General decay for a nonlinear pseudo-parabolic equation with viscoelastic term

Ngo Tran Vu; Dao Bao Dung; Huynh Thi Hoang Dung

doi:10.1515/dema-2022-0164

Article Open Access

General decay for a nonlinear pseudo-parabolic equation with viscoelastic term

Ngo Tran Vu , Dao Bao Dung and Huynh Thi Hoang Dung

Published/Copyright: October 21, 2022

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From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

This work is concerned with a multi-dimensional viscoelastic pseudo-parabolic equation with critical Sobolev exponent. First, with some suitable conditions, we prove that the weak solution exists globally. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literature. More precisely, we consider the kernel g : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) satisfying g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) , where ξ and G are functions satisfying some specific properties.

Keywords: pseudo-parabolic equation; viscoelastic; global existence; decay rate

MSC 2010: 35B40; 35K51

1 Introduction

In this article, we consider the following nonlinear pseudo-parabolic equation:

(1) u t − Δ u − Δ u t + ∫ 0 t g ( t − s ) Δ u ( x , s ) d s = f ( u ) , ( x , t ) ∈ Ω × ( 0 , ∞ ) ,

associated with the homogeneous Dirichlet boundary condition:

(2) u ( x , t ) = 0 , x ∈ ∂ Ω ,

and initial condition:

(3) u ( x , 0 ) = u 0 ( x ) ,

where Ω ⊂ R n is an open bounded domain with a smooth boundary ∂ Ω , u 0 ∈ H 0 1 ( Ω ) , the relaxation function g , and the nonlinear interior source f are given functions satisfying the following assumptions:

f ∈ C 1 ( R ) and satisfies the following conditions:
1. f ( 0 ) = 0 , and there exists an exponent q ∈ [ 2 , 2 ∗ ] such that
  (4) ∣ f ( u ) − f ( v ) ∣ ≲ ( 1 + ∣ u ∣ q − 2 + ∣ v ∣ q − 2 ) ∣ u − v ∣ , ∀ u , v ∈ R ,
  where 2 ∗ is a Sobolev critical exponent for the embedding H 0 1 ( Ω ) ↪ L p ( Ω ) .
2. There exists a constant p > 2 such that
  (5) 0 < p F ( u ) ⩽ u f ( u ) , ∀ u ∈ R ⧹ { 0 } ,
  where F ( u ) = ∫ 0 u f ( z ) d z .
3. There exist constants β 1 > 0 , β 2 > 0 , p 1 , p 2 ∈ [ p , 2 ∗ ] such that
  (6) u f ( u ) ⩽ β 1 ∣ u ∣ p 1 + β 2 ∣ u ∣ p 2 , ∀ u ∈ R .
g : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) is a C 1 decreasing function satisfying
(7) L = 1 − ∫ 0 ∞ g ( t ) d t > 0 ,
and there exists a C 1 function G : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) , which is linear or it is strictly increasing and strictly convex C 2 function on ( 0 , r ] , r ⩽ g ( 0 ) , with G ( 0 ) = G ′ ( 0 ) = 0 such that
(8) g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) , ∀ t ∈ [ 0 , ∞ ) ,
where ξ : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) is a decreasing differentiable function.

Pseudo-parabolic equations are the class of partial differential equations that are characterized by the occurrence of a time derivative appearing in the highest order term. In fact, the interest in the study of these partial differential equations goes beyond the mathematical curiosity. This type of problem comes to real world with many different applications in a quite natural way. These problems have a high profile in the study of physical processes dynamics in recent years, see for example [1,2,3, 4,5,6, 7,8,9, 10,11,12, 13,14,15] and references therein.

In [16], Xu and Su considered the following nonlinear pseudo-parabolic problem:

(9) u t − Δ u − Δ u t = u p , ( x , t ) ∈ Ω × ( 0 , ∞ ) , u ( x , t ) = 0 , x ∈ ∂ Ω , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω ,

where Ω ⊂ R n is a bounded open domain with smooth boundary, with p > 1 if n ∈ { 1 , 2 } and 1 < p < n − 2 n + 2 if n ⩾ 3 . By introducing a family of potential wells, which was first introduced by Yacheng and Junsheng in [17], the whole study is conducted by considering the following three cases according to initial energy: subcritical, critical, and supercritical initial energy cases. Under the condition J ( u 0 ) ⩽ d , where d is a depth of potential well associated with the energy functional

J ( u ) = 1 2 ∥ ∇ u ∥ 2 2 − 1 p + 1 ∥ u ∥ p + 1 p + 1 ,

they proved that if I ( u 0 ) > 0 then the weak solution exists globally and exponential decays at infinity. On the other side, if I ( u 0 ) < 0 then the weak solutions will blow up in finite time, where I is a Nehari functional given by I ( u ) = ∥ ∇ u ∥ 2 2 − ∥ u ∥ p + 1 p + 1 . With the supercritical initial energy level J ( u 0 ) > d , by adopting comparison principle, strong maximum principle, as well as variational methods, they gave a sufficient condition to obtain global existence and blow-up in finite time. After that, in [18], with some suitable assumptions, the authors estimated the upper bound and lower bound for blow-up time.

In [19], the authors proved the results of global existence, finite time blow-up for the solutions, and obtained the upper bound for the blow-up time of the following problem with a linear memory term and a nonlinear source term

u t − Δ u − Δ u t + ∫ 0 t g ( t − τ ) Δ u ( τ ) d τ = ∣ u ∣ p − 2 u , ( x , t ) ∈ Ω × ( 0 , T ) ,

where Ω is a bounded domain of R n with smooth boundary ∂ Ω , p > 2 , u ( 0 ) = u 0 ∈ H 0 1 , and g : R + ⟶ R + is a positive nonincreasing function. The concavity method and the improved potential method were used to have the upper bound for the blow-up time with initial data at the arbitrary energy level.

Motivated by the previous results, we consider in this article an initial boundary value problem given in (1)–(3). Let us explain in some detail our main results. In the previous articles, the authors used Galerkin and potential well methods to study the local existence and long-time behavior of the weak solution. In this framework, the compactness of the embedding H 0 1 ( Ω ) ↪ L p ( Ω ) plays an important role. Therefore, the first aim of this article is to improve this condition. In order to prove the local existence, we will use the Banach fixed point theorem. Hence, we need to lack the compactness of the embedding H 0 1 ( Ω ) ↪ L p ( Ω ) . The second aim of this work is to investigate (1)–(3) for relaxation functions g and the nonlinear source f of more general type than the ones in [19]. In fact, we consider the condition g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) , where G is increasing and convex without any additional constraints, and establish energy decay results that address both the optimality and generality. On the optimality level, the energy decay rates are consistent with the decay rates of g . We use minimal conditions and obtain explicit energy decay formulas, which give the best decay rates expected under this level of generality.

This article is organized as follows. In Section 2, we present some notations and materials needed for our work. In Section 3, we establish the global existence of the solution of the problem. Some technical lemmas and the decay results are presented in this section.

2 Preliminaries, notations, and local Hadamard well-posedness result

First of all, let us recall a few preliminaries about the notations and some useful lemmas. Throughout this article, for a clear presentation, we adopt the conventional notation L p ( Ω ) , with 1 ⩽ p ⩽ ∞ , for the usual Lebesgue space equipped with ∥ ⋅ ∥ L p ( Ω ) norm. For simplicity, we write ∥ ⋅ ∥ p for ∥ ⋅ ∥ L p ( Ω ) , we also denote the inner product on Hilbert space L 2 ( Ω ) by ⟨ ⋅ , ⋅ ⟩ . The notation ∥ ⋅ ∥ H 0 1 and ⟨ ⋅ , ⋅ ⟩ H 0 1 stand for Dirichlet norm and inner product on Hilbert space H 0 1 ( Ω ) , respectively. That means

∥ u ∥ H 0 1 = ∥ u ∥ 2 2 + ∥ ∇ u ∥ 2 2 , ∀ u ∈ H 0 1 ( Ω ) ,

and

⟨ u , v ⟩ H 0 1 = ⟨ u , v ⟩ 2 + ⟨ ∇ u , ∇ v ⟩ 2 , ∀ u , v ∈ H 0 1 ( Ω ) .

We also recall a useful lemma. The interested reader is referred to [20] for more details.

Lemma 2.1

(Poincaré’s inequality) On the space H 0 1 ( Ω ) , two norms u ⟼ ∥ u ∥ H 0 1 and u ⟼ ∥ ∇ u ∥ 2 are equivalent. In fact, we have the estimate

(10) ∥ u ∥ 2 2 ⩽ μ 1 − 1 ∥ ∇ u ∥ 2 2 , ∀ u ∈ H 0 1 ( Ω ) ,

where μ 1 is the first eigenvalue of the operator − Δ with zero Dirichlet conditions. Furthermore, estimate (10) is optimal.

Next, we give the precise definition of a weak solution of Problems (1)–(3).

Definition 2.1

We say that the function u is a weak solution of Problems (1)–(3) on ( 0 , T ) if

(11) u ∈ L ∞ ( 0 , T ; H 0 1 ( Ω ) ) , u t ∈ L 2 ( 0 , T ; H 0 1 ( Ω ) ) ,

and u satisfies (1)–(3) in the following sense:

u verifies the distribution identity
(12) ⟨ u ′ ( t ) , w ⟩ H 0 1 + ⟨ ∇ u ( t ) , ∇ w ⟩ = ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) , ∇ w ⟩ d s + ⟨ f ( u ( t ) ) , w ⟩ ,
for any test functions w ∈ H 0 1 ( Ω ) and for almost all t ∈ ( 0 , T ) ;
u satisfies the initial condition
(13) u ( 0 ) = u 0 .

Remark 2.1

It is easy to see, under the regularity assumptions on u as well as nonlinear source f , that all the terms in (12) make sense. Furthermore, from (11), we also deduce that u ∈ C ( [ 0 , T ] ; H 0 1 ( Ω ) ) . This fact implies that u has a pointwise meaning on time, so also (13) satisfies in the usual sense.

Definition 2.2

(Maximal existence time) Let u be a weak solution of Problems (1)–(3). We define the maximal existence time T ∞ of u as follows:

If t ⟼ u ( t ) exists for all t ∈ [ 0 , ∞ ) , then T ∞ = ∞ ;
If there exists a t 0 ∈ ( 0 , ∞ ) such that t ⟼ u ( t ) exists for all t ∈ [ 0 , t 0 ) , but does not exist at t = t 0 , then T ∞ = t 0 .

By using the same technique as in [15, Theorem 2.1], we have the following theorem.

Theorem 2.1

(Local Hadamard well-posedness result) Let [F] and [G] be in force. For any u 0 ∈ H 0 1 ( Ω ) , Problems (1)–(3) admit a unique weak solution u ∈ C 1 ( [ 0 , T ] ; H 0 1 ( Ω ) ) , with T > 0 small enough. Moreover, the weak solution u can be represented in the following integral form:

(14) u ( t ) = exp ( − t ) u 0 + ∫ 0 t exp ( s − t ) A [ u ] ( s ) d s , ∀ t ∈ [ 0 , T ∞ ) ,

where

(15) A [ u ] ( t ) = ∫ 0 t g ( t − s ) u ( s ) d s + ( Id − Δ ) − 1 u ( t ) + f ( u ( t ) ) − ∫ 0 t g ( t − s ) u ( s ) d s .

Moreover, either u can be extended to the whole of [ 0 , ∞ ) or there is T ∞ < ∞ such that lim t → T ∞ − ∥ ∇ u ( t ) ∥ 2 2 = ∞ .

3 Global existence and decay estimates

3.1 Global existence

In order to study the global existence and prove decay estimates, we need to consider the following functional:

(16) E ( t ) = 1 2 1 − ∫ 0 t g ( s ) d s ∥ ∇ u ( t ) ∥ 2 2 + 1 2 ( g ⋆ ∇ u ) ( t ) − ∫ Ω F ( u ( x , t ) ) d x , ∀ t ∈ [ 0 , T ∞ ) ,

where

(17) ( g ⋆ u ) ( t ) = ∫ 0 t g ( t − s ) ∥ u ( s ) − u ( t ) ∥ 2 2 d s , ∀ t ∈ [ 0 , T ∞ ) .

The following lemma states one of the most important properties of the functional E .

Lemma 3.1

Let the assumptions [F] and [G] be in force. Then we have the following estimate:

(18) E ′ ( t ) = − ∥ u ′ ( t ) ∥ H 0 1 2 + 1 2 ( g ′ ⋆ ∇ u ) ( t ) − 1 2 g ( t ) ∥ ∇ u ( t ) ∥ 2 2 ⩽ 0 , ∀ t ∈ [ 0 , T ∞ ) .

In particular, the functional E is decreasing.

Proof of Lemma 3.1

Multiplying equation (1) by u t , integrating over Ω , and using the following identity:

∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) , ∇ u ′ ( t ) ⟩ d s = 1 2 ( g ′ ⋆ ∇ u ) ( t ) − 1 2 g ( t ) ∥ ∇ u ( t ) ∥ 2 2 − 1 2 d d t ( g ⋆ ∇ u ) ( t ) − ∫ 0 t g ( s ) d s ∥ ∇ u ( t ) ∥ 2 2 ,

we deduce that

E ′ ( t ) = 1 2 ( g ′ ⋆ ∇ u ) ( t ) − 1 2 g ( t ) ∥ ∇ u ( t ) ∥ 2 2 ⩽ 0 , ∀ t ∈ [ 0 , T ∞ ) .

The desired conclusions can be followed immediately from the above estimate. Lemma 3.1 is proved.□

We put

(19) ϱ ( t ) = 1 − ∫ 0 t g ( s ) d s ∥ ∇ u ( t ) ∥ 2 2 + ( g ⋆ ∇ u ) ( t ) , ∀ t ∈ [ 0 , T ∞ ) .

Since [ G ] holds, we have

(20) ϱ ( t ) ⩾ L ∥ ∇ u ( t ) ∥ 2 , ∀ t ∈ [ 0 , T ∞ ) .

Furthermore, it follows from [ F ], (16), and Lemma 3.1 that

(21) E ( 0 ) ⩾ E ( t ) ⩾ 1 2 ( ϱ ( t ) ) 2 − β 1 p ∥ u ( t ) ∥ p 1 p 1 − β 2 p ∥ u ( t ) ∥ p 2 p 2 ⩾ 1 2 ( ϱ ( t ) ) 2 − β 1 S p 1 p 1 p ∥ ∇ u ( t ) ∥ 2 p 1 − β 2 S p 2 p 2 p ∥ ∇ u ( t ) ∥ 2 p 2 ⩾ 1 2 ( ϱ ( t ) ) 2 − β 1 S p 1 p 1 p L p 1 2 ( ϱ ( t ) ) p 1 − β 2 S p 2 p 2 p L p 2 2 ( ϱ ( t ) ) p 2 = H ( ϱ ( t ) ) ,

where S σ = sup u ∈ H 0 1 ( Ω ) ⧹ { 0 } ∥ u ∥ σ ∥ ∇ u ∥ 2 > 0 , and the function H : [ 0 , ∞ ) ⟶ R defined by

(22) H ( λ ) = λ 2 2 − β 1 S p 1 p 1 p L p 1 2 λ p 1 − β 2 S p 2 p 2 p L p 2 2 λ p 2 .

Then, we have the following lemma, its proof is not difficult, so we omit it.

Lemma 3.2

With the function H : [ 0 , ∞ ) ⟶ R defined by (22), we have the following statements:

H ( 0 ) = 0 and lim λ → ∞ H ( λ ) = − ∞ .
The equation H ′ ( λ ) = 0 has a unique positive solution λ 0 satisfying
(23) 1 − β 1 p 1 S p 1 p 1 p L p 1 2 λ 0 p 1 − 2 − β 2 p 2 S p 2 p 2 p L p 2 2 λ 0 p 2 − 2 = 0 .
The function λ ⟼ H ( λ ) is strictly increasing on [ 0 , λ 0 ) , strictly decreasing on ( λ 0 , ∞ ) , takes the maximum at λ = λ 0 , and
(24) H ( λ 0 ) = β 1 ( p 1 − 2 ) S p 1 p 1 2 p L p 1 2 λ 0 p 1 + β 2 ( p 2 − 2 ) S p 2 p 2 2 p L p 2 2 λ 0 p 2 = E 1 .

The following lemma will play an essential role in the proof of our main result. It is similar to a lemma used first by Vitillaro [21].

Lemma 3.3

Support that assumptions [F] and [G] hold. For any u 0 ∈ H 0 1 ( Ω ) such that

(25) 0 ⩽ E ( 0 ) < β 1 ( p 1 − 2 ) S p 1 p 1 2 p L p 1 2 λ 0 p 1 + β 2 ( p 2 − 2 ) S p 2 p 2 2 p L p 2 2 λ 0 p 2 = E 1 ,

if ∥ ∇ u 0 ∥ 2 < λ 0 , then there exists a unique λ 1 ∈ ( 0 , λ 0 ) such that ϱ ( t ) ⩽ λ 1 for all t ∈ [ 0 , T ∞ ) .

Proof of Lemma 3.3

First, we have ϱ ( 0 ) = ∥ ∇ u 0 ∥ 2 < λ 0 . It follows from Lemma 3.2 that there exists a unique constant λ 1 ∈ ( 0 , λ 0 ) such that H ( λ 1 ) = E ( 0 ) . Thus, from (21), we deduce that

H ( λ 1 ) = E ( 0 ) ⩾ H ( ϱ ( 0 ) ) .

Thus, we have ϱ ( 0 ) ⩽ λ 1 . We will claim that ϱ ( t ) ⩽ λ 1 for all t ∈ [ 0 , T ∞ ) . Conversely, suppose that there exists t ∗ ∈ ( 0 , T ∞ ) such that ϱ ( t ∗ ) > λ 1 . By the continuity of the function t ⟼ ϱ ( t ) , without loss of generality, we may assume that ϱ ( t ∗ ) ∈ ( λ 1 , λ 0 ) . Recalling (21), it may be concluded that

H ( λ 1 ) = E ( 0 ) ⩾ E ( t ∗ ) ⩾ H ( ϱ ( t ∗ ) ) > H ( λ 1 ) ,

which is impossible. Hence, we have ϱ ( t ) ⩽ λ 1 for all t ∈ [ 0 , T ∞ ) . Lemma 3.3 is proved completely.□

It follows immediately from (16) and (19) that

(26) E ( t ) = p − 2 2 p ( ϱ ( t ) ) 2 + I ( t ) p , ∀ t ∈ [ 0 , T ∞ ) ,

where

(27) I ( t ) = 1 − ∫ 0 t g ( s ) d s ∥ ∇ u ( t ) ∥ 2 2 + ( g ⋆ ∇ u ) ( t ) − p ∫ Ω F ( u ( x , t ) ) d x , ∀ t ∈ [ 0 , T ∞ ) .

Analogously to (21), we also deduce that

(28) I ( t ) ⩾ 1 − β 1 S p 1 p 1 L p 1 2 ( ϱ ( t ) ) p 1 − 2 − β 2 S p 2 p 2 L p 2 2 ( ϱ ( t ) ) p 2 ( ϱ ( t ) ) 2 ⩾ 1 − β 1 S p 1 p 1 λ 1 p 1 − 2 L p 1 2 − β 2 S p 2 p 2 λ 1 p 2 − 2 L p 2 2 ( ϱ ( t ) ) 2 = η ( ϱ ( t ) ) 2 ⩾ 0 ,

which is clear from Lemma 3.3, and the fact that

η = 1 − β 1 S p 1 p 1 λ 1 p 1 − 2 L p 1 2 − β 2 S p 2 p 2 λ 1 p 2 − 2 L p 2 2 > 1 − β 1 p 1 S p 1 p 1 λ 0 p 1 − 2 p L p 1 2 − β 2 p 2 S p 2 p 2 λ 0 p 2 − 2 p L p 2 2 = 0 .

Hence, we have

(29) E ( 0 ) ⩾ E ( t ) ⩾ p − 2 2 p ( ϱ ( t ) ) 2 ⩾ ( p − 2 ) L 2 p ∥ ∇ u ( t ) ∥ 2 2 , ∀ t ∈ [ 0 , T ∞ ) .

Therefore, by using a standard continuous argument, we have the following theorem.

Theorem 3.1

Under the same assumptions as in Lemma 3.3, the weak solution of Problems (1)–(3) exists globally. Furthermore, we have

(30) ∥ ∇ u ( t ) ∥ 2 2 ⩽ 2 p E ( t ) ( p − 2 ) L ⩽ 2 p E ( 0 ) ( p − 2 ) L , ∀ t ∈ [ 0 , ∞ ) .

3.2 Decay estimates

We begin by introducing the following technical lemmas, which play a crucial role in the following development.

Lemma 3.4

Assume that [G] holds. Then there exist two positive constants d and t 1 such that

(31) g ′ ( t ) ⩽ − d g ( t ) , ∀ t ∈ [ 0 , t 1 ] .

Proof of Lemma 3.4

Since g is decreasing and ∫ 0 ∞ g ( t ) d t < ∞ , we have lim t → ∞ g ( t ) = 0 . Thus, there exists t 1 > 0 such that g ( t 1 ) = r and g ( t ) ⩽ r for all t ∈ [ t 1 , ∞ ) . As g and ξ are positive decreasing continuous and G is a positive continuous function, we discover

0 < g ( t 1 ) ⩽ g ( t ) ⩽ g ( 0 ) , 0 < ξ ( t 1 ) ⩽ ξ ( t ) ⩽ ξ ( 0 ) .

This fact implies that there are two positive constants a and b such that

a ⩽ ξ ( t ) G ( g ( t ) ) ⩽ b , ∀ t ∈ [ 0 , t 1 ] .

Therefore, we have

g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) ⩽ − a g ( 0 ) g ( 0 ) ⩽ − a g ( 0 ) g ( t ) = − d g ( t ) , ∀ t ∈ [ 0 , t 1 ] .

Lemma 3.4 is proved.□

Lemma 3.5

Under the assumption [G], the functional

(32) ψ 1 ( t ) = ∫ 0 t r ( t − s ) ∥ ∇ u ( s ) ∥ 2 2 d s , ∀ t ∈ [ 0 , T ∞ ) ,

satisfies the following estimate:

(33) ψ 1 ′ ( t ) ⩽ − 1 2 ( g ⋆ ∇ u ) ( t ) + 3 ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 , ∀ t ∈ [ 0 , T ∞ ) ,

where r ( t ) = ∫ t ∞ g ( s ) d s .

Proof of Lemma 3.5

By straightforward calculation, we have

ψ 1 ′ ( t ) = r ( 0 ) ∥ ∇ u ( t ) ∥ 2 2 − ∫ 0 t g ( t − s ) ∥ ∇ u ( s ) ∥ 2 2 d s = r ( t ) ∥ ∇ u ( t ) ∥ 2 2 − ∫ 0 t g ( t − s ) ∥ ∇ u ( s ) − ∇ u ( t ) ∥ 2 2 d s − 2 ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) − ∇ u ( t ) , ∇ u ( t ) ⟩ d s ⩽ ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 − ( g ⋆ ∇ u ) ( t ) − 2 ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) − ∇ u ( t ) , ∇ u ( t ) ⟩ d s .

Adopting the Cauchy-Schwarz inequality, we can easily conclude that

− 2 ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) − ∇ u ( t ) , ∇ u ( t ) ⟩ d s ⩽ 2 ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 + ∫ 0 t g ( s ) d s 2 ( 1 − L ) ( g ⋆ ∇ u ) ( t ) ⩽ 2 ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 + 1 2 ( g ⋆ ∇ u ) ( t ) .

Combining the aforementioned results, we obtain (33). Lemma 3.5 is proved completely.□

Let us consider the following notation:

(34) ( g ⊙ ∇ u ) ( t ) = ∫ Ω ∫ 0 t g ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s 2 d x .

Lemma 3.6

For any α ∈ ( 0 , 1 ) , we have the following estimate:

(35) ( g ⊙ ∇ u ) ( t ) ⩽ C α ( h ⋆ ∇ u ) ( t ) , ∀ t ∈ [ 0 , T ∞ ) ,

where

(36) C α = ∫ 0 ∞ g 2 ( s ) α g ( s ) − g ′ ( s ) d s , h ( t ) = α g ( t ) − g ′ ( t ) .

Proof of Lemma 3.6

Using the Cauchy-Schwarz inequality, we obtain

( g ⊙ ∇ u ) ( t ) = ∫ Ω ∫ 0 t g ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s 2 d x = ∫ Ω ∫ 0 t g ( t − s ) α g ( t − s ) − g ′ ( t − s ) α g ( t − s ) − g ′ ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s 2 d x ⩽ ∫ 0 t g 2 ( s ) α g ( s ) − g ′ ( s ) d s ∫ 0 t h ( t − s ) ∥ ∇ u ( s ) − ∇ u ( t ) ∥ 2 2 d s ⩽ C α ( h ⋆ ∇ u ) ( t ) ,

which is the desired conclusion.□

Remark 3.1

We note that α g 2 ( s ) α g ( s ) − g ′ ( s ) ⩽ g ( s ) for all s ∈ [ 0 , ∞ ) , using the Lebesgue dominated convergence theorem, it is easily seen that

(37) lim α ↘ 0 α C α = lim α ↘ 0 ∫ 0 ∞ α g 2 ( s ) α g ( s ) − g ′ ( s ) d s = 0 .

Lemma 3.7

For any ε > 0 , the functional

(38) ψ 2 ( t ) = C ( ε ) E ( t ) , ∀ t ∈ [ 0 , T ∞ ) ,

satisfies the following estimate:

(39) ψ 2 ′ ( t ) ⩽ − L 2 ∥ ∇ u ( t ) ∥ 2 2 + C α 2 L ( h ⋆ ∇ u ) ( t ) + ( ε + η ∗ ) E ( t ) , ∀ t ∈ [ 0 , T ∞ ) ,

where

(40) C ( ε ) = p ( μ 1 + 1 ) 2 ε ( p − 2 ) μ 1 L , η ∗ = 2 p p − 2 β 1 S p 1 p 1 L p 1 2 2 p E ( 0 ) p − 2 p 1 − 2 2 + β 2 S p 2 p 2 L p 2 2 2 p E ( 0 ) p − 2 .

In addition, if L > η ∗ and ε > 0 small enough, then there exists a positive constant ϖ such that

(41) ψ 2 ′ ( t ) ⩽ − ϖ E ( t ) + 1 − L + L 2 2 L ( g ⋆ ∇ u ) ( t ) , ∀ t ∈ [ 0 , T ∞ ) .

Proof of Lemma 3.7

Multiplying equation (1) by u and integrating over Ω , we obtain

(42) ∥ ∇ u ( t ) ∥ 2 2 = − ⟨ u ′ ( t ) , u ( t ) ⟩ H 0 1 + ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) , ∇ u ( t ) ⟩ d s + ⟨ f ( u ( t ) ) , u ( t ) ⟩ .

We shall estimate, respectively, the following integrals on the right-hand side of (42) as follows.

Estimate: I 1 = − ⟨ u ′ ( t ) , u ( t ) ⟩ H 0 1 .

Recalling (30) and Lemma 3.1, we deduce that

(43) I 1 = − ⟨ u ′ ( t ) , u ( t ) ⟩ H 0 1 ⩽ ∥ u ( t ) ∥ H 0 1 ∥ u ′ ( t ) ∥ H 0 1 ⩽ ∥ ∇ u ( t ) ∥ 2 μ 1 + 1 μ 1 ∥ u ′ ( t ) ∥ H 0 1 ⩽ E ( t ) ( μ 1 + 1 ) 2 p ( p − 2 ) μ 1 L ∥ u ′ ( t ) ∥ H 0 1 2 ⩽ ε E ( t ) − C ( ε ) E ′ ( t ) .

Estimate: I 2 = ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) , ∇ u ( t ) ⟩ d s .

We have

(44) I 2 = ∫ 0 t g ( t − s ) ⟨ ∇ u ( s ) , ∇ u ( t ) ⟩ d s ⩽ 1 2 ∥ ∇ u ( t ) ∥ 2 2 + ∫ Ω ∫ 0 t g ( t − s ) ( ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ + ∣ ∇ u ( x , t ) ∣ ) d s 2 d x .

We then use Cauchy-Schwarz and Young’s inequalities, and the fact that ∫ 0 t g ( s ) d s ⩽ 1 − L , to obtain, for any ζ > 0 ,

(45) ∫ Ω ∫ 0 t g ( t − s ) ( ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ + ∣ ∇ u ( x , t ) ∣ ) d s 2 d x = ∫ Ω ∫ 0 t g ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s 2 d x + ∫ Ω ∫ 0 t g ( t − s ) ∣ ∇ u ( x , t ) ∣ d s 2 d x + 2 ∫ Ω ∫ 0 t g ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s ∫ 0 t g ( t − s ) ∣ ∇ u ( x , t ) ∣ d s d x ⩽ ( 1 + ζ ) ( 1 − L ) 2 ∥ ∇ u ( t ) ∥ 2 2 + 1 + 1 ζ ( g ⊙ ∇ u ) ( t ) .

With ζ = L 1 − L , it follows from (44) and (45) that

(46) I 2 ⩽ 2 − L 2 ∥ ∇ u ( t ) ∥ 2 2 + 1 2 L ( g ⊙ ∇ u ) ( t ) .

Moreover, adopting Lemma 3.6, we have

(47) I 2 ⩽ 2 − L 2 ∥ ∇ u ( t ) ∥ 2 2 + C α 2 L ( h ⋆ ∇ u ) ( t ) .

Estimate: I 3 = ⟨ f ( u ( t ) ) , u ( t ) ⟩ .

Using (6) and (30), we obtain

(48) I 3 = ⟨ f ( u ( t ) ) , u ( t ) ⟩ ⩽ β 1 ∥ u ( t ) ∥ p 1 p 1 + β 2 ∥ u ( t ) ∥ p 2 p 2 ⩽ β 1 S p 1 p 1 ∥ ∇ u ( t ) ∥ 2 p 1 + β 2 S p 2 p 2 ∥ ∇ u ( t ) ∥ 2 p 2 ⩽ β 1 S p 1 p 1 2 p E ( t ) ( p − 2 ) L p 1 2 + β 2 S p 2 p 2 2 p E ( t ) ( p − 2 ) L p 2 2 ⩽ 2 p p − 2 β 1 S p 1 p 1 L p 1 2 2 p E ( 0 ) p − 2 p 1 − 2 2 + β 2 S p 2 p 2 L p 2 2 2 p E ( 0 ) p − 2 E ( t ) = η ∗ E ( t ) .

Combining (44), (47), (48) with (42), we deduce that

ψ 2 ′ ( t ) ⩽ − L 2 ∥ ∇ u ( t ) ∥ 2 2 + C α 2 L ( h ⋆ ∇ u ) ( t ) + ( ε + η ∗ ) E ( t ) , ∀ t ∈ [ 0 , T ∞ ) .

Then the estimate (39) holds. In order to establish (40), we use Hölder’s inequality and the fact that ∫ 0 t g ( s ) d s ⩽ 1 − L , to obtain

(49) ( g ⊙ ∇ u ) ( t ) = ∫ Ω ∫ 0 t g ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s d x = ∫ Ω ∫ 0 t g ( t − s ) g ( t − s ) ∣ ∇ u ( x , s ) − ∇ u ( x , t ) ∣ d s d x ⩽ ( 1 − L ) ( g ⋆ ∇ u ) ( t ) .

Hence, combining (44), (46), (48), (49) with (42), we deduce that

(50) ψ 2 ′ ( t ) ⩽ − L 2 ∥ ∇ u ( t ) ∥ 2 2 + 1 − L 2 L ( g ⋆ ∇ u ) ( t ) + ( ε + η ∗ ) E ( t ) ⩽ − ( L − η ∗ − ε ) E ( t ) + 1 − L + L 2 2 L ( g ⋆ ∇ u ) ( t ) .

Choose ε > 0 such that

(51) ϖ = L − η ∗ − ε > 0 .

Thus, (50) leads to (41). Lemma 3.7 is proved.□

Now, we consider the following functional

(52) ϕ ( t ) = ψ 1 ( t ) + δ 1 ψ 2 ( t ) + δ 2 E ( t ) , ∀ t ∈ [ 0 , T ∞ ) ,

where δ 1 , δ 2 are two positive constants that will be chosen later.

Lemma 3.8

Assume that the assumptions [F] and [G] be in force. If

(53) η ∗ < min L 2 , L 32 ( 1 − L ) ,

then the function t ⟼ ∫ 0 t E ( s ) d s is uniform bounded.

Proof of Lemma 3.8

Recalling Lemmas 3.1, 3.5, and 3.7, we deduce that

(54) ϕ ′ ( t ) ⩽ − 1 2 ( g ⋆ ∇ u ) ( t ) + 3 ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 − δ 1 L 2 ∥ ∇ u ( t ) ∥ 2 2 + δ 1 ( ε + η ∗ ) E ( t ) + δ 2 2 ( g ′ ⋆ ∇ u ) ( t ) = − δ 1 L 2 − 3 ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 − δ 2 2 − δ 1 C α 2 L ( h ⋆ ∇ u ) ( t ) − 1 2 − α δ 2 2 ( g ⋆ ∇ u ) ( t ) + δ 1 ( ε + η ∗ ) E ( t ) .

We choose

(55) δ 1 = 16 ( 1 − L ) L , α = 1 3 δ 2 .

Furthermore, since lim α ↘ 0 α C α = 0 , there exists α 0 > 0 such that α C α < L 4 δ 1 for all α ∈ ( 0 , α 0 ) . With δ 2 > 0 large enough, then α = 1 3 δ 2 < α 0 . Thus, we have δ 2 2 − δ 1 C α 2 L ⩾ δ 2 4 . Therefore, we obtain

(56) ϕ ′ ( t ) ⩽ − 4 ( 1 − L ) ∥ ∇ u ( t ) ∥ 2 2 − 1 4 ( g ⋆ ∇ u ) ( t ) + 16 ( 1 − L ) L ( ε + η ∗ ) E ( t ) ⩽ − 2 κ ∗ E ( t ) + 16 ( 1 − L ) L ( ε + η ∗ ) E ( t ) = − 2 κ ∗ − 8 ( 1 − L ) L ( ε + η ∗ ) E ( t ) ,

where κ ∗ = min 4 ( 1 − L ) , 1 4 . With ε > 0 and small enough, we have (51) and κ ∗ − 8 ( 1 − L ) η ∗ L > 0 hold. Hence, there exists ς > 0 , such that

ς ∫ 0 t E ( s ) d s ⩽ ϕ ( 0 ) − ϕ ( t ) ⩽ ϕ ( 0 ) , ∀ t ∈ [ 0 , T ∞ ) ,

and the proof is complete.□

Theorem 3.2

Let the assumptions [F] and [G] be in force. For any u 0 ∈ H 0 1 ( Ω ) such that 0 < E ( 0 ) < E 1 , ∥ ∇ u 0 ∥ 2 < λ 0 , and

(57) η ∗ = 2 p p − 2 β 1 S p 1 p 1 L p 1 2 2 p E ( 0 ) p − 2 p 1 − 2 2 + β 2 S p 2 p 2 L p 2 2 2 p E ( 0 ) p − 2 < min L 2 , L 32 ( 1 − L ) ,

then we have the following decay properties:

If G is linear, then there exists a positive constant ω such that
(58) E ( t ) ≲ exp − ω ∫ 0 t ξ ( s ) d s , ∀ t ∈ [ 0 , ∞ ) .
If G is nonlinear, then there exists a positive constant ω such that
(59) E ( t ) ≲ G 1 − 1 ω ∫ t 1 t ξ ( s ) d s , ∀ t ∈ [ t 1 , ∞ ) ,
where G 1 ( z ) = ∫ z r d s s G ′ ( s ) is strictly decreasing and convex on ( 0 , r ] , and lim z ↘ 0 G 1 ( z ) = ∞ .

Proof of Theorem 3.2

We put W ( t ) = ψ 2 ( t ) + E ( t ) . It follows from Lemmas 3.1, 3.4, and 3.7 that

(60) W ′ ( t ) = ψ 2 ′ ( t ) + E ′ ( t ) ⩽ − ϖ E ( t ) + 1 − L + L 2 2 L ︸ ≔ ϖ 1 > 0 ∫ t 1 t g ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s , ∀ t ∈ [ t 1 , ∞ ) .

Now, we estimate the second term in the right-hand side of (60). For this purpose, we need to distinguish two cases:

Case 1: G is linear.

Multiplying (60) by ξ and using [ G ], we deduce that

ξ ( t ) W ′ ( t ) ⩽ − ϖ ξ ( t ) E ( t ) + ϖ 1 ξ ( t ) ∫ t 1 t g ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩽ − ϖ ξ ( t ) E ( t ) − ϖ 1 ∫ t 1 t g ′ ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩽ − ϖ ξ ( t ) E ( t ) − ϖ 1 ∫ 0 t g ′ ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩽ − ϖ ξ ( t ) E ( t ) − 2 ϖ 1 E ′ ( t ) .

Using the fact ξ ′ ( t ) ⩽ 0 , the functional Φ ( t ) = ξ ( t ) W ( t ) + 2 ϖ 1 E ( t ) satisfies Φ ∼ E and

Φ ′ ( t ) ⩽ − ϖ ξ ( t ) E ( t ) ≲ ξ ( t ) Φ ( t ) , ∀ t ∈ [ t 1 , ∞ ) .

By direct calculation, we obtain

E ( t ) ≲ Φ ( t ) ≲ exp − ω ∫ 0 t ξ ( s ) d s , ∀ t ∈ [ t 1 , ∞ ) .

Case 2: G is nonlinear.

First, we note that

(61) ∫ t 1 t ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ≲ ∫ 0 t ( ∥ ∇ u ( t ) ∥ 2 2 + ∥ ∇ u ( t − s ) ∥ 2 2 ) d s ≲ ∫ 0 t ( E ( t ) + E ( t − s ) ) d s ≲ ∫ 0 t E ( s ) d s ≲ ∫ 0 ∞ E ( s ) d s < ∞ .

Hence, there exists ε 1 > 0 such that

(62) 0 < ℐ ( t ) = ε 1 ∫ t 1 t ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s < 1 , ∀ t ∈ [ t 1 , ∞ ) .

We also need to consider the functional χ ( t ) = − ∫ t 1 t g ′ ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s . It follows from Lemma 3.1 that χ ( t ) ⩽ − 2 E ′ ( t ) . Furthermore, since G is strictly convex on ( 0 , r ] and G ( 0 ) = 0 , then

G ( θ z ) ⩽ θ G ( z ) , ∀ θ ∈ [ 0 , 1 ] , z ∈ ( 1 , r ] .

Therefore, by adopting the Jensen inequality, we have

χ ( t ) = 1 ε 1 ℐ ( t ) ∫ t 1 t ℐ ( t ) ( − g ′ ( s ) ) ε 1 ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩾ 1 ε 1 ℐ ( t ) ∫ t 1 t ℐ ( t ) ξ ( t ) G ( g ( s ) ) ε 1 ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩾ ξ ( t ) ε 1 ℐ ( t ) ∫ t 1 t G ( ℐ ( t ) g ( s ) ) ε 1 ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s

⩾ ξ ( t ) ε 1 ℐ ( t ) ∫ t 1 t G ¯ ( ℐ ( t ) g ( s ) ) ε 1 ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩾ ξ ( t ) ε 1 G ¯ ε 1 ∫ t 1 t g ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ,

where G ¯ is an extension of G such that G ¯ is strictly increasing and strictly convex C 2 function on ( 0 , ∞ ) . Thus, we may conclude that

∫ t 1 t g ( s ) ∥ ∇ u ( t ) − ∇ u ( t − s ) ∥ 2 2 d s ⩽ 1 ε 1 ( G ¯ ) − 1 ε 1 χ ( t ) ξ ( t ) , ∀ t ∈ [ t 1 , ∞ ) .

This fact implies

(63) W ′ ( t ) ⩽ − ϖ E ( t ) + ϖ 1 ε 1 ( G ¯ ) − 1 ε 1 χ ( t ) ξ ( t ) , ∀ t ∈ [ t 1 , ∞ ) .

Now, with ε 0 ∈ ( 0 , r ) , we put

(64) W 1 ( t ) = G ¯ ′ ε 0 E ( t ) E ( 0 ) W ( t ) + E ( t ) , ∀ t ∈ [ t 1 , ∞ ) ,

which is equivalent to E . Using the fact that E ′ ( t ) ⩽ 0 , G ¯ ′ ( s ) > 0 , G ¯ ″ ( s ) > 0 , we deduce from (64) that

(65) W 1 ′ ( t ) = ε 0 E ′ ( t ) E ( 0 ) G ¯ ″ ε 0 E ( t ) E ( 0 ) W ( t ) − ϖ G ¯ ′ ε 0 E ( t ) E ( 0 ) E ( t ) + ϖ 1 ε 1 G ¯ ′ ε 0 E ( t ) E ( 0 ) ( G ¯ ) − 1 ε 1 χ ( t ) ξ ( t ) + E ′ ( t ) ⩽ − ϖ G ¯ ′ ε 0 E ( t ) E ( 0 ) E ( t ) + ϖ 1 ε 1 G ¯ ′ ε 0 E ( t ) E ( 0 ) ( G ¯ ) − 1 ε 1 χ ( t ) ξ ( t ) .

Let ( G ¯ ) ∗ be the convex conjugate of G ¯ in the sense of Young, then

( G ¯ ) ∗ ( s ) = s ( G ¯ ′ ) − 1 ( s ) − G ¯ ( ( G ¯ ′ ) − 1 ( s ) ) , ∀ s ∈ ( 0 , ( G ¯ ′ ) ( r ) ] ,

and ( G ¯ ) ∗ satisfies the following generalized Young’s inequality:

(66) A B ⩽ ( G ¯ ) ∗ ( A ) + G ¯ ( B ) , ∀ A ∈ ( 0 , ( G ¯ ′ ) ( r ) ] , B ∈ ( 0 , r ] .

So, with A = G ¯ ′ ε 0 E ( t ) E ( 0 ) , B = ( G ¯ ) − 1 ε 1 χ ( t ) ξ ( t ) , it follows from (65) and (66) that

W 1 ′ ( t ) ⩽ − ϖ G ¯ ′ ε 0 E ( t ) E ( 0 ) E ( t ) + ϖ 1 ε 1 G ¯ ∗ G ¯ ′ ε 0 E ( t ) E ( 0 ) + ϖ 1 χ ( t ) ξ ( t ) ⩽ − ϖ G ¯ ′ ε 0 E ( t ) E ( 0 ) E ( t ) + ϖ 1 ε 0 E ( t ) ε 1 E ( 0 ) G ¯ ′ ε 0 E ( t ) E ( 0 ) + ϖ 1 χ ( t ) ξ ( t ) = − ϖ G ′ ε 0 E ( t ) E ( 0 ) E ( t ) + ϖ 1 ε 0 E ( t ) ε 1 E ( 0 ) G ′ ε 0 E ( t ) E ( 0 ) + ϖ 1 χ ( t ) ξ ( t ) .

Thus, we have

(67) ξ ( t ) W 1 ′ ( t ) ⩽ − ϖ ξ ( t ) E ( t ) G ′ ε 0 E ( t ) E ( 0 ) + ϖ 1 ε 0 ε 1 ξ ( t ) E ( t ) E ( 0 ) G ′ ε 0 E ( t ) E ( 0 ) − 2 ϖ E ′ ( t ) .

We put W 2 ( t ) = ξ ( t ) W 1 ( t ) + 2 ϖ E ( t ) . It is easy to check that W 2 ∼ E . That means there exist two positive constants ν 1 , ν 2 such that

ν 1 W 2 ( t ) ⩽ E ( t ) ⩽ ν 2 W 2 ( t ) , ∀ t ∈ [ t 1 , ∞ ) .

Furthermore, an easy computation shows that

(68) W 2 ′ ( t ) ≲ − ξ ( t ) E ( t ) E ( 0 ) G ′ ε 0 E ( t ) E ( 0 ) = − ξ ( t ) G 2 E ( t ) E ( 0 ) , ∀ t ∈ [ t 1 , ∞ ) ,

where G 2 ( z ) = z G ′ ( ε 0 z ) with z ∈ [ 0 , 1 ] . Since G 2 ′ ( z ) = G ′ ( ε 0 z ) + ε 0 z G ″ ( ε 0 z ) > 0 , then, making use of the strict convexity of G on ( 0 , r ] , it may be concluded that G 2 ( z ) , G 2 ′ ( z ) > 0 for all z ∈ ( 0 , 1 ] . Thus,

(69) ϑ ( t ) = ν 1 W 2 ( t ) E ( 0 ) ∼ E ( t )

and

ϑ ′ ( t ) ≲ − ξ ( t ) G 2 ε 0 E ( t ) E ( 0 ) ⩽ − ξ ( t ) G 2 ( ϑ ( t ) ) , ∀ t ∈ [ t 1 , ∞ ) .

Then, integrating over [ t 1 , t ] and using a change of variables, we obtain

G 1 ( ε 0 ϑ ( t ) ) ⩾ ∫ ε 0 ϑ ( t ) ε 0 ϑ ( t 1 ) d s s G ′ ( s ) ≳ ∫ t 1 t ξ ( s ) d s , ∀ t ∈ [ t 1 , ∞ ) .

That means there exists a positive constant ϖ such that

E ( t ) ≲ ϑ ( t ) ≲ G 1 − 1 ϖ ∫ t 1 t ξ ( s ) d s , ∀ t ∈ [ t 1 , ∞ ) .

Theorem 3.2 is proved completely.□

4 Conclusion

In this article, we consider a multi-dimensional viscoelastic pseudo-parabolic equation with critical Sobolev exponent. The first aim of our article is proving the existence of global solution with small data. Next, we show that the stability of the system holds for a much larger class of kernels than the ones considered in previous literature. More precisely, we consider the kernel g : [ 0 , ∞ ) ⟶ ( 0 , ∞ ) satisfying g ′ ( t ) ⩽ − ξ ( t ) G ( g ( t ) ) , where ξ and G are functions satisfying some specific properties.

Acknowledgments

The authors would like to thank the referees for the patience they had in reading the originally submitted manuscript and for their most useful remarks, which eventually led to an improved version of the article. The authors also would like to offer their special thanks to Dr. Truong Thi Nhan for her generous support and encouragement during the preparation this manuscript. Finally, this research is funded by University of Economics Ho Chi Minh City, Vietnam.

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submissions.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-06-09

Revised: 2022-08-24

Accepted: 2022-08-26

Published Online: 2022-10-21

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0164

Keywords for this article

pseudo-parabolic equation; viscoelastic; global existence; decay rate

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