partial t} - u - \frac{\partial u}{\partial t}, \end{matrix}$ together with the boundary condition $\begin{matrix} (3.23) & \frac{\partial u}{\partial t} = 0 on Γ . \end{matrix}$ Multiplying (3.22) by $\frac{\partial u}{\partial t}$ , one has, owing to (2.9), $\begin{matrix} (3.24) & \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}^{2} + c {‖ \frac{\partial u}{\partial t} ‖}_{H^{1} (Ω)}^{2} ⩽ c^{'} (‖ u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing (3.13) (for $k = 1$ ), $δ_{2}$ times (3.21) and $δ_{3}$ times (3.24), where $δ_{2}, δ_{3} > 0$ are chosen small enough, one obtains a differential inequality of the form $\begin{matrix} (3.25) & \frac{d E_{3}}{d t} + c (E_{3} + ‖ u ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}_{H^{1} (Ω)}^{2}) ⩽ c^{'}, c > 0, \end{matrix}$ where $\begin{matrix} E_{3} = E_{2} + δ_{2} {‖ A^{\frac{1}{2}} u ‖}^{2} + δ_{3} {‖ \frac{\partial u}{\partial t} ‖}^{2} \end{matrix}$ satisfies $\begin{matrix} (3.26) & E_{3} ⩾ c (‖ u ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + \int_{Ω} F (u) d x + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) - c^{'}, c > 0 . \end{matrix}$

One finally rewrites (3.17) as an elliptic equation, for $t > 0$ fixed, $\begin{matrix} (3.27) & A u + f (u) = - \frac{\partial u}{\partial t} + \frac{\partial α}{\partial t}, u = 0 on Γ . \end{matrix}$ Multiplying (3.27) by $A u$ , one finds, owing to (2.9) and a classical elliptic regularity result, $\begin{matrix} (3.28) & {‖ u (t) ‖}_{H^{2} (Ω)}^{2} ⩽ c E_{3} (t) + c^{'}, t ⩾ 0 . \end{matrix}$ In particular, it follows from (3.25) (for $k = 1$ ) and Gronwall’s lemma that $\begin{matrix} (3.29) & E_{3} (t) ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{'}, c > 0, t ⩾ 0, \end{matrix}$ where one has used the continuous embedding $H^{2} (Ω) \subset C (\overline{Ω})$ to deduce that $\begin{matrix} (3.30) & | \int_{Ω} F (u_{0}) d x | ⩽ Q (‖ u_{0} ‖_{H^{2} (Ω)}) \end{matrix}$ and the fact that $\begin{matrix} \frac{\partial u}{\partial t} (0) = - A u_{0} - f (u_{0}) + α_{1}, \end{matrix}$ which yields $\begin{matrix} (3.31) & ‖ \frac{\partial u}{\partial t} (0) ‖ ⩽ Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ α_{1} ‖) . \end{matrix}$

Combining (3.28) and (3.29), one finally finds, in view of (3.26), $\begin{array}{l} {‖ u (t) ‖}_{H^{2} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} (t) ‖}^{2} + {‖ α (t) ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} (t) ‖}^{2} \\ (3.32) & ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{'}, c > 0, t ⩾ 0 . \end{array}$

3.2. Second case: $k ⩾ 2$

One multiplies (2.7) by $A_{k} u$ and one obtains, owing to the interpolation inequality (3.3), $\begin{matrix} (3.33) & \frac{d}{d t} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + c ‖ u ‖_{H^{2 k} (Ω)}^{2} ⩽ c^{'} (‖ u ‖^{2} + {‖ f (u) ‖}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$ It follows from the continuity of f and F, the continuous embedding $H^{k} (Ω) \subset C (\overline{Ω})$ (one recalls that $k ⩾ 2$ ) and (3.15) that $\begin{array}{l} ‖ u ‖^{2} + {‖ f (u) ‖}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2} & ⩽ Q (‖ u ‖_{H^{k} (Ω)}) + {‖ \frac{\partial α}{\partial t} ‖}^{2} \\ (3.34) & ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{'}, c > 0, t ⩾ 0, \end{array}$ so that $\begin{array}{l} \frac{d}{d t} {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + c ‖ u ‖_{H^{2 k} (Ω)}^{2} \\ (3.35) & ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{″}, c, c^{'} > 0, t ⩾ 0 . \end{array}$

One then differentiates (2.7) with respect to time and noting that $\begin{matrix} \frac{\partial^{2} α}{\partial t^{2}} = Δ α - \frac{\partial α}{\partial t} - u - \frac{\partial u}{\partial t}, \end{matrix}$ one obtains $\begin{array}{l} (3.36) & \frac{\partial}{\partial t} \frac{\partial u}{\partial t} + A_{k} \frac{\partial u}{\partial t} + B_{k} \frac{\partial u}{\partial t} + f^{'} (u) \frac{\partial u}{\partial t} = Δ α - \frac{\partial α}{\partial t} - u - \frac{\partial u}{\partial t}, \\ (3.37) & D^{β} \frac{\partial u}{\partial t} = 0 on Γ, | β | ⩽ k - 1, \\ (3.38) & \frac{\partial u}{\partial t} |_{t = 0} = - A_{k} u_{0} - B_{k} u_{0} - f (u_{0}) + α_{1} . \end{array}$ One can note that, if $u_{0} \in H^{2 k} (Ω) \cap H_{0}^{k} (Ω) (= D (A_{k}))$ and $α_{1} \in L^{2} (Ω)$ , then $\frac{\partial u}{\partial t} (0) \in L^{2} (Ω)$ and $\begin{matrix} (3.39) & ‖ \frac{\partial u}{\partial t} (0) ‖ ⩽ Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ α_{1} ‖) . \end{matrix}$

One multiplies (3.36) by $\frac{\partial u}{\partial t}$ and one finds, owing to (2.9) and the interpolation inequality (3.3), $\begin{matrix} (3.40) & \frac{d}{d t} {‖ \frac{\partial u}{\partial t} ‖}^{2} + c {‖ \frac{\partial u}{\partial t} ‖}_{H^{k} (Ω)}^{2} ⩽ c^{'} (‖ u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing (3.13) (for $k ⩾ 2$ ), (3.35) and $δ_{4}$ times (3.40), where $δ_{4} > 0$ is chosen small enough, one obtains a differential inequality of the form $\begin{array}{l} \frac{d E_{4}}{d t} + c (E_{4} + ‖ u ‖_{H^{2 k} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}_{H^{k} (Ω)}^{2}) \\ (3.41) & ⩽ e^{- c^{'} t} Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{″}, c, c^{'} > 0, \end{array}$ where $\begin{matrix} E_{4} = E_{2} + {‖ A_{k}^{\frac{1}{2}} u ‖}^{2} + δ_{4} {‖ \frac{\partial u}{\partial t} ‖}^{2} \end{matrix}$ satisfies $\begin{matrix} (3.42) & E_{4} ⩾ c (‖ u ‖_{H^{k} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + \int_{Ω} F (u) d x + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}) - c^{'}, c > 0 . \end{matrix}$ In particular, it follows from (3.41)–(3.42) that $\begin{matrix} (3.43) & {‖ \frac{\partial u}{\partial t} (t) ‖}^{2} ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{'}, c > 0, t ⩾ 0 \end{matrix}$ and $\begin{array}{l} \int_{t}^{t + r} (‖ u ‖_{H^{2 k} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}_{H^{k} (Ω)}^{2}) d s \\ (3.44) & ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{'} (r), c > 0, t ⩾ 0, r > 0 given . \end{array}$

Next, one rewrites (2.7) as an elliptic equation, for $t > 0$ fixed, $\begin{matrix} (3.45) & A_{k} u = - \frac{\partial u}{\partial t} - B_{k} u - f (u) + \frac{\partial α}{\partial t}, D^{β} u = 0 on Γ, | β | ⩽ k - 1 . \end{matrix}$ Multiplying (3.45) by $A_{k} u$ , one has, owing to the interpolation inequality (3.3), $\begin{matrix} (3.46) & ‖ A_{k} u ‖^{2} ⩽ c (‖ u ‖^{2} + {‖ f (u) ‖}^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), \end{matrix}$ hence, owing to (3.34) and (3.43), $\begin{matrix} (3.47) & {‖ u (t) ‖}_{H^{2 k} (Ω)}^{2} ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) + c^{'}, c > 0, t ⩾ 0 . \end{matrix}$

One now multiplies (2.2) by $- Δ \frac{\partial α}{\partial t}$ and by $δ_{5}$ times $- Δ α$ , where $δ_{5} > 0$ is chosen small enough enough so that $\begin{matrix} {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + δ_{5} [‖ \nabla α ‖^{2} + 2 ((\nabla \frac{\partial α}{\partial t}, \nabla α))] ⩾ c ({‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + ‖ \nabla α ‖^{2}), c > 0, \end{matrix}$ one integrates over Ω and by parts, and one sums the resulting relations to obtain $\begin{array}{l} \frac{d}{d t} [‖ Δ α ‖^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + δ_{5} ‖ \nabla α ‖^{2} + 2 δ_{5} ((\nabla \frac{\partial α}{\partial t}, \nabla α))] + c (‖ Δ α ‖^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2}) \\ ⩽ c^{'} (‖ u ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}_{H^{1} (Ω)}^{2}) . \end{array}$ In particular, $\begin{array}{l} \frac{d}{d t} [‖ Δ α ‖^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2} + δ_{5} ‖ \nabla α ‖^{2} + δ_{5} ((\nabla \frac{\partial α}{\partial t}, \nabla α))] \\ (3.48) & ⩽ c (‖ u ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial u}{\partial t} ‖}_{H^{1} (Ω)}^{2}) . \end{array}$ This yields, owing to (3.44) ( $k ⩾ 2$ ) and the uniform Gronwall’s lemma that $\begin{array}{l} {‖ α (t) ‖}_{H^{2} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} (t) ‖}_{H^{1} (Ω)}^{2} \\ (3.49) & ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ α_{0} ‖_{H^{2} (Ω)}, ‖ α_{1} ‖_{H^{1} (Ω)}) + c^{'} (r), c > 0, t ⩾ r, r > 0 given . \end{array}$ Furthermore, on $[0, r]$ , if one assumes that $u_{0} \in H^{2 k} (Ω) \cap H_{0}^{k} (Ω)$ and $(α_{0}, α_{1}) \in (H^{2} (Ω) \cap H_{0}^{1}) \times H_{0}^{1} (Ω)$ , one deduces from (3.48), (3.44) and Gronwall’s lemma an $H^{2} (Ω) \times H^{1} (Ω)$ -estimate on $(α, \frac{\partial α}{\partial t})$ on $[0, r]$ which, combined with (3.49) and (3.47), gives an $H^{2 k} (Ω) \times H^{2} (Ω) \times H^{1} (Ω)$ -estimate on $(u, α, \frac{\partial α}{\partial t})$ , for all times, namely $\begin{array}{l} {‖ u (t) ‖}_{H^{2 k} (Ω)}^{2} + {‖ α (t) ‖}_{H^{2} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} (t) ‖}_{H^{1} (Ω)}^{2} \\ (3.50) & ⩽ e^{- c t} Q (‖ u_{0} ‖_{H^{2 k} (Ω)}, ‖ α_{0} ‖_{H^{2} (Ω)}, ‖ α_{1} ‖_{H^{1} (Ω)}) + c^{'}, c > 0, t ⩾ 0 . \end{array}$

Remark 3.1.
One can note that one has estimate which is similar to (3.49) when $k = 1$ .

4. The dissipative semigroup

This section is devoted to the existence and uniqueness of soltions to the problem (2.1)–(2.4), and of bounded absording sets to the semigroup.

One first has the following theorem.

Theorem 4.1.

One assumes that $u_{0} \in H_{0}^{k} (Ω)$ , with $\int_{Ω} F (u_{0}) d x < + \infty$ when $k = 1$ , and $(α_{0}, α_{1}) \in H_{0}^{1} (Ω) \times L^{2} (Ω)$ . Then, ( 2.1 )–( 2.4 ) possesses a unique solution $(u, α, \frac{\partial α}{\partial t})$ such that, $\forall T > 0$ , $u (0) = u_{0}$ , $α (0) = α_{0}$ , $\frac{\partial α}{\partial t} (0) = α_{1}$ , $\begin{array}{l} u \in L^{\infty} (R^{+}; H_{0}^{k} (Ω)) \cap L^{2} (0, T; H^{2 k} (Ω) \cap H_{0}^{k} (Ω), \\ \frac{\partial u}{\partial t} \in L^{2} (0, T; L^{2} (Ω)), \\ α \in L^{\infty} (R^{+}; H_{0}^{1} (Ω)), \\ \frac{\partial α}{\partial t} \in L^{\infty} (R^{+}; L^{2} (Ω)) \end{array}$ and $\begin{array}{l} \frac{d}{d t} ((u, v)) + ((A_{k}^{\frac{1}{2}} u, A_{k}^{\frac{1}{2}} v)) + B_{k}^{\frac{1}{2}} [u, v] + ((f (u), v)) \\ = \frac{d}{d t} ((α, v)), \forall v \in C_{c}^{\infty} (Ω), \\ \frac{d}{d t} [((\frac{\partial α}{\partial t}, w)) + ((α, w))] + ((\nabla α, \nabla w)) \\ = - ((u, w)) - \frac{d}{d t} ((u, w)), \forall w \in C_{c}^{\infty} (Ω), \end{array}$ where $B_{1}^{\frac{1}{2}} [u, v] = 0$ and, for $k ⩾ 2$ , $\begin{matrix} B_{k}^{\frac{1}{2}} [u, v] = \sum_{i = 1}^{k - 1} \sum_{| β | = i} a_{β} ((D^{β} u, D^{β} v)) . \end{matrix}$

If one further assumes that $u_{0} \in H^{2 k} (Ω) \cap H_{0}^{k} (Ω)$ and $(α_{0}, α_{1}) \in (H^{2} (Ω) \cap H_{0}^{1} (Ω)) \times H_{0}^{1} (Ω)$ , then $\begin{array}{l} u \in L^{\infty} (R^{+}; H^{2 k} (Ω) \cap H_{0}^{k} (Ω)), \\ \frac{\partial u}{\partial t} \in L^{\infty} (R^{+}; L^{2} (Ω)) \cap L^{2} (0, T; H_{0}^{k} (Ω)), \forall T > 0, \\ α \in L^{\infty} (R^{+}; H^{2} (Ω) \cap H_{0}^{1} (Ω)) \end{array}$ and $\begin{matrix} \frac{\partial α}{\partial t} \in L^{\infty} (R^{+}; H_{0}^{1} (Ω)) . \end{matrix}$

Proof.

Existence:
The proofs of existence and regulararity in (i) and (ii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.
Uniqueness:
Let now $(u^{(1)}, α^{(1)}, \frac{\partial α^{(1)}}{\partial t})$ and $(u^{(2)}, α^{(2)}, \frac{\partial α^{(2)}}{\partial t})$ be two solutions to (2.1)–(2.3) with initial data $(u_{0}^{(1)}, α_{0}^{(1)}, α_{1}^{(1)})$ and $(u_{0}^{(2)}, α_{0}^{(2)}, α_{1}^{(2)})$ , respectively. One sets $\begin{matrix} (u, α, \frac{\partial α}{\partial t}) = (u^{(1)}, α^{(1)}, \frac{\partial α^{(1)}}{\partial t}) - (u^{(2)}, α^{(2)}, \frac{\partial α^{(2)}}{\partial t}) \end{matrix}$ and $\begin{matrix} (u_{0}, α_{0}, α_{1}) = (u_{0}^{(1)}, α_{0}^{(1)}, α_{1}^{(1)}) - (u_{0}^{(2)}, α_{0}^{(2)}, α_{1}^{(2)}) . \end{matrix}$ Then, $(u, α)$ satisfies $\begin{array}{l} (4.1) & \frac{\partial u}{\partial t} + A_{k} u + B_{k} u + f (u^{(1)}) - f (u^{(2)}) = \frac{\partial α}{\partial t}, \\ (4.2) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \\ (4.3) & D^{β} u = α = 0 on Γ, | β | ⩽ k - 1, \\ (4.4) & u |_{t = 0} = u_{0}, α |_{t = 0} = α_{0}, \frac{\partial α}{\partial t} |_{t = 0} = α_{1} . \end{array}$

Multiplying (4.1) by u, one obtains, owing to (2.9) and the interpolation inequality (3.3), $\begin{matrix} (4.5) & \frac{d}{d t} ‖ u ‖^{2} + c ‖ u ‖_{H^{k} (Ω)}^{2} ⩽ c^{'} (‖ u ‖^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Next, multiplying (4.2) by $u + \frac{\partial α}{\partial t}$ , one finds $\begin{matrix} (4.6) & \frac{d}{d t} (‖ \nabla α ‖^{2} + {‖ u + \frac{\partial α}{\partial t} ‖}^{2}) + c {‖ u + \frac{\partial α}{\partial t} ‖}^{2} ⩽ c^{'} (‖ u ‖_{H^{1} (Ω)}^{2} + ‖ \nabla α ‖^{2}), c > 0 . \end{matrix}$

Summing then (4.5) and $δ_{6}$ times (4.6), where $δ_{6} > 0$ is small enough, one has a differential inequality of the form (one notes that $k ⩾ 1$ ) $\begin{matrix} (4.7) & \frac{d E_{5}}{d t} ⩽ c E_{5}, \end{matrix}$ where $\begin{matrix} E_{5} = ‖ u ‖^{2} + δ_{6} (‖ \nabla α ‖^{2} + {‖ u + \frac{\partial α}{\partial t} ‖}^{2}) \end{matrix}$ satisfies $\begin{matrix} (4.8) & E_{5} ⩾ c (‖ u ‖^{2} + ‖ α ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$ It follows from (4.7), (4.8) and Gronwall’s lemma that $\begin{matrix} (4.9) & {‖ u (t) ‖}^{2} + {‖ α (t) ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial α}{\partial t} (t) ‖}^{2} ⩽ c^{'} e^{c t} (‖ u_{0} ‖^{2} + ‖ α_{0} ‖_{H^{1} (Ω)}^{2} + ‖ α_{1} ‖^{2}), t ⩾ 0, \end{matrix}$ hence the uniqueness, as well as the continuous dependence with respect to the initial data in the $L^{2} \times H^{1} \times L^{2}$ -norm. □

One sets $Φ = H_{0}^{k} (Ω) \times H_{0}^{1} (Ω) \times L^{2} (Ω)$ . It follows from Theorem 4.1 that one can define the family of solving operators $\begin{matrix} S (t) : Φ \to Φ, (u_{0}, α_{0}, α_{1}) \mapsto (u (t), α (t), \frac{\partial α}{\partial t} (t)), t ⩾ 0 \end{matrix}$ (i.e., $S (0) = I$ (identity operator) and $S (t + τ) = S (t) \circ S (τ)$ , t, $τ ⩾ 0$ ) which are continuous with respect to the $L^{2} \times H^{1} \times L^{2}$ -topology. One then deduces from (3.15) the Theorem 4.2.
The semigroup $S (t)$ is dissipative in Φ, in the sense that it possesses a bounded absording set $B_{0} \subset Φ$ (i.e., for every $B \subset Φ$ bounded, there exists $t_{0} = t_{0} (B) ⩾ 0$ such that $t ⩾ t_{0}$ implies $S (t) B \subset B_{0}$ ).
Remark 4.1.
It is easy to see that one can assume, without loss of generality, that $B_{0}$ is positively invariant by $S (t)$ , i.e. $S (t) B_{0} \subset B_{0}$ , for every $t ⩾ 0$ .

5. Existence of the global attractor

This section is devoted to the existence of the global attractor and one takes the coefficients $a_{β} ⩾ 0$ .

One now states the following existence result of the global attractor.

Theorem 5.1.
One assumes that the assumptions of Theorem 4.1 , and ( 2.8 ) and ( 2.9 ) hold. Then, the semigroup $S (t)$ possesses the global attractor $A \subset B_{0}$ which is compact in Φ and bounded in $Φ_{1}$ .
Proof.
The equality (2.2) is a damped wave equation, then one uses a semigroup decomposition argument consisting in splitting the semigroup $S (t)$ , $t ⩾ 0$ , into the sum of two operators families: $S (t) = S_{1} (t) + S_{2} (t)$ , where operators $S_{1} (t)$ go to zero as t tends to infinity while operators $S_{2} (t)$ are compacts. In order to do that, one writes $\begin{matrix} (u, α, \frac{\partial α}{\partial t}) = (v, γ, \frac{\partial γ}{\partial t}) + (ϕ, ψ, \frac{\partial ψ}{\partial t}) \end{matrix}$ where $(v, γ, \frac{\partial γ}{\partial t})$ is the solution to $\begin{array}{l} (5.1) & \frac{\partial v}{\partial t} + A_{k} v + B_{k} v = \frac{\partial γ}{\partial t}, \\ (5.2) & \frac{\partial^{2} γ}{\partial t^{2}} + \frac{\partial γ}{\partial t} - Δ γ = - v - \frac{\partial v}{\partial t}, \\ (5.3) & v = γ = 0 on Γ, \\ (5.4) & v /_{t = 0} = u_{0}, γ /_{t = 0} = α_{0}, \frac{\partial γ}{\partial t} /_{t = 0} = α_{1} \end{array}$ and $(ϕ, ψ)$ satisfies $\begin{array}{l} (5.5) & \frac{\partial ϕ}{\partial t} + A_{k} ϕ + B_{k} ϕ + f (u) = \frac{\partial ψ}{\partial t}, \\ (5.6) & \frac{\partial^{2} ψ}{\partial t^{2}} + \frac{\partial ψ}{\partial t} - Δ ψ = - ϕ - \frac{\partial ϕ}{\partial t}, \\ (5.7) & ϕ = ψ = 0 on Γ, \\ (5.8) & ϕ /_{t = 0} = ξ /_{t = 0} = \frac{\partial ψ}{\partial t} /_{t = 0} = 0 . \end{array}$

Multiplying (5.1) by $v + \frac{\partial v}{\partial t}$ and (5.2) by $\frac{\partial γ}{\partial t}$ , and summing the resulting relations, one obtains $\begin{array}{l} \frac{d}{d t} ({‖ A_{k}^{\frac{1}{2}} v ‖}^{2} + B_{k}^{\frac{1}{2}} [v] + ‖ v ‖^{2} + ‖ \nabla γ ‖^{2} + {‖ \frac{\partial γ}{\partial t} ‖}^{2}) \\ (5.9) & + 2 {‖ A_{k}^{\frac{1}{2}} v ‖}^{2} + 2 B_{k}^{\frac{1}{2}} [v] + 2 {‖ \frac{\partial v}{\partial t} ‖}^{2} + 2 ‖ v ‖^{2} + 2 {‖ \frac{\partial γ}{\partial t} ‖}^{2} = 0, \end{array}$ where $\begin{matrix} B_{k}^{\frac{1}{2}} [v] = \sum_{i = 1}^{k - 1} \sum_{| β | = i} a_{β} {‖ D^{β} v ‖}^{2}, a_{β} ⩾ 0, \forall | β | ⩽ k - 1 . \end{matrix}$

One then mutiplies (5.2) by γ, one has $\begin{matrix} (5.10) & \frac{d}{d t} [‖ γ ‖^{2} + 2 ((\frac{\partial γ}{\partial t}, γ))] + c ‖ \nabla γ ‖^{2} ⩽ c^{'} (‖ v ‖^{2} + {‖ \frac{\partial v}{\partial t} ‖}^{2} + {‖ \frac{\partial γ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing (5.9) and $δ_{7}$ times (5.10), where $δ_{7} > 0$ is chosen small enough so that $\begin{matrix} {‖ \frac{\partial γ}{\partial t} ‖}^{2} + δ_{7} [‖ γ ‖^{2} + 2 ((\frac{\partial γ}{\partial t}, γ))] ⩾ c ({‖ \frac{\partial γ}{\partial t} ‖}^{2} + ‖ γ ‖^{2}), c > 0, \end{matrix}$ and one obtains adifferential inequality of the form $\begin{matrix} (5.11) & \frac{d E_{6}}{d t} + c E_{6} ⩽ 0, c > 0, \end{matrix}$ where $\begin{matrix} E_{6} = {‖ A_{k}^{\frac{1}{2}} v ‖}^{2} + B_{k}^{\frac{1}{2}} [v] + ‖ v ‖^{2} + ‖ \nabla γ ‖^{2} + {‖ \frac{\partial γ}{\partial t} ‖}^{2} + δ_{7} [‖ γ ‖^{2} + 2 ((\frac{\partial γ}{\partial t}, γ))] \end{matrix}$ satisfies $\begin{matrix} (5.12) & E_{6} ⩾ c (‖ v ‖_{H^{k} (Ω)}^{2} + ‖ γ ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial γ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$ It follows from (5.11), (5.12) and Gronwall’s lemma that $\begin{array}{l} {‖ v (t) ‖}_{H^{k} (Ω)}^{2} + {‖ γ (t) ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial γ}{\partial t} (t) ‖}^{2} \\ (5.13) & ⩽ c e^{- c^{'} t} (‖ u_{0} ‖_{H^{k} (Ω)}^{2} + ‖ α_{0} ‖_{H^{1} (Ω)}^{2} + ‖ α_{1} ‖^{2}), c, c^{'} > 0, t ⩾ 0 . \end{array}$ One can see that $S_{1} (t) (u_{0}, α_{0}, α_{1}) = (v (t), γ (t), \frac{\partial γ}{\partial t} (t))$ tends to zero when t tends to infinity.

Now, one considers (5.5)–(5.8). One multiplies (5.5) by $ϕ + \frac{\partial ϕ}{\partial t}$ and (5.6) by $\frac{\partial ψ}{\partial t}$ , one integrates over Ω and by parts, and summing the resulting relations, one then writes $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ ϕ ‖^{2} + {‖ A_{k}^{\frac{1}{2}} ϕ ‖}^{2} + B_{k}^{\frac{1}{2}} [ϕ] + ‖ \nabla ψ ‖^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2}) + c (‖ ϕ ‖_{H^{k} (Ω)}^{2} + {‖ \frac{\partial ϕ}{\partial t} ‖}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2}) \\ ⩽ | \int_{Ω} f (u) ϕ d x | + | \int_{Ω} f (u) \frac{\partial ϕ}{\partial t} d x |, c > 0 . \end{array}$ One then has $\begin{matrix} | \int_{Ω} f (u) \frac{\partial ϕ}{\partial t} d x | ⩽ \int_{Ω} (| u |^{2} + 1) | u | | \frac{\partial ϕ}{\partial t} | d x \end{matrix}$ Furthemore, one has, owing to Hölder’s inequality ( $k ⩾ 1$ ), $\begin{array}{l} | \int_{Ω} f (u) \frac{\partial ϕ}{\partial t} d x | d x & ⩽ (‖ u ‖_{L^{6} (Ω)}^{2} + 1) ‖ u ‖_{L^{6} (Ω)} ‖ \frac{\partial ϕ}{\partial t} ‖ \\ ⩽ c (‖ u ‖_{H^{1} (Ω)}^{2} + 1) ‖ u ‖_{H^{1} (Ω)} ‖ \frac{\partial ϕ}{\partial t} ‖ \\ (5.14) & ⩽ c (‖ u ‖_{H^{k} (Ω)}^{2} + 1) ‖ u ‖_{H^{k} (Ω)} ‖ \frac{\partial ϕ}{\partial t} ‖ . \end{array}$ Analogously, one finds $\begin{matrix} (5.15) & | \int_{Ω} f (u) ϕ d x | ⩽ c (‖ u ‖_{H^{k} (Ω)}^{2} + 1) ‖ u ‖_{H^{k} (Ω)} ‖ ϕ ‖ . \end{matrix}$ One deduce from (5.14) and (5.15) an inequality of the form $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ ϕ ‖^{2} + {‖ A_{k}^{\frac{1}{2}} ϕ ‖}^{2} + B_{k}^{\frac{1}{2}} [ϕ] + ‖ \nabla ψ ‖^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2}) \\ + c (‖ ϕ ‖_{H^{k} (Ω)}^{2} + {‖ \frac{\partial ϕ}{\partial t} ‖}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2}) \\ (5.16) & ⩽ c^{'} (‖ u ‖_{H^{k} (Ω)}^{4} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, c > 0 . \end{array}$

One then multiplies (5.6) by ψ and integrating over Ω and by parts, one has $\begin{matrix} (5.17) & \frac{d}{d t} [‖ ψ ‖^{2} + 2 ((\frac{\partial ψ}{\partial t}, ψ))] + c ‖ \nabla ψ ‖^{2} ⩽ c^{'} (‖ ϕ ‖^{2} + {‖ \frac{\partial ϕ}{\partial t} ‖}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing (5.16) and $δ_{8}$ times (5.17), where $δ_{8} > 0$ is chosen small enough so that $\begin{matrix} {‖ \frac{\partial ψ}{\partial t} ‖}^{2} + δ_{8} [‖ ψ ‖^{2} + 2 ((\frac{\partial ψ}{\partial t}, \nabla ψ))] ⩾ c ({‖ \frac{\partial ψ}{\partial t} ‖}^{2} + ‖ ψ ‖^{2}), c > 0, \end{matrix}$ one obtains a differential inequality of the form $\begin{matrix} (5.18) & \frac{d E_{7}}{d t} + c (E_{7} + {‖ \frac{\partial ϕ}{\partial t} ‖}^{2}) ⩽ c^{'} (‖ u ‖_{H^{k} (Ω)}^{4} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, c > 0, \end{matrix}$ where $\begin{matrix} E_{7} = ‖ ϕ ‖^{2} + {‖ A_{k}^{\frac{1}{2}} ϕ ‖}^{2} + B_{k}^{\frac{1}{2}} [ϕ] + ‖ \nabla ψ ‖^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2} + δ_{8} [‖ ψ ‖^{2} + 2 ((\frac{\partial ψ}{\partial t}, ψ))] \end{matrix}$ satisfies $\begin{matrix} (5.19) & E_{7} ⩾ c (‖ ϕ ‖_{H^{k} (Ω)}^{2} + ‖ ψ ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Multiplying (5.5) by $- Δ (ϕ + \frac{\partial ϕ}{\partial t})$ and (5.6) by $- Δ \frac{\partial ψ}{\partial t}$ , integrating over Ω and by parts, and summing the two resulting relations to obtain, using Hölder’s inequality, $\begin{array}{l} \frac{d}{d t} (‖ \nabla ϕ ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ϕ ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [ϕ] + ‖ Δ ψ ‖^{2} + {‖ \nabla \frac{\partial ψ}{\partial t} ‖}^{2}) \\ + c (‖ ϕ ‖_{H^{k + 1} (Ω)}^{2} + {‖ \frac{\partial ϕ}{\partial t} ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}_{H^{1} (Ω)}^{2}) \\ (5.20) & ⩽ ‖ f^{'} (u) \nabla u ‖ ‖ \nabla ϕ ‖ + ‖ f^{'} (u) \nabla u ‖ ‖ \nabla \frac{\partial ϕ}{\partial t} ‖, \end{array}$ where $\begin{matrix} {\overline{B}}_{k}^{\frac{1}{2}} [ϕ] = \sum_{i = 1}^{k - 1} \sum_{| β | = i} a_{β} {‖ \nabla D^{β} ϕ ‖}^{2} . \end{matrix}$

One multiplies then (5.6) by $- Δ ψ$ , to have $\begin{array}{l} \frac{d}{d t} [‖ \nabla ψ ‖^{2} + 2 ((\nabla \frac{\partial ψ}{\partial t}, \nabla ψ))] + c ‖ Δ ψ ‖^{2} \\ (5.21) & ⩽ c^{'} (‖ ϕ ‖^{2} + {‖ \frac{\partial ϕ}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial ψ}{\partial t} ‖}^{2}), c > 0 . \end{array}$ One deduces summing (5.20) and $δ_{9}$ times (5.21), where $δ_{9} > 0$ is chosen small enough so that $\begin{array}{l} {‖ \nabla \frac{\partial ψ}{\partial t} ‖}^{2} + δ_{9} [‖ \nabla ψ ‖^{2} + 2 ((\nabla \frac{\partial ψ}{\partial t}, \nabla ψ))] ⩾ c ({‖ \nabla \frac{\partial ψ}{\partial t} ‖}^{2} + ‖ \nabla ψ ‖^{2}), c > 0, \\ (5.22) & \frac{d}{d t} [‖ \nabla ϕ ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ϕ ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [ϕ] + ‖ Δ ψ ‖^{2} + {‖ \nabla \frac{\partial ψ}{\partial t} ‖}^{2} \\ + δ_{9} ‖ \nabla ψ ‖^{2} + 2 δ_{9} ((\nabla \frac{\partial ψ}{\partial t}, \nabla ψ))] \\ + c (‖ ϕ ‖_{H^{k + 1} (Ω)}^{2} + ‖ ψ ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ϕ}{\partial t} ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}_{H^{1} (Ω)}^{2}) ⩽ {‖ f^{'} (u) \nabla u ‖}^{2} . \end{array}$

Finally, summing (5.18) and (5.22), in particular, one finds $\begin{matrix} (5.23) & \frac{d E_{8}}{d t} ⩽ c (‖ u ‖_{H^{k} (Ω)}^{4} + 1) ‖ u ‖_{H^{k} (Ω)}^{2} + {‖ f^{'} (u) \nabla u ‖}^{2}, \end{matrix}$ where $\begin{matrix} E_{8} = E_{7} + [‖ \nabla ϕ ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ϕ ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [ϕ] + ‖ Δ ψ ‖^{2} + {‖ \nabla \frac{\partial ψ}{\partial t} ‖}^{2} + δ_{9} ‖ \nabla ψ ‖^{2} + 2 δ_{9} ((\nabla \frac{\partial ψ}{\partial t}, \nabla ψ))] \end{matrix}$ satisfies $\begin{matrix} (5.24) & E_{8} ⩾ c (‖ ϕ ‖_{H^{k + 1} (Ω)}^{2} + ‖ ψ ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ψ}{\partial t} ‖}_{H^{1} (Ω)}^{2}), c > 0 . \end{matrix}$ It follows from the assumptions of the theorem that (see [14]) $f^{'} (u) \nabla u \in L^{2} (0, T; L^{2} {(Ω)}^{3})$ , for every $T > 0$ , and $\begin{matrix} (5.25) & {‖ f^{'} (u) \nabla u ‖}_{L^{2} (0, T; L^{2} {(Ω)}^{3})} ⩽ Q (T, ‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖) \end{matrix}$ for some functions Q. One thus deduces from (5.23), (5.24) and (5.25) that $\begin{array}{l} {‖ ϕ (t) ‖}_{H^{k + 1} (Ω)}^{2} + {‖ ψ (t) ‖}_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ψ}{\partial t} (t) ‖}_{H^{1} (Ω)}^{2} \\ (5.26) & ⩽ Q (t, ‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖), t ⩾ 0 . \end{array}$ Hence, the operator $S_{2} (t) (u_{0}, α_{0}, α_{1}) = (ϕ (t), ψ (t), \frac{\partial ψ}{\partial t})$ is asymptotically compact in the sense of Kuratowski measure of noncompactness (see [20]), which completes the proof. □
Remark 5.1.
One recalls that the global attractor $A$ is the smallest (for the inclusion) compact set of the phase space which is invariant by the flow (i.e., $S (t) A = A$ , for $t ⩾ 0$ ) and attracts all bounded sets of initial data as time goes to infinity; it thus appears as a suitable object in view of the study of the asymptotic behavior of the system. Furthermore, the finite-dimensionality means, very roughly speaking, that, even though the initial phase space has infinite dimension, the reduced dynamics can be described by a finite number of parameters. One refers the interested reader to, e.g., [21,24] for more details and discussions on this.

Now that the existence of the global attractor is proved, a naturel question is to know it this attractor has finite dimension in terms of the Hausdorff or fractal dimension. That is the aim of the next section.
6. Existence of exponential attractors

In this section, one derives several estimates on the difference of solutions of problem (2.1)–(2.3) which are of fundamental significance for the study of exponential attracors.

One starts by stating an abstract result that will be useful in what follows (see [21]).

Theorem 6.1.

Let V and H be two Banach spaces such that V is compactly embedded into H and $S (t) : X \to X$ be a semigroup acting on a closed subset X of H. One assumes that

$\begin{matrix} S (t) u - S (t) v = φ^{1} (t) + φ^{2} (t), for every u, v \in X, \end{matrix}$ with $\begin{matrix} {‖ φ^{1} (t) ‖}_{H}^{2} ⩽ d (t) ‖ u - v ‖_{H}^{2}, \end{matrix}$ d is continuous, $t ⩾ 0$ , $d (t) \to 0$ as $t \to + \infty$ , and $\begin{matrix} {‖ φ^{2} (t) ‖}_{V}^{2} ⩽ h (t) ‖ u - v ‖_{H}^{2}, t > 0, h is continuous; \end{matrix}$

$(t, x) \mapsto S (t) x$ is uniformly Hölder continuous on $[0, T] \times B$ , for every $T > 0$ , and $B \subset X$ bounded.

Then $S (t)$ possesses an exponential attractor $M$ ( $M$ contains the global attractor $A$ ) on X.

In order to get the existence of exponential attractors in this case, Theorem 6.1 can be used. One has the following result.

Proposition 6.1.

Assume that the assumptions of Theorem 4.1 hold. Then, the semigroup $S (t)$ , $t ⩾ 0$ correspoding to equations ( 2.1 )–( 2.4 ) defined from Φ into itself satisfies a decomposition as in Theorem 6.1 .

Proof.

Let now $(u^{(1)}, α^{(1)}, \frac{\partial α^{(1)}}{\partial t})$ and $(u^{(2)}, α^{(2)}, \frac{\partial α^{(2)}}{\partial t})$ be two solutions to (2.1)–(2.3) with initial data $(u_{0}^{(1)}, α_{0}^{(1)}, α_{1}^{(1)})$ and $(u_{0}^{(2)}, α_{0}^{(2)}, α_{1}^{(2)})$ , respectively. One sets $\begin{matrix} (u, α, \frac{\partial α}{\partial t}) = (u^{(1)} - u^{(2)}, α^{(1)} - α^{(2)}, \frac{\partial α^{(1)}}{\partial t} - \frac{\partial α^{(2)}}{\partial t}) . \end{matrix}$ Then, $(u, α)$ satisfies $\begin{array}{l} (6.1) & \frac{\partial u}{\partial t} + A_{k} u + B_{k} u + f (u^{(1)}) - f (u^{(2)}) = \frac{\partial α}{\partial t}, \\ (6.2) & \frac{\partial^{2} α}{\partial t^{2}} + \frac{\partial α}{\partial t} - Δ α = - u - \frac{\partial u}{\partial t}, \\ (6.3) & u = α = 0 on Γ, \\ (6.4) & u (0) = u_{0}, α (0) = α_{0}, \frac{\partial α}{\partial t} (0) = α_{1}, \end{array}$ where $\begin{matrix} (u_{0}, α_{0}, α_{1}) = (u_{0}^{(1)} - u_{0}^{(2)}, α_{0}^{(1)} - α_{0}^{(2)}, α_{1}^{(1)} - α_{1}^{(2)}) . \end{matrix}$ Now one decomposes the solution $(u, α, \frac{\partial α}{\partial t})$ as follows: $\begin{matrix} (u, α, \frac{\partial α}{\partial t}) = (ν, η, \frac{\partial η}{\partial t}) + (ω, ξ, \frac{\partial ξ}{\partial t}), \end{matrix}$ where $\begin{matrix} (ν, η, \frac{\partial η}{\partial t}) and (ω, ξ, \frac{\partial ξ}{\partial t}) \end{matrix}$ are solutions to $\begin{array}{l} (6.5) & \frac{\partial ν}{\partial t} + A_{k} ν + B_{k} ν = \frac{\partial η}{\partial t}, \\ (6.6) & \frac{\partial^{2} η}{\partial t^{2}} + \frac{\partial η}{\partial t} - Δ η = - ν - \frac{\partial ν}{\partial t}, \\ (6.7) & ν = η = 0 on Γ, \\ (6.8) & ν /_{t = 0} = u_{0}, η /_{t = 0} = α_{0}, \frac{\partial η}{\partial t} /_{t = 0} = α_{1}, \end{array}$ and $\begin{array}{l} (6.9) & \frac{\partial ω}{\partial t} + A_{k} ω + B_{k} ω + f (u^{(1)}) - f (u^{(2)}) = \frac{\partial ξ}{\partial t}, \\ (6.10) & \frac{\partial^{2} ξ}{\partial t^{2}} + \frac{\partial ξ}{\partial t} - Δ ξ = - ω - \frac{\partial ω}{\partial t}, \\ (6.11) & ω = ξ = 0 on Γ, \\ (6.12) & ω /_{t = 0} = ξ /_{t = 0} = \frac{\partial ξ}{\partial t} /_{t = 0} = 0, \end{array}$ respectively. Repeating for (6.5)–(6.8) estimates which led us to (5.11), one writes $\begin{matrix} (6.13) & \frac{d E_{9}}{d t} + c E_{9} ⩽ 0, c > 0, \end{matrix}$ where $\begin{matrix} E_{9} = {‖ A_{k}^{\frac{1}{2}} ν ‖}^{2} + B_{k}^{\frac{1}{2}} [ν] + ‖ ν ‖^{2} + ‖ \nabla η ‖^{2} + {‖ \frac{\partial η}{\partial t} ‖}^{2} + δ_{7} [‖ η ‖^{2} + 2 ((\frac{\partial η}{\partial t}, η))] \end{matrix}$ satisfies as (5.12) $\begin{matrix} (6.14) & E_{9} ⩾ c (‖ ν ‖_{H^{k} (Ω)}^{2} + ‖ η ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial η}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$ It follows from (6.13), (6.14) and Gronwall’s lemma that $\begin{array}{l} {‖ ν (t) ‖}_{H^{k} (Ω)}^{2} + {‖ η (t) ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial η}{\partial t} (t) ‖}^{2} \\ (6.15) & ⩽ c e^{- c^{'} t} (‖ u_{0} ‖_{H^{k} (Ω)}^{2} + ‖ α_{0} ‖_{H^{1} (Ω)}^{2} + ‖ α_{1} ‖^{2}), c, c^{'} > 0, t ⩾ 0 . \end{array}$ Setting $d (t) = c e^{- c^{'} t}$ , $t ⩾ 0$ , d is continuous and $d (t) \to 0$ as $t \to + \infty$ .

One now considers (6.9)–(6.12). One multiplies (6.9) by $ω + \frac{\partial ω}{\partial t}$ and (6.10) by $\frac{\partial ξ}{\partial t}$ , one integrates over Ω and by parts, and summing the resulting relations, one then writes $\begin{array}{l} \frac{1}{2} \frac{d}{d t} (‖ ω ‖^{2} + {‖ A_{k}^{\frac{1}{2}} ω ‖}^{2} + B_{k}^{\frac{1}{2}} [ω] + ‖ \nabla ξ ‖^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2}) + c (‖ ω ‖_{H^{k} (Ω)}^{2} + {‖ \frac{\partial ω}{\partial t} ‖}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2}) \\ ⩽ | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) ω d x | + | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) \frac{\partial ω}{\partial t} d x |, c > 0 . \end{array}$ From to (2.10) and owing to the Hölder’s inequality, one has ( $k ⩾ 1$ ) $\begin{array}{l} | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) \frac{\partial ω}{\partial t} d x | & ⩽ \int_{Ω} | f (u^{(1)}) - f (u^{(2)}) | | \frac{\partial ω}{\partial t} | d x \\ ⩽ c \int_{Ω} ({| u^{(1)} |}^{2} + {| u^{(2)} |}^{2} + 1) | u | | \frac{\partial ω}{\partial t} | d x \\ (6.16) & ⩽ c ({‖ u^{(1)} ‖}_{H^{k} (Ω)}^{2} + {‖ u^{(2)} ‖}_{H^{k} (Ω)}^{2} + 1) ‖ u ‖_{H^{k} (Ω)} ‖ \frac{\partial ω}{\partial t} ‖ . \end{array}$

Analogously, one finds $\begin{matrix} (6.17) & | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) ω d x | ⩽ c ({‖ u^{(1)} ‖}_{H^{k} (Ω)}^{2} + {‖ u^{(2)} ‖}_{H^{k} (Ω)}^{2} + 1) ‖ u ‖_{H^{k} (Ω)} ‖ ω ‖ . \end{matrix}$ One deduces from (6.16) and (6.17) that $\begin{array}{l} \frac{d}{d t} (‖ ω ‖^{2} + {‖ A_{k}^{\frac{1}{2}} ω ‖}^{2} + B_{k}^{\frac{1}{2}} [ω] + ‖ \nabla ξ ‖^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2}) + c (‖ ω ‖_{H^{k} (Ω)}^{2} + {‖ \frac{\partial ω}{\partial t} ‖}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2}) \\ (6.18) & ⩽ c^{'} ({‖ u^{(1)} ‖}_{H^{k} (Ω)}^{4} + {‖ u^{(2)} ‖}_{H^{k} (Ω)}^{4} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, c > 0 . \end{array}$

One then multiplies (6.10) by ξ and integrating over Ω and by parts, one has $\begin{matrix} (6.19) & \frac{d}{d t} [‖ ξ ‖^{2} + 2 ((\frac{\partial ξ}{\partial t}, ξ))] + c ‖ \nabla ξ ‖^{2} ⩽ c^{'} (‖ ω ‖^{2} + {‖ \frac{\partial ω}{\partial t} ‖}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing (6.18) and $δ_{10}$ times (6.19), where $δ_{10} > 0$ is chosen small enough so that $\begin{matrix} {‖ \frac{\partial ξ}{\partial t} ‖}^{2} + δ_{10} [‖ ξ ‖^{2} + 2 ((\frac{\partial ξ}{\partial t}, ξ))] ⩾ c ({‖ \frac{\partial ξ}{\partial t} ‖}^{2} + ‖ ξ ‖^{2}), c > 0, \end{matrix}$ one obtains a differential inequality of the form $\begin{matrix} (6.20) & \frac{d E_{10}}{d t} + c (E_{10} + {‖ \frac{\partial ω}{\partial t} ‖}^{2}) ⩽ c^{'} ({‖ u^{(1)} ‖}_{H^{k} (Ω)}^{4} + {‖ u^{(2)} ‖}_{H^{k} (Ω)}^{4} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, c > 0, \end{matrix}$ where $\begin{matrix} E_{10} = ‖ ω ‖^{2} + {‖ A_{k}^{\frac{1}{2}} ω ‖}^{2} + B_{k}^{\frac{1}{2}} [ω] + ‖ \nabla ξ ‖^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2} + δ_{10} [‖ ξ ‖^{2} + 2 ((\frac{\partial ξ}{\partial t}, ξ))] \end{matrix}$ satisfies $\begin{matrix} (6.21) & E_{10} ⩾ c (‖ ω ‖_{H^{k} (Ω)}^{2} + ‖ ξ ‖_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Next, multiplying (6.9) by $- Δ (ω + \frac{\partial ω}{\partial t})$ and (6.10) by $- Δ \frac{\partial ξ}{\partial t}$ , integrating over Ω and by parts, and summing the two resulting relations to obtain $\begin{array}{l} \frac{d}{d t} (‖ \nabla ω ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ω ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [u] + ‖ Δ ξ ‖^{2} + {‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2}) \\ + c (‖ ω ‖_{H^{k + 1} (Ω)}^{2} + {‖ \frac{\partial ω}{\partial t} ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{1} (Ω)}^{2}) \\ (6.22) & ⩽ | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) Δ ω d x | + | \int_{Ω} (f^{'} (u^{(1)}) \nabla u^{(1)} - f^{'} (u^{(2)}) \nabla u^{(2)}) \nabla \frac{\partial ω}{\partial t} d x | . \end{array}$

One now multiplies (6.10) by and $- Δ ξ$ , to have $\begin{matrix} (6.23) & \frac{d}{d t} [‖ \nabla ξ ‖^{2} + 2 ((\nabla \frac{\partial ξ}{\partial t}, \nabla ξ))] + c ‖ Δ ξ ‖^{2} ⩽ c^{'} (‖ ω ‖^{2} + {‖ \frac{\partial ω}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2}), c > 0 . \end{matrix}$

Summing (6.22) and $δ_{11}$ times (6.23), where $δ_{11} > 0$ is chosen small enough so that $\begin{matrix} {‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2} + δ_{11} [‖ \nabla ξ ‖^{2} + 2 ((\nabla \frac{\partial ξ}{\partial t}, \nabla ξ))] ⩾ c ({‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2} + ‖ \nabla ξ ‖^{2}), c > 0, \end{matrix}$ one obtains $\begin{array}{l} \frac{d}{d t} [‖ \nabla ω ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ω ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [u] + ‖ Δ ξ ‖^{2} + {‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2} + δ_{11} ‖ \nabla ξ ‖^{2} + 2 δ_{11} ((\nabla \frac{\partial ξ}{\partial t}, \nabla ξ))] \\ + c (‖ ω ‖_{H^{k + 1} (Ω)}^{2} + ‖ ξ ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ω}{\partial t} ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{1} (Ω)}^{2}) \\ ⩽ | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) Δ ω d x | + | \int_{Ω} (f^{'} (u^{(1)}) \nabla u^{(1)} - f^{'} (u^{(2)}) \nabla u^{(2)}) \nabla \frac{\partial ω}{\partial t} d x | . \end{array}$

As above, one has $\begin{array}{l} | \int_{Ω} (f (u^{(1)}) - f (u^{(2)})) Δ ω d x | & ⩽ c \int_{Ω} ({| u^{(1)} |}^{2} + {| u^{(2)} |}^{2} + 1) | u | | Δ ω | d x \\ (6.24) & ⩽ c ({‖ u^{(1)} ‖}_{H^{k} (Ω)}^{2} + {‖ u^{(2)} ‖}_{H^{k} (Ω)}^{2} + 1) ‖ u ‖_{H^{k} (Ω)} ‖ Δ ω ‖^{2} \end{array}$ and $\begin{array}{l} | \int_{Ω} (f^{'} (u^{(1)}) \nabla u^{(1)} - f^{'} (u^{(2)}) \nabla u^{(2)}) \nabla \frac{\partial ω}{\partial t} d x | \\ ⩽ \int_{Ω} | f^{'} (u^{(1)}) - f^{'} (u^{(2)}) | | \nabla u^{(1)} | | \nabla \frac{\partial ω}{\partial t} | d x + \int_{Ω} | f^{'} (u^{(2)}) | | \nabla u | | \nabla \frac{\partial ω}{\partial t} | d x . \end{array}$ The assumption (2.10) and Hölder’s inequality imply $\begin{array}{l} \int_{Ω} | f^{'} (u^{(1)}) - f^{'} (u^{(2)}) | | \nabla u^{(1)} | | \nabla \frac{\partial ω}{\partial t} | d x \\ ⩽ c \int_{Ω} (| u^{(1)} | + | u^{(2)} | + 1) | u | | \nabla u^{(1)} | | \nabla \frac{\partial ω}{\partial t} | d x \\ ⩽ c ({‖ u^{(1)} ‖}_{L^{\infty} (Ω)} + {‖ u^{(2)} ‖}_{L^{\infty} (Ω)} + 1) ‖ u ‖_{H^{k} (Ω)} {‖ u^{(1)} ‖}_{H^{2 k} (Ω)} ‖ \nabla \frac{\partial ω}{\partial t} ‖ \\ ⩽ c ({‖ u^{(1)} ‖}_{H^{2 k} (Ω)} + {‖ u^{(2)} ‖}_{H^{2 k} (Ω)} + 1) ‖ u ‖_{H^{k} (Ω)} {‖ u^{(1)} ‖}_{H^{2 k} (Ω)} ‖ \nabla \frac{\partial ω}{\partial t} ‖ . \end{array}$ Furthermore, (2.10) yields $\begin{array}{l} \int_{Ω} | f^{'} (u^{(2)}) | | \nabla u | | \nabla \frac{\partial ω}{\partial t} | d x & ⩽ c \int_{Ω} ({| u^{(2)} |}^{2} + 1) | \nabla u | | \nabla \frac{\partial ω}{\partial t} | \\ ⩽ c ({‖ u^{(2)} ‖}_{H^{2 k} (Ω)}^{2} + 1) ‖ u ‖_{H^{k} (Ω)} ‖ \nabla \frac{\partial ω}{\partial t} ‖ \end{array}$ One arrives at the following estimate $\begin{array}{l} | \int_{Ω} (f^{'} (u^{(1)}) \nabla u^{(1)} - f^{'} (u^{(2)}) \nabla u^{(2)}) \nabla \frac{\partial ω}{\partial t} d x | \\ (6.25) & ⩽ c ({‖ u^{(1)} ‖}_{H^{2 k} (Ω)}^{r_{1}} + {‖ u^{(2)} ‖}_{H^{2 k} (Ω)}^{r_{1}} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, r_{1} ⩾ 0 . \end{array}$

From (6.24) and (6.25), One has $\begin{array}{l} \frac{d}{d t} [‖ \nabla ω ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ω ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [u] + ‖ Δ ξ ‖^{2} + {‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2} + δ_{10} ‖ \nabla ξ ‖^{2} + 2 δ_{10} ((\nabla \frac{\partial ξ}{\partial t}, \nabla ξ))] \\ + c (‖ ω ‖_{H^{k + 1} (Ω)}^{2} + ‖ ξ ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ω}{\partial t} ‖}_{H^{1} (Ω)}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{1} (Ω)}^{2}) \\ (6.26) & ⩽ c^{'} ({‖ u^{(1)} ‖}_{H^{2 k} (Ω)}^{r} + {‖ u^{(2)} ‖}_{H^{2 k} (Ω)}^{r} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, r ⩾ 0, c > 0 . \end{array}$

Finally, summing (6.20) and (6.26), one obtains a differential inequality of the form $\begin{array}{l} \frac{d E_{11}}{d t} + c (E_{11} + {‖ \frac{\partial ω}{\partial t} ‖}_{H^{1} (Ω)}^{2}) \\ (6.27) & ⩽ c^{'} ({‖ u^{(1)} ‖}_{H^{2 k} (Ω)}^{r} + {‖ u^{(2)} ‖}_{H^{2 k} (Ω)}^{r} + 1) ‖ u ‖_{H^{k} (Ω)}^{2}, r ⩾ 0, c > 0, \end{array}$ where $\begin{array}{rcl} E_{11} & = & E_{10} + ‖ \nabla ω ‖^{2} + {‖ {\overline{A}}_{k}^{\frac{1}{2}} ω ‖}^{2} + {\overline{B}}_{k}^{\frac{1}{2}} [u] + ‖ Δ ξ ‖^{2} + {‖ \nabla \frac{\partial ξ}{\partial t} ‖}^{2} \\ + δ_{11} ‖ \nabla ξ ‖^{2} + 2 δ_{10} ((\nabla \frac{\partial ξ}{\partial t}, \nabla ξ)) \end{array}$ satisfies $\begin{matrix} (6.28) & E_{11} ⩾ c (‖ ω ‖_{H^{k + 1} (Ω)}^{2} + ‖ ξ ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{1} (Ω)}^{2}), c > 0 . \end{matrix}$ From previous estimates, especially (3.10)–(3.11), one can see that $\begin{matrix} ‖ u ‖_{H^{k} (Ω)}^{2} ⩽ Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖), \end{matrix}$ where Q is a monotonic function that depends only on time. Therefore, thanks to (6.27), (6.28) and the Gronwall’s lemma, one has the following inequality: $\begin{matrix} ‖ ω ‖_{H^{k + 1} (Ω)}^{2} + ‖ ξ ‖_{H^{2} (Ω)}^{2} + {‖ \frac{\partial ξ}{\partial t} ‖}_{H^{1} (Ω)}^{2} ⩽ h (t) Q (‖ u_{0} ‖_{H^{k} (Ω)}, ‖ α_{0} ‖_{H^{1} (Ω)}, ‖ α_{1} ‖), \end{matrix}$ where $\begin{matrix} V = H^{k + 1} (Ω) \times H^{2} (Ω) \times H^{1} (Ω), H = H^{k} (Ω) \times H^{1} (Ω) \times L^{2} (Ω) \end{matrix}$ and $\begin{matrix} h (t) = c^{'} \int_{0}^{t} e^{- c (t - s)} ({‖ u^{(1)} (s) ‖}_{H^{2 k} (Ω)}^{r} + {‖ u^{(2)} (s) ‖}_{H^{2 k} (Ω)}^{r} + 1) d s . \end{matrix}$ The function h is well continuous, which completes the proof of the proposition. □

Thus, the first condition of Theorem 6.1 is proved and it is sufficient to prove the second condition in order to obtain the existence of an exponential attractor.

Proposition 6.2.

Let $B_{R} \subset Φ$ bounded and $T > 0$ . The semigroup $S (t)$ , $t ⩾ 0$ generated by the problem ( 2.1 )–( 2.4 ) is Hölder continuous on $[0, T] \times B_{R}$ .

Proof.

The lipschitz continuity in space being a consequence of (4.9), it just remains to prove the continuity in time (actually, a Hölder condition in time for the semigroup $S (t)$ , $t ⩾ 0$ ). One assumes that the initial data belong to $B_{R}$ . For every $t_{1} ⩾ 0$ and $t_{2} ⩾ 0$ , owing to the above estimates, one gets $\begin{array}{l} {‖ S (t_{1}) (u_{0}, α_{0}, α_{1}) - S (t_{2}) (u_{0}, α_{0}, α_{1}) ‖}_{Φ} \\ = {‖ (u (t_{1}) - u (t_{2}), α (t_{1}) - α (t_{2}), \frac{\partial α}{\partial t} (t_{1}) - \frac{\partial α}{\partial t} (t_{2})) ‖}_{Φ} \\ ⩽ ‖ (u (t_{1}) - u (t_{2}) ‖_{H^{k} (Ω)} + ‖ α (t_{1}) - α (t_{2}) ‖_{H^{1} (Ω)} \\ + ‖ \frac{\partial α}{\partial t} (t_{1}) - \frac{\partial α}{\partial t} (t_{2}) ‖ \\ = {‖ \int_{t_{1}}^{t_{2}} \frac{\partial u}{\partial t} (τ) d τ ‖}_{H^{k} (Ω)} + {‖ \int_{t_{1}}^{t_{2}} \frac{\partial α}{\partial t} (τ) d τ ‖}_{H^{1} (Ω)} \\ + ‖ \int_{t_{1}}^{t_{2}} \frac{\partial^{2} α}{\partial t^{2}} (τ) d τ ‖ \\ ⩽ \int_{t_{1}}^{t_{2}} {‖ \frac{\partial u}{\partial t} (τ) ‖}_{H^{k} (Ω)} d τ + \int_{t_{1}}^{t_{2}} {‖ \frac{\partial α}{\partial t} (τ) ‖}_{H^{1} (Ω)} d τ \\ + \int_{t_{1}}^{t_{2}} ‖ \frac{\partial^{2} α}{\partial t^{2}} (τ) ‖ d τ \\ ⩽ c | t_{1} - t_{2} |^{\frac{1}{2}} + {(\int_{t_{1}}^{t_{2}} {‖ \frac{\partial^{2} α}{\partial t^{2}} (τ) ‖}^{2} d τ)}^{\frac{1}{2}} | t_{1} - t_{2} |^{\frac{1}{2}}, \end{array}$ where c depends on T. One multiplies (2.2) by $\frac{\partial^{2} α}{\partial t^{2}}$ to obtain $\begin{array}{l} \frac{d}{d t} [{‖ \frac{\partial α}{\partial t} ‖}^{2} + 2 ((\nabla α, \nabla \frac{\partial α}{\partial t}))] + c {‖ \frac{\partial^{2} α}{\partial t^{2}} ‖}^{2} \\ (6.29) & ⩽ c^{'} (‖ u ‖^{2} + {‖ \frac{\partial u}{\partial t} ‖}^{2} + {‖ \nabla \frac{\partial α}{\partial t} ‖}^{2}), c > 0 . \end{array}$ Integrating (6.29) between $t_{1}$ and $t_{2}$ , one deduces from the above estimates that $\begin{matrix} (6.30) & \int_{t_{1}}^{t_{2}} ‖ \frac{\partial^{2} α}{\partial t^{2}} (τ) ‖ d τ ⩽ c, \end{matrix}$ where c depends on T and $B_{R}$ . Finally, one has $\begin{matrix} (6.31) & {‖ S (t_{1}) (u_{0}, α_{0}, α_{1}) - S (t_{2}) (u_{0}, α_{0}, α_{1}) ‖}_{Φ} ⩽ c | t_{1} - t_{2} |^{\frac{1}{2}}, \end{matrix}$ where c depends on T and $B_{R}$ , which finishes the proof of the Hölder continuity with respect to t. □

One finally deduces from Proposition 6.1 and Proposition 6.2 the following result [13,14].

Theorem 6.2.

The semigroup $S (t)$ possesses an exponential attractor $M \subset B_{0}$ , i.e.,

$M$ has a finite fractal dimension, $\dim_{F} (M) < + \infty$ ;

$M$ is positively invariant, $S (t) M \subset M$ , for every $t ⩾ 0$ ;

$M$ attracts exponentially fast the bounded subsets of Φ, $\begin{matrix} (6.32) & \forall B \subset Φ bounded, {dist}_{Φ} (S (t) B, M) ⩽ c_{1} e^{- c_{2} t}, c_{1}, c_{2} > 0, t ⩾ 0, \end{matrix}$ where ${dist}_{X}$ denotes the Hausdorff semidistance (for the norm of X) defined as $\begin{matrix} {dist}_{X} (A, B) = sup_{a \in A} inf_{b \in B} ‖ a - b ‖_{X} . \end{matrix}$

Remark 6.1.

The exponential attractor $M$ being finite-dimensional and $A \subset M$ , this ensures the boundedness of the fractal dimension of the global attractor.

Remark 6.2.

Compared to the global attractor, an exponential attractor is expected to be more robust under perturbations. Indeed, the rate of attraction of trajectories to the global attractor may be slow and it is very difficult, if not impossible, to estimate this rate of attraction with respect to the physical parameters of the problem in general. As a consequence, global attractors may change drastically under small perturbations. One refers the reader to [21,24] for discussions on this subject.

Footnotes

Acknowledgements

The authors wish to thank Alain Miranville for useful discussions. They also wish to thank an anonymous referees for their careful reading of the paper and useful comments.

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Long-time behavior of the higher-order anisotropic Caginalp phase-field systems based on the Cattaneo law

Abstract

Keywords

1. Introduction

2. Setting of the problem

3. A priori estimates

3.1. First case: k = 1

3.2. Second case: k ⩾ 2

Remark 3.1. One can note that one has estimate which is similar to (3.49) when k = 1 . 4. The dissipative semigroup

Footnotes

Acknowledgements

References

3.1. First case: $k = 1$

3.2. Second case: $k ⩾ 2$

Remark 3.1.
One can note that one has estimate which is similar to (3.49) when $k = 1$ .

4. The dissipative semigroup