Convergence estimates for some abstract linear second order differential equations with two small parameters

Abstract

In a real Hilbert space H we consider the following singularly perturbed Cauchy problem $\begin{array}{l} (P_{ε δ}) & \{\begin{matrix} ε u_{ε δ}^{″} (t) + δ u_{ε δ}^{'} (t) + A u_{ε δ} (t) = f (t), & t \in (0, T), \\ u_{ε δ} (0) = u_{0}, u_{ε δ}^{'} (0) = u_{1}, \end{matrix} \end{array}$ where $u_{0}, u_{1} \in H$ , $f : [0, T] \mapsto H$ and ε, δ are two small parameters.

We study the behavior of the solutions $u_{ε δ}$ to the problem ( P ε δ ) in two different cases:

when $ε \to 0$ and $δ ⩾ δ_{0} > 0$ ;

when $ε \to 0$ and $δ \to 0$ .

We obtain a priori estimates of the solutions to the perturbed problem, which are uniform with respect to the parameters, and a relationship between the solutions to both problems. We establish that the solution to the unperturbed problem has a singular behavior with respect to the parameters in the neighborhood of

t = 0

. We describe the boundary layer and the boundary layer function in both cases.

Keywords

singular perturbation abstract second order Cauchy problem boundary layer function a priori estimate

Let H and V be two real Hilbert spaces endowed with norms $| \cdot |$ and $‖ \cdot ‖$ , respectively. Denote by $(\cdot, \cdot)$ the scalar product in H. The framework of our study is determined by the following assumptions:

$V \subset H$ densely and continuously, i.e. $\begin{matrix} (1) & ‖ u ‖ ⩾ ω_{0} | u |, \forall u \in V, ω_{0} > 0 . \end{matrix}$

$A : D (A) = V \mapsto H$ is a linear, self-adjoint and positive definite operator, i.e. $\begin{matrix} (2) & (A u, u) ⩾ ω | u |^{2}, \forall u \in V, ω > 0 . \end{matrix}$

Consider the following singularly perturbed Cauchy problem $\begin{array}{l} (P_{ε δ}) & \{\begin{matrix} ε u_{ε δ}^{″} (t) + δ u_{ε δ}^{'} (t) + A u_{ε δ} (t) = f (t), & t \in (0, T), \\ u_{ε δ} (0) = u_{0}, u_{ε δ}^{'} (0) = u_{1}, \end{matrix} \end{array}$ where $u_{0}, u_{1} \in H$ , $f : [0, T] \mapsto H$ and ε, δ are two small parameters.

We study the behavior of the solutions $u_{ε δ}$ to the problem ( P ε δ ) in two different cases:

$ε \to 0$ and $δ ⩾ δ_{0} > 0$ , relative to the following unperturbed system: $\begin{array}{l} (P_{δ}) & \{\begin{matrix} δ l_{δ}^{'} (t) + A l_{δ} (t) = f (t), & t \in (0, T), \\ l_{δ} (0) = u_{0}, \end{matrix} \end{array}$

$ε \to 0$ and $δ \to 0$ , relative to the following unperturbed system: $\begin{array}{l} (P_{0}) & \{\begin{matrix} A v (t) = f (t), t \in (0, T), \\ v (0) = A^{- 1} f (0) . \end{matrix} \end{array}$

The problem ( P ε δ ) is the abstract model of singularly perturbed problems of hyperbolic–parabolic type.

Many physical processes are described by systems of type ( P ε δ ). For example in [3], is considered the equation $\begin{matrix} ρ v_{t t} + γ v_{t} = σ Δ v, \end{matrix}$ where ρ, γ, σ are the mass density per unit area of the membrane, the coefficient of viscosity of the medium, and the tension of the membrane, respectively. This equation characterizes the vibration of a membrane in a viscous medium and can be rewritten as $\begin{matrix} ε^{2} u_{t t} + u_{t} = Δ u, \end{matrix}$ with $ε = {(ρ σ)}^{1 / 2} / γ$ .

In the case when the medium is highly viscous ( $γ ≫ 1$ ), or the density ρ is very small, we have $ε \to 0$ and the formal “limit” of this equation will be the following first order equation $\begin{matrix} u_{t} = Δ u . \end{matrix}$

Without pretending to do a complete analysis, let us mention some works dedicated to the study of singularly perturbed Cauchy problems for linear or nonlinear differential equations of second order of type ( P ε δ ). The case when $δ = 1$ was widely studied, see, e.g. [4,5,7,12] and the bibliography therein. In [8] the asymptotic behavior of solutions to singular perturbation problems for second order equations, as $ε \to 0$ and $δ \to 0$ , is studied. In [2,9,14], numerical results concerning the singular behavior of solutions to the problem ( P ε δ ) for some ordinary differential equations and their applicability in modeling different physical and engineering processes, are presented.

Let X be a Banach space. For $k \in N^{⋆}$ , $1 ⩽ p ⩽ + \infty$ , $(a, b) \subset (- \infty, + \infty)$ , we denote by $W^{k, p} (a, b; X)$ the Banach space of all vectorial distributions $u \in D^{'} (a, b; X)$ , $u^{(j)} \in L^{p} (a, b; X)$ , $j = 0, 1, \dots, k$ , endowed with the norm $\begin{matrix} (3) & ‖ u ‖_{W^{k, p} (a, b; X)} = \{\begin{matrix} {(\sum_{j = 0}^{k} ‖ u^{(j)} ‖_{L^{p} (a, b; X)}^{p})}^{1 / p}, & p \in [1, \infty), \\ {max}_{0 ⩽ j ⩽ k} ‖ u^{(j)} ‖_{L^{\infty} (a, b; X)}, & p = \infty . \end{matrix} \end{matrix}$ If $p = 2$ , and X is a Hilbert space, then $W^{k, 2} (a, b; X)$ is also a Hilbert space with the inner product $\begin{matrix} {(u, v)}_{H^{k} (a, b; X)} = \sum_{j = 0}^{k} \int_{a}^{b} {(u^{(j)} (t), v^{(j)} (t))}_{X} d t . \end{matrix}$ For $s \in R$ , $k \in N$ and $p \in [1, \infty]$ , we define the Banach space $\begin{matrix} W_{s}^{k, p} (a, b; X) = {f : (a, b) \to H; f^{(l)} (\cdot) e^{- s t} \in L^{p} (a, b; X), l = 0, \dots, k}, \end{matrix}$ with the norm $\begin{matrix} ‖ f ‖_{W_{s}^{k, p} (a, b; X)} = {‖ f e^{- s t} ‖}_{W^{k, p} (a, b; X)} . \end{matrix}$

Definition 1.
Let $T > 0$ and $f \in L^{2} (0, T; H)$ , $A : D (A) \subseteq H \to H$ . The function $u \in L^{2} (0, T; D (A))$ with $u^{'} \in L^{2} (0, T; H)$ and $u^{″} \in L^{2} (0, T; H)$ is called strong solution to the Cauchy problem $\begin{array}{rcl} (4) & u^{″} (t) + u^{'} (t) + A u (t) = f (t), \forall t \in (0, T), \\ (5) & u (0) = u_{0}, u^{'} (0) = u_{1}, \end{array}$ if u satisfies equation (4) in the sense of distributions a.e. $t \in (0, T)$ , and the initial conditions (5).
Definition 2.
Let $T > 0$ and $f \in L^{2} (0, T; H)$ , $A : D (A) \subseteq H \to H$ . The function $l \in L^{2} (0, T; D (A))$ with $l^{'} \in L^{2} (0, T; H)$ is called strong solution to the Cauchy problem $\begin{array}{rcl} (6) & l^{'} (t) + A l (t) = f (t), \forall t \in (0, T), \\ (7) & l (0) = u_{0} \end{array}$ if l verifies equation (6) in the sense of distributions a.e. $t \in (0, T)$ , and the initial condition (7).
Theorem 1 ([1]).

Let $T > 0$ . Let us assume that the operator $A : D (A) \subset H \to H$ is linear, self-adjoint and positive definite, i.e. condition ( 2 ) is fulfilled. If $u_{0} \in D (A)$ , $u_{1} \in H$ and $f \in W^{1, 1} (0, T; H)$ , then there exists a unique strong solution to problem ( 4 ) , ( 5 ), such that $u \in W^{2, \infty} (0, T; H)$ , $A^{1 / 2} u \in W^{1, \infty} (0, T; H)$ , $A u \in L^{\infty} (0, T; H)$ .

Theorem 2 ([1]).

Let $T > 0$ . Let us assume that the operator $A : D (A) \subset H \to H$ is linear, self-adjoint, positive definite. If $u_{0} \in D (A)$ and $f \in W^{1, 1} (0, T; H)$ , then there exists a unique strong solution to the problem ( 6 ) , ( 7 ), such that $l \in W^{1, \infty} (0, T; H)$ . The following estimates are valid: $\begin{array}{rcl} (8) & ‖ l ‖_{C ([0, t]; H)} + {‖ A^{1 / 2} l ‖}_{L^{2} (0, t; H)} ⩽ C M_{0} (t), \forall t \in [0, T], \\ (9) & M_{0} (t) = | u_{0} | + \int_{0}^{t} | f (s) | d s . \end{array}$

The problems ( P ε δ ) and ( P δ ) can be rewritten as follows: $\begin{array}{l} (P_{μ}) & \{\begin{matrix} μ U_{μ}^{″} (s) + U_{μ}^{'} (s) + A U_{μ} (s) = F (s), & s \in (0, T / δ), \\ U_{μ} (0) = u_{0}, U_{μ}^{'} (0) = δ u_{1}, \end{matrix} \end{array}$ and $\begin{array}{l} (P_{0}) & \{\begin{matrix} L^{'} (s) + A L (s) = F (s), s \in (0, T / δ), \\ L (0) = u_{0}, \end{matrix} \end{array}$ where $U_{μ} (s) = u_{ε δ} (δ s)$ , $L (s) = l_{δ} (s δ)$ , $F (s) = f (s δ)$ and $μ = ε / δ^{2}$ .

Using the results obtained in [13] for the solutions of problems ( P μ ) and ( P 0 ), we get the following two theorems:

Theorem 3.
Let $T > 0$ , $δ ⩾ δ_{0} > 0$ and $p \in (1, \infty)$ . Let us assume that the operator $A : D (A) = V \mapsto H$ verifies conditions (IH) and (IA). If $u_{0} \in D (A)$ , $u_{1} \in H$ , $f \in W^{1, p} (0, T; H)$ , then there exist constants $C = C (T, p, ω_{0}, ω, δ_{0}) > 0$ , $ε_{0} = ε_{0} (ω_{0}, ω)$ , $ε_{0} \in (0, 1)$ , such that $\begin{matrix} (10) & ‖ u_{ε δ} - l_{δ} ‖_{C ([0, T]; H)} ⩽ C ε^{β} (| A^{1 / 2} u_{0} | + | u_{1} | + ‖ f ‖_{W^{1, p} (0, T; H)}), \forall ε \in (0, ε_{0}], \end{matrix}$ where $u_{ε δ}$ and $l_{δ}$ are strong solutions to the problems ( P ε δ ) and ( P δ ) respectively, and $β = min {1 / 4, (p - 1) / 2 p}$ .
Theorem 4.
Let $T > 0$ , $δ ⩾ δ_{0} > 0$ and $p \in (1, \infty)$ . Let us assume that the operator $A : D (A) = V \mapsto H$ satisfies conditions (IH) and (IA). If $u_{0}, A u_{0}, u_{1}, f (0) \in D (A)$ , $f \in W^{2, p} (0, T; H)$ , then there exist constants $C = C (T, p, ω_{0}, ω, δ_{0}) > 0$ , $ε_{0} = ε_{0} (ω_{0}, ω)$ , $ε_{0} \in (0, 1)$ , such that $\begin{matrix} (11) & \begin{matrix} {‖ u_{ε δ}^{'} - l_{δ}^{'} + H_{ε δ} exp {- \frac{δ^{2} t}{ε}} ‖}_{C ([0, T]; H)} \\ ⩽ C ε^{(p - 1) / 2 p} (| A^{1 / 2} u_{0} | + | A u_{1} | + | A H_{ε δ} | + ‖ f ‖_{W^{2, p} (0, T; H)}), \forall ε \in (0, ε_{0}], \end{matrix} \end{matrix}$ where $u_{ε δ}$ and $l_{δ}$ are strong solutions to the problems ( P ε δ ) and ( P δ ) respectively, and $\begin{matrix} H_{ε δ} = δ^{- 1} f (0) - u_{1} - δ^{- 1} A u_{0} . \end{matrix}$

The main result of this paper is the following theorem:
Theorem 5.
Let $T > 0$ and $p \in (1, \infty)$ . Let us assume that the operator $A : D (A) = V \mapsto H$ satisfies conditions (IH) and (IA). If $u_{0} \in V$ , $u_{1} \in H$ , $f \in W^{1, p} (0, T; H)$ , then there exist constants $C = C (p, T, ω_{0}, ω) > 0$ and $ε_{0} = ε_{0} (ω_{0}, ω)$ , $ε_{0} \in (0, 1)$ , such that $\begin{matrix} (12) & \begin{matrix} ‖ u_{ε δ} - v - h_{δ} ‖_{C ([0, T]; H)} \\ ⩽ C (| A^{1 / 2} u_{0} | + | u_{1} | + ‖ f ‖_{W^{1, p} (0, T; H)}) Θ (ε, δ), \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \end{matrix} \end{matrix}$ where $u_{ε δ}$ and v are strong solutions to the problems ( P ε δ ) and ( P 0 ) respectively, and $\begin{array}{rcl} Θ (ε, δ) = \frac{ε^{β}}{δ^{1 + 1 / p}} + \sqrt{δ}, \\ β = min {\frac{1}{4}, \frac{p - 1}{2 p}} . \end{array}$ The function $h_{δ}$ is the solution to the problem $\begin{matrix} (13) & \{\begin{matrix} δ h_{δ}^{'} (t) + A h_{δ} (t) = 0, & t \in (0, T), \\ h_{δ} (0) = u_{0} - A^{- 1} f (0), \end{matrix} \end{matrix}$ and $\begin{matrix} | h_{δ} (t) | ⩽ | u_{0} - A^{- 1} f (0) | e^{- δ t / ω}, t \in [0, T] . \end{matrix}$ If in addition, $u_{1} \in D (A^{1 / 2})$ , then $\begin{matrix} (14) & \begin{matrix} ‖ u_{ε δ} - v - h_{δ} ‖_{C ([0, T]; H)} \\ ⩽ C (| A^{1 / 2} u_{0} | + | A^{1 / 2} u_{1} | + ‖ f ‖_{W^{1, p} (0, T; H)}) Θ_{1} (ε, δ), \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \end{matrix} \end{matrix}$ where $\begin{matrix} Θ_{1} (ε, δ) = \frac{ε^{\frac{p - 1}{2 p}}}{δ^{2 + \frac{1}{p}}} + \sqrt{δ} . \end{matrix}$

The proof of this theorem is based on the following two key points:

A priori estimates of solutions to the perturbed problem ( P μ ), which are uniform with respect to the small parameter μ.

The relationship between the solutions to problems ( P μ ) and ( P 0 ).

In what follows we will present some results obtained in our previous works, they will be used to prove the last theorem.
Lemma 1 ([11]).

Let $S > 0$ . Let us assume that the operator $A : D (A) = V \mapsto H$ satisfies conditions (IH) and (IA). If $u_{0} \in D (A)$ , $u_{1} \in H$ and $F \in W^{1, 1} (0, S; H)$ , then exists a constant $C (ω_{0}, ω) > 0$ such that for every strong solution $U_{μ}$ to the problem ( P μ ) the following estimate holds: $\begin{matrix} (15) & {‖ A^{1 / 2} U_{μ} ‖}_{C ([0, s]; H)} + {‖ U_{μ}^{'} ‖}_{L^{2} (0, s; H)} ⩽ C M_{0} (s), \forall μ \in (0, 1 / 2], \forall s \in [0, S] . \end{matrix}$ If in addition, $u_{1} \in D (A^{1 / 2})$ then $\begin{matrix} (16) & \begin{matrix} μ {‖ U_{μ}^{″} ‖}_{C ([0, s]; H)} + {‖ U_{μ}^{'} ‖}_{C ([0, s]; H)} + {‖ A^{1 / 2} U_{μ}^{'} ‖}_{L^{2} (0, s; H)} + {‖ A U_{μ} (t) ‖}_{C ([0, s]; H)} \\ ⩽ C M_{1} (s), \forall μ \in (0, 1 / 2], \forall s \in [0, S], \end{matrix} \end{matrix}$ where $\begin{array}{rcl} M_{0} (s) = | A^{1 / 2} u_{0} | + | u_{1} | + | F (0) | + ‖ F ‖_{W^{1, 1} (0, s; H)}, \\ M_{1} (s) = | A^{1 / 2} u_{0} | + | A^{1 / 2} u_{1} | + | F (0) | + ‖ F ‖_{W^{1, 1} (0, s; H)} . \end{array}$

In what follows we introduce the notation $\begin{matrix} (17) & K (s, τ, μ) = \frac{1}{2 \sqrt{π} μ} (K_{1} (s, τ, μ) + 3 K_{2} (s, τ, μ) - 2 K_{3} (s, τ, μ)), \forall μ > 0, \end{matrix}$ where $\begin{array}{rcl} (18) & K_{1} (s, τ, μ) = exp {\frac{3 s - 2 τ}{4 μ}} λ (\frac{2 s - τ}{2 \sqrt{μ s}}), \\ (19) & K_{2} (s, τ, μ) = exp {\frac{3 s + 6 τ}{4 μ}} λ (\frac{2 s + τ}{2 \sqrt{μ s}}), \\ (20) & K_{3} (s, τ, μ) = exp {\frac{τ}{μ}} λ (\frac{s + τ}{2 \sqrt{μ s}}), \end{array}$ where $\begin{matrix} λ (s) = \int_{s}^{\infty} e^{- η^{2}} d η . \end{matrix}$ The properties of the function $K (t, τ, ε)$ are collected in the next lemma.

Lemma 2 ([10]).

The function $K (t, τ, ε)$ possesses the following properties:

$K \in C ([0, \infty) \times [0, \infty)) \cap C^{2} ((0, \infty) \times (0, \infty))$ .

$K_{t} (t, τ, ε) = ε K_{τ τ} (t, τ, ε) - K_{τ} (t, τ, ε)$ , $\forall t > 0$ , $\forall τ > 0$ .

$ε K_{τ} (t, 0, ε) - K (t, 0, ε) = 0$ , $\forall t ⩾ 0$ .

$K (0, τ, ε) = \frac{1}{2 ε} exp {- \frac{τ}{2 ε}}$ , $\forall τ ⩾ 0$ .

For every $t > 0$ fixed and every $q, s \in N$ there exist constants $C_{1} (q, s, t, ε) > 0$ and $C_{2} (q, s, t) > 0$ such that $\begin{matrix} | \partial_{t}^{s} \partial_{τ}^{q} K (t, τ, ε) | ⩽ C_{1} (q, s, t, ε) exp {- C_{2} (q, s, t) τ / ε}, \forall τ > 0; \end{matrix}$

Moreover, for $γ \in R$ there exist $C_{1}$ , $C_{2}$ and $ε_{0}$ , all of them positive and depending on γ, such that the following estimates are fulfilled: $\begin{array}{rcl} \int_{0}^{\infty} e^{γ τ} | K_{t} (t, τ, ε) | d τ ⩽ C_{1} ε^{- 1} e^{C_{2} t}, \forall ε \in (0, ε_{0}], \forall t ⩾ 0, \\ \int_{0}^{\infty} e^{γ τ} | K_{τ} (t, τ, ε) | d τ ⩽ C_{1} ε^{- 1} e^{C_{2} t}, \forall ε \in (0, ε_{0}], \forall t ⩾ 0, \\ \int_{0}^{\infty} e^{γ τ} | K_{τ τ} (t, τ, ε) | d τ ⩽ C_{1} ε^{- 2} e^{C_{2} t}, \forall ε \in (0, ε_{0}], \forall t ⩾ 0 . \end{array}$

$K (t, τ, ε) > 0$ , $\forall t ⩾ 0$ , $\forall τ ⩾ 0$ .

For every continuous function $φ : [0, \infty) \to H$ with $| φ (t) | ⩽ M exp {γ t}$ the following equality holds true: $\begin{matrix} lim_{t \to 0} | \int_{0}^{\infty} K (t, τ, ε) φ (τ) d τ - \int_{0}^{\infty} e^{- τ} φ (2 ε τ) d τ | = 0, for every ε \in (0, {(2 γ)}^{- 1}) . \end{matrix}$

$\begin{matrix} \int_{0}^{\infty} K (t, τ, ε) d τ = 1, \forall t ⩾ 0 . \end{matrix}$

Let $γ > 0$ and $q \in [0, 1]$ . There exist $C_{1}$ , $C_{2}$ and $ε_{0}$ all of them positive and depending on γ and q, such that the following estimates are fulfilled: $\begin{matrix} \int_{0}^{\infty} K (t, τ, ε) e^{γ τ} | t - τ |^{q} d τ ⩽ C_{1} e^{C_{2} t} ε^{q / 2}, \forall ε \in (0, ε_{0}], \forall t > 0 . \end{matrix}$ If $γ ⩽ 0$ and $q \in [0, 1]$ , then $\begin{matrix} \int_{0}^{\infty} K (t, τ, ε) e^{γ τ} | t - τ |^{q} d τ ⩽ C ε^{q / 2} {(1 + \sqrt{t})}^{q}, \forall ε \in (0, 1], \forall t ⩾ 0 . \end{matrix}$

Let $p \in (1, \infty]$ and $f : [0, \infty) \to H$ , $f (t) \in W_{γ}^{1, p} (0, \infty; H)$ . If $γ > 0$ , then there exist $C_{1}$ , $C_{2}$ and $ε_{0}$ all of them positive and depending on γ and p, such that $\begin{matrix} | f (t) - \int_{0}^{\infty} K (t, τ, ε) f (τ) d τ | ⩽ C_{1} e^{C_{2} t} {‖ f^{'} ‖}_{L_{γ}^{p} (0, \infty; H)} ε^{(p - 1) / 2 p}, \forall ε \in (0, ε_{0}], \forall t ⩾ 0 . \end{matrix}$

If $γ ⩽ 0$ , then $\begin{array}{rcl} | f (t) - \int_{0}^{\infty} K (t, τ, ε) f (τ) d τ | \\ ⩽ C (γ, p) {‖ f^{'} ‖}_{L_{γ}^{p} (0, \infty; H)} {(1 + \sqrt{t})}^{\frac{p - 1}{p}} ε^{(p - 1) / 2 p}, \forall ε \in (0, 1], \forall t ⩾ 0 . \end{array}$

For every $q > 0$ and $α ⩾ 0$ there exists a constant $C (q, α) > 0$ , such that $\begin{matrix} \int_{0}^{t} \int_{0}^{\infty} K (τ, θ, ε) e^{- q θ / ε} | τ - θ |^{α} d θ d τ ⩽ C (q, α) ε^{1 + α}, \forall ε > 0, \forall t ⩾ 0 . \end{matrix}$

Let $f \in W_{γ}^{1, \infty} (0, \infty; H)$ with $γ ⩾ 0$ . There exist $C_{1}$ , $C_{2}$ and $ε_{0}$ all of them positive and depending on γ, such that $\begin{matrix} | \int_{0}^{\infty} K_{t} (t, τ, ε) f (τ) d τ | ⩽ C_{1} e^{C_{2} t} {‖ f^{'} ‖}_{L_{γ}^{\infty} (0, \infty; H)}, \forall ε \in (0, ε_{0}], \forall t ⩾ 0 . \end{matrix}$

Lemma 3 ([10]).

Assume that $A : D (A) \subset H \to H$ is a linear, self-adjoint, positive definite operator and $F \in L_{γ}^{\infty} (0, \infty; H)$ for some $γ ⩾ 0$ . If $U_{μ}$ is the strong solution to the problem ( P μ ) with $U_{μ} \in W_{γ}^{2, \infty} (0, \infty; H) \cap L_{γ}^{\infty} (0, \infty; H)$ , $A U_{μ} \in L_{γ}^{\infty} (0, \infty; H)$ , then for every $0 < μ < {(4 γ)}^{- 1}$ the function $W_{μ}$ , defined by $\begin{matrix} (21) & W_{μ} (s) = \int_{0}^{\infty} K (s, τ, μ) U_{μ} (τ) d τ, \end{matrix}$ is the strong solution in H to the problem $\begin{matrix} (22) & \{\begin{matrix} W_{μ}^{'} (s) + A W_{μ} (s) = F_{0} (s, μ), & a . e . s > 0 in H, \\ W_{μ} (0) = φ_{μ}, \end{matrix} \end{matrix}$ where $\begin{matrix} (23) & \begin{matrix} F_{0} (s, μ) = \frac{1}{\sqrt{π}} [2 exp {\frac{3 s}{4 μ}} λ (\sqrt{\frac{s}{μ}}) - λ (\frac{1}{2} \sqrt{\frac{s}{μ}})] δ u_{1} + \int_{0}^{\infty} K (s, τ, μ) F (τ) d τ, \\ φ_{μ} = \int_{0}^{\infty} e^{- τ} U_{μ} (2 μ τ) d τ . \end{matrix} \end{matrix}$

Proof of Theorem 5.

In all the proof, C denotes various constants $C (T, p, ω_{0}, ω)$ . Introduce the notation, $\begin{array}{rcl} M_{0} = | A^{1 / 2} u_{0} | + | u_{1} | + ‖ f ‖_{W^{1, p} (0, T; H)}, \\ M_{1} = | A^{1 / 2} u_{0} | + | A^{1 / 2} u_{1} | + ‖ f ‖_{W^{1, p} (0, T; H)} . \end{array}$ Consider the function $f \in W^{1, p} (0, T; H)$ . Define on $[0, \infty)$ the function $\tilde{f}$ as follows: $\begin{matrix} \tilde{f} (t) = \{\begin{matrix} f (t), & 0 ⩽ t ⩽ T, \\ \frac{2 T - t}{T} f (T), & T ⩽ t ⩽ 2 T, \\ 0, & t ⩾ 2 T . \end{matrix} \end{matrix}$ It is easily seen that $\begin{matrix} (24) & ‖ \tilde{f} ‖_{W^{1, p} (0, \infty; H)} ⩽ C (p, T) ‖ f ‖_{W^{1, p} (0, T; H)} . \end{matrix}$ If we denote by ${\tilde{U}}_{μ}$ the unique strong solution to the problem ( P μ ), defined on $(0, \infty)$ instead of $(0, S)$ with $S = T / δ$ and $\tilde{f}$ instead of f, then from Lemma 1, it follows that ${\tilde{U}}_{μ} \in W_{γ}^{2, \infty} (0, \infty; H) \cap W_{γ}^{1, 2} (0, \infty; V)$ , $A^{1 / 2} {\tilde{U}}_{μ} \in L_{γ}^{\infty} (0, \infty; H)$ , $A {\tilde{U}}_{μ} \in L_{γ}^{\infty} (0, \infty; H)$ with $γ = γ (ω_{0}, ω)$ .

Moreover, estimate (24) implies $\begin{matrix} (25) & \begin{matrix} ‖ \tilde{F} ‖_{L^{p} (0, \infty; H)} ⩽ C (p, T) δ^{- 1 / p} ‖ f ‖_{L^{p} (0, T; H)}, p \in (1, \infty), \forall δ \in (0, 1], \\ {‖ {\tilde{F}}^{'} ‖}_{L^{p} (0, \infty; H)} ⩽ C (p, T) δ^{1 - 1 / p} {‖ f^{'} ‖}_{L^{p} (0, T; H)}, p \in (1, \infty), \forall δ \in (0, 1], \\ ‖ \tilde{F} ‖_{W^{1, p} (0, \infty; H)} ⩽ C (p, T) δ^{- 1 / p} ‖ f ‖_{W^{1, p} (0, T; H)}, p \in (1, \infty), \forall δ \in (0, 1] . \end{matrix} \end{matrix}$ Due to these estimates and to Lemma 1, the following estimates hold true: $\begin{matrix} (26) & {‖ A^{1 / 2} {\tilde{U}}_{μ} ‖}_{C ([0, s]; H)} + {‖ {\tilde{U}}_{μ}^{'} ‖}_{L^{2} (0, s; H)} ⩽ C M_{0} δ^{- 1 / p}, \end{matrix}$ and $\begin{array}{rcl} μ {‖ {\tilde{U}}_{μ}^{″} ‖}_{L^{\infty} (0, s; H)} + {‖ {\tilde{U}}_{μ}^{'} ‖}_{C ([0, s]; H)} + {‖ A^{1 / 2} {\tilde{U}}_{μ}^{'} ‖}_{L^{2} (0, s; H)} + {‖ A {\tilde{U}}_{μ} (t) ‖}_{C ([0, s]; H)} \\ (27) & ⩽ C M_{1} δ^{- 1 / p}, \forall μ \in (0, 1 / 2], \forall s \in [0, S] . \end{array}$ By Lemma 3, the function $W_{μ}$ , defined by $\begin{matrix} (28) & W_{μ} (s) = \int_{0}^{\infty} K (s, τ, μ) {\tilde{U}}_{μ} (τ) d τ, \end{matrix}$ is the strong solution in H to the problem $\begin{matrix} (29) & \{\begin{matrix} W_{μ}^{'} (s) + A W_{μ} (s) = {\tilde{F}}_{0} (s, μ), & a.e. s > 0 in H, \\ W_{μ} (0) = φ_{μ}, \end{matrix} \end{matrix}$ for every $ε \in (0, ε_{0}]$ , where $\begin{matrix} (30) & \begin{matrix} {\tilde{F}}_{0} (s, μ) = δ f_{0} (s, μ) u_{1} + \int_{0}^{\infty} K (s, τ, μ) \tilde{F} (τ) d τ, \\ f_{0} (s, μ) = \frac{1}{\sqrt{π}} [2 exp {\frac{3 s}{4 μ}} λ (\sqrt{\frac{s}{μ}}) - λ (\frac{1}{2} \sqrt{\frac{s}{μ}})], \\ φ_{μ} = \int_{0}^{\infty} e^{- τ} {\tilde{U}}_{μ} (2 μ τ) d τ . \end{matrix} \end{matrix}$ Proof of estimate (12).

Using properties (vi), (viii), (x) from Lemma 2, as well as (26), we obtain $\begin{matrix} (31) & ‖ {\tilde{U}}_{μ} - W_{μ} ‖ |_{C ([0, s]; H)} ⩽ C M_{0} μ^{1 / 4} δ^{- 1 / p} \sqrt{1 + \sqrt{s}} ⩽ C M_{0} \frac{ε^{1 / 4}}{δ^{3 / 4 + 1 / p}} \end{matrix}$ for all $ε \in (0, ε_{0}]$ , $δ \in (0, 1]$ and $s \in [0, S]$ .

Denote by $R (s, μ) = \tilde{L} (s) - W_{μ} (s)$ , where $\tilde{L}$ is the strong solution to the problem ( P 0 ) with $\tilde{f}$ instead of f, $T = \infty$ and $W_{μ}$ is the strong solution of (29). Then, due to Theorem 2, $R (\cdot, μ) \in W_{γ}^{1, \infty} (0, \infty; H)$ and R is the strong solution in H to the problem $\begin{matrix} (32) & \{\begin{matrix} R^{'} (s, μ) + A R (s, μ) = F (s, μ), & a.e. t > 0, \\ R (0, μ) = R_{0}, \end{matrix} \end{matrix}$ where $R_{0} = u_{0} - W_{μ} (0)$ and $\begin{matrix} (33) & F (s, μ) = \tilde{F} (s) - \int_{0}^{\infty} K (s, τ, μ) {\tilde{F}}_{ε} (τ) d τ - δ f_{0} (s, μ) u_{1} . \end{matrix}$

Taking the inner product in H by R and integrating afterwards, we obtain $\begin{matrix} {| R (s, μ) |}^{2} + 2 \int_{0}^{s} {| A^{1 / 2} R (ξ, μ) |}^{2} d ξ ⩽ {| R (0, μ) |}^{2} + 2 \int_{0}^{s} | F (ξ, μ) | | R (ξ, μ) | d ξ, \forall s ⩾ 0 . \end{matrix}$

Applying lemma of Brezis (see, e.g., [6]), we get $\begin{matrix} (34) & | R (s, μ) | + {(\int_{0}^{s} {| A^{1 / 2} R (ξ, μ) |}^{2} d ξ)}^{1 / 2} ⩽ | R_{0} | + \int_{0}^{s} | F (ξ, μ) | d ξ, \forall s ⩾ 0 . \end{matrix}$

Using (26), we obtain $\begin{array}{rcl} | R_{0} | & ⩽ & \int_{0}^{\infty} e^{- τ} | {\tilde{U}}_{μ} (2 μ τ) - u_{0} | d τ \\ ⩽ & \int_{0}^{\infty} e^{- τ} \int_{0}^{2 μ τ} | {\tilde{U}}_{μ}^{'} (ξ) | d ξ d τ \\ (35) & ⩽ & C μ^{1 / 2} M_{0} δ^{- 1 / p} = C M_{0} \frac{ε^{1 / 2}}{δ^{1 / p + 1}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1] . \end{array}$

In what follows, we will estimate $| F (s, μ) |$ . Using the property (x) from Lemma 2 and (25), we have $\begin{array}{rcl} | \tilde{F} (s) - \int_{0}^{\infty} K (s, τ, μ) \tilde{F} (τ) d τ | & ⩽ & \int_{0}^{\infty} K (s, τ, μ) \int_{s}^{τ} | {\tilde{F}}^{'} (ξ) | d ξ d τ \\ (36) & ⩽ & C {‖ {\tilde{F}}^{'} ‖}_{L^{p} (0, \infty; H)} {(1 + \sqrt{s})}^{\frac{p - 1}{p}} μ^{\frac{p - 1}{2 p}} ⩽ C M_{0} \frac{ε^{\frac{p - 1}{2 p}}}{δ^{\frac{p - 1}{2 p}}}, \end{array}$ for all $ε \in (0, ε_{0}]$ , $δ \in (0, 1]$ , $s \in [0, S]$ .

Since $\begin{matrix} e^{ξ} λ (\sqrt{ξ}) ⩽ C, \forall ξ ⩾ 0, \end{matrix}$ the following estimates hold true: $\begin{matrix} \int_{0}^{s} exp {\frac{3 ξ}{4 μ}} λ (\sqrt{\frac{ξ}{μ}}) d ξ ⩽ C μ \int_{0}^{\infty} e^{- ξ / 4} d ξ ⩽ C μ, \forall s ⩾ 0, \end{matrix}$ and $\begin{matrix} \int_{0}^{s} λ (\frac{1}{2} \sqrt{\frac{ξ}{μ}}) d ξ ⩽ μ \int_{0}^{\infty} λ (\frac{1}{2} \sqrt{ξ}) d ξ ⩽ C μ, \forall s ⩾ 0 . \end{matrix}$

Then $\begin{matrix} (37) & | δ \int_{0}^{s} f_{0} (ξ, μ) u_{1} d ξ | ⩽ C δ μ | u_{1} | ⩽ C M_{0} \frac{ε}{δ}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s ⩾ 0 . \end{matrix}$

Using (36) and (37), from (33) we obtain $\begin{matrix} (38) & \int_{0}^{s} | F (ξ, μ) | d τ ⩽ C M_{0} \frac{ε^{\frac{p - 1}{2 p}}}{δ}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S] . \end{matrix}$ From (34), using (35) and (38), we get the estimate $\begin{matrix} (39) & ‖ R ‖_{C ([0, s]; H)} ⩽ C M_{0} \frac{ε^{\frac{p - 1}{2 p}}}{δ^{1 + \frac{1}{p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S] . \end{matrix}$

Consequently, from (31) and (39), we deduce $\begin{array}{rcl} ‖ {\tilde{U}}_{μ} - \tilde{L} ‖_{C ([0, s]; H)} & ⩽ & ‖ {\tilde{U}}_{μ} - W_{μ} ‖_{C ([0, s]; H)} + ‖ R ‖_{C ([0, s]; H)} \\ (40) & ⩽ & C M_{0} \frac{ε^{β}}{δ^{1 + \frac{1}{p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S], \end{array}$ with $β = min {\frac{1}{4}, \frac{p - 1}{p}}$ . Since for all $s \in [0, S]$ , $U_{μ} (s) = {\tilde{U}}_{μ} (s)$ , $L (s) = \tilde{L} (s)$ , $U_{μ} (s) = u_{ε δ} (δ s)$ and $L (s) = l_{δ} (δ s)$ , from (40) we get $\begin{matrix} (41) & ‖ u_{ε δ} - l_{δ} ‖_{C ([0, T]; H)} ⩽ C M_{0} \frac{ε^{β}}{δ^{1 + \frac{1}{p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1] . \end{matrix}$

In what follows, introduce the notation $R_{1} (t, δ) = l_{δ} (t) - v (t) - h_{δ} (t)$ , where $l_{δ}$ is the solution to the problem ( P δ ), v is the solution to the problem ( P 0 ) and $h_{δ}$ is the solution to the problem $\begin{matrix} (42) & \{\begin{matrix} δ h_{δ}^{'} (t) + A h_{δ} (t) = 0, t \in (0, T), \\ h_{δ} (0) = u_{0} - A^{- 1} f (0) . \end{matrix} \end{matrix}$

In this case we observe that $R_{1} (t, δ) = l_{δ} (t) - A^{- 1} f (t) - h_{δ} (t)$ and $R_{1}^{'} (t, δ) = l_{δ}^{'} (t) - A^{- 1} f^{'} (t) - h_{δ}^{'} (t)$ . Taking into account the last statements, we deduce that $R_{1}$ is the solution to the problem $\begin{matrix} (43) & \{\begin{matrix} δ R_{1}^{'} (t, δ) + A R_{1} (t, δ) = - δ A^{- 1} f^{'} (t), & t \in (0, T), \\ R_{1} (0) = 0 . \end{matrix} \end{matrix}$

From (43), in the same way as estimate (34) was obtained from (32), we get $\begin{array}{rcl} | R_{1} (t, δ) | & ⩽ & \sqrt{δ} \int_{0}^{t} | A^{- 1} f^{'} (τ) | d τ \\ ⩽ & \sqrt{δ} C {(\int_{0}^{T} {| f^{'} (τ) |}^{p} d τ)}^{1 / p} \cdot T^{\frac{p - 1}{p}} \\ (44) & ⩽ & C M_{0} \sqrt{δ}, \forall t \in [0, T], δ \in (0, 1] . \end{array}$

Thus, the estimate (12) is a simple consequence of (41) and (44). □

Proof of estimate (14).

Since $u_{1} \in D (A^{1 / 2})$ , using (27), in the same way as estimates (31), (35), (36) and (37) were obtained, we get successively $\begin{array}{rcl} ‖ {\tilde{U}}_{μ} - W_{μ} ‖_{C ([0, s]; H)} ⩽ C M_{1} μ^{1 / 2} δ^{- 1 / p} (1 + \sqrt{s}) \\ (45) & ⩽ C M_{1} \frac{ε^{1 / 2}}{δ^{3 / 2 + 1 / p}}, \forall ε \in (0, ε_{0}], δ \in (0, 1], \forall s \in [0, S] . \\ (46) & | R_{0} | ⩽ C μ M_{1} δ^{- 1 / p} ⩽ C M_{1} \frac{ε}{δ^{1 / p + 2}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1] . \\ | \tilde{F} (s) - \int_{0}^{\infty} K (s, τ, μ) \tilde{F} (τ) d τ | \\ (47) & ⩽ C M_{1} \frac{ε^{\frac{p - 1}{2 p}}}{δ^{\frac{p - 1}{2 p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S] \end{array}$ and $\begin{matrix} (48) & | δ \int_{0}^{s} f_{0} (ξ, μ) u_{1} d ξ | ⩽ C δ μ | u_{1} | ⩽ C M_{1} \frac{ε}{δ}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s ⩾ 0 . \end{matrix}$

Using (47) and (48), from (33) we obtain $\begin{matrix} (49) & \int_{0}^{s} | F (ξ, μ) | d τ ⩽ C M_{1} \frac{ε^{\frac{p - 1}{2 p}}}{δ}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S] . \end{matrix}$

From (34), using (46) and (49), we get the estimate $\begin{matrix} (50) & ‖ R ‖_{C ([0, s]; H)} ⩽ C M_{1} \frac{ε^{\frac{p - 1}{2 p}}}{δ^{2 + \frac{1}{p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S] . \end{matrix}$

Consequently, from (45) and (50), we deduce $\begin{matrix} (51) & ‖ {\tilde{U}}_{μ} - \tilde{L} ‖_{C ([0, s]; H)} ⩽ C M_{1} \frac{ε^{\frac{p - 1}{2 p}}}{δ^{2 + \frac{1}{p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \forall s \in [0, S] . \end{matrix}$

Since for all $s \in [0, S]$ , $U_{μ} (s) = {\tilde{U}}_{μ} (s)$ , $L (s) = \tilde{L} (s)$ , $U_{μ} (s) = u_{ε δ} (δ s)$ and $L (s) = l_{δ} (δ s)$ , from (51) we get $\begin{matrix} ‖ u_{ε δ} - l_{δ} ‖_{C ([0, T]; H)} ⩽ C M_{1} \frac{ε^{\frac{p - 1}{2 p}}}{δ^{2 + \frac{1}{p}}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1] . \end{matrix}$

Taking into account that $M_{0} ⩽ M_{1}$ , estimate (14) is a simple consequence of the last estimate and of (44). Theorem 5 is proved. □

Remark 1.

From estimates (12) and (14) it follows that $‖ u_{ε δ} - v ‖_{C ([0, T]; H)} \to 0$ only if $u_{0} - A^{- 1} f (0) = 0$ . In the opposite case, the solution $u_{ε δ}$ has a singular behavior with respect to the parameters ε and δ, as $Θ (ε, δ) \to 0$ or $Θ_{1} (ε, δ) \to 0$ in the neighborhood of $t = 0$ . This behavior is defined by the boundary function $h_{δ}$ , which is solution to the problem (13).

Remark 2.

If in the statement of Theorem 5 one assumes $f \equiv 0$ , then estimates (12) and (14) take the following form: $\begin{matrix} ‖ u_{ε δ} - v - h_{δ} ‖_{C ([0, T]; H)} ⩽ C (| A^{1 / 2} u_{0} | + | u_{1} |) \frac{ε^{1 / 4}}{δ}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1], \end{matrix}$ and $\begin{matrix} ‖ u_{ε δ} - v - h_{δ} ‖_{C ([0, T]; H)} ⩽ C (| A^{1 / 2} u_{0} | + | A^{1 / 2} u_{1} |) \frac{ε^{1 / 2}}{δ^{2}}, \forall ε \in (0, ε_{0}], \forall δ \in (0, 1] . \end{matrix}$

Footnotes

Acknowledgement

Researches in part supported by the Program 12.839.08.06F.

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