Continuous dependence on initial data for damped fourth-order wave equation with strain term

Jiangbo Han; Weipeng Wu

doi:10.1515/anona-2025-0104

Article Open Access

Continuous dependence on initial data for damped fourth-order wave equation with strain term

Jiangbo Han and Weipeng Wu

Published/Copyright: August 30, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Advances in Nonlinear Analysis Volume 14 Issue 1

Abstract

This article investigates the fourth-order wave equation with general nonlinear strain term as well as both strong and weak damping terms. By deriving and utilizing energy decay estimate independent of the initial data in the stable manifold, we derive results regarding the continuous dependence of the global solution on initial data for the model equation with such a structure. These conclusions reflect the dissipative nature of the model.

Keywords: continuous dependence; fourth-order wave equation; general nonlinear strain term; strong damping; weak damping; energy decay estimate

MSC 2010: 35L05; 35B30

1 Introduction

We consider the following initial boundary value problem of the fourth-order wave equation with general nonlinear strain term, as well as strong and weak dampings:

(1.1) u t t + u x x x x − u x x − ω u x x t + γ u t + ( σ ( u x ) ) x = 0 , x ∈ ( 0 , L ) , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , x ∈ ( 0 , L ) , u ( 0 , t ) = u ( L , t ) = u x ( 0 , t ) = u x ( L , t ) = 0 , t > 0 ,

where ω > 0 , γ ≥ 0 ,

u 0 ( x ) ∈ H 0 2 ( Ω ) , u 1 ( x ) ∈ L 2 ( Ω ) ,

and L is a positive constant. For the case ω = 0 , γ = 0 , i.e., without considering the damping effect, the original model

(1.2) u t t + u x x x x = a ( u x 2 ) x + f ( x )

of such equation with the strain term ( σ ( u x ) ) x in (1.1) comes from the elasto-plastic microstructure models that depict the longitudinal motion of some elasto-plastic bars [2,3]. Here, the more general case σ ( s ) = ν s + χ ∣ s ∣ m − 2 s of the stress function in such model equations was considered in [8]. For the case γ > 0 , i.e., with the weak damping u t , the two-dimensional version of the equation

(1.3) u t t + u x x x x + γ u t + ( σ ( u x ) ) x = 0

is also related to the limit in Mindlin-Timoshenko equations, which describe the dynamics of a plate accounting for transverse shear effects [7,15]. For the case ω > 0 , i.e., with the strong damping u x x t , the equation

(1.4) u t t + u x x x x − ω u x x t + ( σ ( u x ) ) x = 0

corresponds to some viscoelastic materials of second grade [1]. And this model is a nonlinear one-dimensional beam equation with viscosity arising in the analysis of mechanical movements of shape memory alloys, where u and σ ( u x ) are the displacement and the stress of distributed loads, respectively [23]. The term u x x t also brings the effect of the viscosity to one-dimensional elasticity model u t t = ( σ ( u x ) ) x to make it become u t t = ( σ ( u x ) + ω u x t ) x that describes the one-dimensional motion of a viscoelastic bar, where u x is the strain and u x t is the strain rate [1,20].

In this article, we focus on the model equation with both the general nonlinear strain term and strong and weak dampings in problem (1.1) to investigate the stability of the global solution, i.e., its continuous dependence on the initial data. To quickly delve into the research topic of this article, we review some global well-posedness results for problem (1.1). For the case ω = 0 and γ = 0 , where damping terms are not considered, Chen and Yang [4] devoted their attention to model (1.2) by incorporating a general strain term ( σ ( u x ) ) x rather than a ( u x 2 ) x . Their objective was to study the existence as well as the non-existence of the global solution. Subsequently, Esquivel-Avila [8] investigated the n -dimensional version model

(1.5) u t t + Δ 2 u − Δ u + a ∑ i = 1 n ( ∣ u x i ∣ m − 2 u x i ) x i − ∣ u ∣ p − 2 u = 0

and gave the initial conditions that cause the global existence and the finite-time blowup. Different from the case without damping, the case ω > 0 or γ > 0 endows the energy of the model with dissipative properties. For ω = 0 and γ > 0 , i.e., with the weak damping but without the strong damping, Yang [30] studied the model equation (1.3) and proved the global existence, the finite-time blowup, and the asymptotic behavior of the solution. Furthermore, Wang and Wang [25] and Yang and Yang [32] extended these results to the context of diverse stress functions σ ( s ) . For ω > 0 and γ = 0 , i.e., with the strong damping but without the weak damping, Racke and Shang [23] considered the model equation (1.4) and studied the existence of both the global solution and global attractors. For ω > 0 and γ > 0 , i.e., with both the strong and weak dampings, Lian et al. [16] investigated the following n-dimensional version equation:

(1.6) u t t + Δ 2 u − Δ u − Δ u t + ∣ u t ∣ r − 2 u t + ∑ i = 1 n ( σ i ( u x i ) ) x i + f ( u ) = 0 .

They proved the local existence of solution and gave a sharp threshold to distinguish between the global existence and the finite-time blowup while also showing the long time behavior of global solution.

By reviewing the aforementioned research related to problem (1.1), it is indicated that the different initial data can result in distinct behaviors of the solution, i.e., the global existence and the finite-time blowup. And, the structural terms of the model equation are instrumental in shaping the relationship between the initial data and the resulting solution. Indeed, the presence of some nonlinear terms means that global existence is no longer guaranteed for all initial data, resulting in the finite-time blowup of the solution under some initial conditions. As the potential well theory was proposed by Payne and Sattinger [21,24], the sharp criterion, i.e., the classification of the initial data, was established for the classical semilinear wave equation u t t − Δ u = f ( u ) , distinguishing between global existence and finite-time blowup. Subsequently, through the many efforts of Xu et al. [5,6,10,11,17–19,22,26–29,31,33] to deepen and develop this theory, suitable potential well structures were established for various complex nonlinear evolution equations, which makes this theory a cornerstone technique to study the classification problem of the initial data for general nonlinear evolution equations and the influence of the models’ structures on such classification. Under the framework of the potential well theory, for the wave models, strong and weak dampings contribute to the decay of energy, thereby facilitating the global well-posedness. In contrast, the nonlinear source term emerges as a pivotal factor driving the finite-time blowup. It is reasonable to conjecture that the interplay between these two opposing factors in the model equations exerts a significant influence on the stability of solution, i.e., continuous dependence of the solution on the initial data. However, the commonly used strategy shown in [9,14] for studying the continuous dependence of solutions to nonlinear evolution equations is to apply the boundedness estimations of solutions without deeply taking advantage of the structural characteristics of the models, resulting in that the conclusions obtained by this strategy, namely, the growth estimates of the distance between the solutions, cannot reflect the structural nature of the models. In other words, the similar conclusions are given regardless of whether the model equation has a dissipative structure, leading to the results inherently imprecise for the case with damping. In [12], it has been validated that the wave models with dissipative structures in the one-dimensional case exhibit the stronger continuous dependence of the global solution on the initial data. To propel this research forward, in the consideration of the model u t t − Δ u − ω Δ u t + μ ∣ u t ∣ m − 2 u t = ∣ u ∣ p − 2 u , a new approach has been proposed in [13] for the general n-dimensional case, which applies specific energy decay estimates to enhance the stability conclusions derived from the existing strategy within [9,14]. Here, such energy decay estimates have also been obtained under the framework of potential well theory. By effectively capturing the dissipative nature of the model, the new conclusions reveal that the dissipation characteristics generated by the damping of the wave equation can substantially enhance the stability of the global solution. And by observing the proof of this new technique, the crucial step lies in providing an appropriate control for the nonlinear term associated with the source term ∣ u ∣ p − 2 u , which is the factor leading to the finite-time blowup. Thus, through careful analysis, it is revealed that such nonlinear terms are detrimental to the stability of the solution. Therefore, investigating the combined impact of the two opposing factors, i.e., the damping and the nonlinear terms, on the stability of the solution represents an important and intriguing research. For the model equation in problem (1.1), the structural term that leads to finite-time blowup is the general nonlinear strain term ( σ ( u x ) ) x , which is distinct from the nonlinear source term ∣ u ∣ p − 2 u considered in [13]. Moreover, from the aforementioned studies, it is evident that the nonlinear stress term σ ( u x ) holds the significant importance in both physics and mathematics. In this article, we study a model that has both the general nonlinear strain term ( σ ( u x ) ) x and two classes of damping, i.e., strong damping ω u x x t and weak damping γ u t . The objective of this article is to investigate the stability of the global solution, i.e., continuous dependence of the global solution on the initial data to propel this research. Within this new structural framework, it is necessary to address this problem in a new setting, with a particular focus on obtaining some rigorous estimates for the general nonlinear strain term. First, we give some energy decay estimate independent of the initial data in the stable manifold for such a model equation. Subsequently, we shall use two distinct strategies to derive two conclusions of continuous dependence of the global solution on the initial data, which can both reflect the dissipative characteristics of such model.

The rest of this article is organized as follows. In Section 2, we state some notations, assumptions regarding the nonlinear stress function, as well as the functionals and the manifolds in the potential well theory. In Section 3, by establishing and utilizing the specific energy decay estimate, we investigate the continuous dependence of the global solution on the initial data.

2 Preliminaries

To facilitate the discussion and analysis in this article, we use the notations ( u , v ) ≔ ∫ 0 L u v d x and ‖ u ‖ r ≔ ∫ 0 L ∣ u ∣ r d x 1 r to represent the L 2 -inner product and the norm of L r ( 0 , L ) with 1 ≤ r < + ∞ , respectively. Taking 1 ≤ q < + ∞ , we define

(2.1) R q ≔ sup u ∈ H 0 1 ( 0 , L ) \ { 0 } ‖ u ‖ q ‖ u x ‖ 2 ,

which means

(2.2) ‖ u ‖ q ≤ R q ‖ u x ‖ 2

for u ∈ H 0 1 ( 0 , L ) . Here, from H 0 1 ( 0 , L ) ↪ L q ( 0 , L ) with 1 ≤ q < + ∞ , we know 0 < R q < + ∞ . And for u ∈ H 0 2 ( 0 , L ) , due to u x ∈ H 0 1 ( 0 , L ) , we have

(2.3) ‖ u x ‖ q ≤ R q ‖ u x x ‖ 2 ,

where 1 ≤ q < + ∞ . We introduce the following assumptions on the nonlinear stress function σ ( s ) :

(2.4) ( H ) ( i ) σ ( s ) ∈ C 1 ( R ) , σ ( 0 ) = σ ′ ( 0 ) = 0 , σ ( s ) is monotone, and is convex for s > 0 , and concave for s < 0 ; ( i i ) there exist constants 1 < p < + ∞ and C 0 > 0 , such that ( p + 1 ) G ( s ) ≤ s σ ( s ) , ∣ σ ( s ) ∣ ≤ C 0 ∣ s ∣ p , for all s ∈ R and ∣ σ ( s 1 ) − σ ( s 2 ) ∣ ≤ C 0 ( 1 + ∣ s 1 ∣ p − 1 + ∣ s 2 ∣ p − 1 ) ∣ s 1 − s 2 ∣ , for all s 1 , s 2 ∈ R ,

where

(2.5) G ( u ) ≔ ∫ 0 u σ ( s ) d s .

In order to study the stability of the global solution, i.e., its continuous dependence on the initial data, we use the notation

(2.6) D ( u ( t ) , v ( t ) ) ≔ 1 2 ‖ u t ( t ) − v t ( t ) ‖ 2 2 + 1 2 ‖ u x x ( t ) − v x x ( t ) ‖ 2 2 + 1 2 ‖ u x ( t ) − v x ( t ) ‖ 2 2

to describe the distance between the two solutions u and v with respect to the initial data u 0 , u 1 , and v 0 , v 1 , respectively. And we use

(2.7) D ( u ( 0 ) , v ( 0 ) ) = 1 2 ‖ u 1 − v 1 ‖ 2 2 + 1 2 ‖ ( u 0 ) x x − ( v 0 ) x x ‖ 2 2 + 1 2 ‖ ( u 0 ) x − ( v 0 ) x ‖ 2 2

to show the distance between the initial data u 0 , u 1 , and v 0 , v 1 .

Next, we recall the potential energy functional

(2.8) J ( u ) ≔ 1 2 ‖ u x x ‖ 2 2 + 1 2 ‖ u x ‖ 2 2 − ∫ 0 L G ( u x ) d x

and the Nehari functional

I ( u ) ≔ ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 − ∫ 0 L σ ( u x ) u x d x

in the potential well theory, where G ( u x ) is defined by (2.5). Then, the energy functional is defined by

(2.9) E ( u ( t ) , u t ( t ) ) ≔ 1 2 ‖ u t ‖ 2 2 + J ( u ) ,

where E ( u 0 , u 1 ) is used to denote the initial energy E ( u ( 0 ) , u t ( 0 ) ) . And the Nehari manifold can be defined as

N ≔ { u ∈ H 0 2 ( 0 , L ) \ { 0 } ∣ I ( u ) = 0 } .

Subsequently, we define the depth of the potential well

(2.10) d ≔ inf u ∈ N J ( u ) .

Based on the aforementioned preparations, we present the stable manifold

W ≔ { u ∈ H 0 2 ( 0 , L ) ∣ J ( u ) < d , I ( u ) > 0 } ∪ { 0 } .

Since we need to establish some energy decay estimate independent of the initial data in the stable manifold in order to study the continuous dependence of the global solution on the initial data, we require the condition E ( u 0 , u 1 ) ≤ ϱ d 1 , where ϱ is any constant satisfying 0 < ϱ < 1 and

(2.11) d 1 ≔ p − 1 2 ( p + 1 ) 1 C 0 R p + 1 p + 1 2 p − 1 .

According to the definition of d 1 , i.e., (2.11), we can obtain the following conclusion.

Lemma 2.1

Let Assumption ( H ) hold. Then, one has d 1 ≤ d , where d is defined by (2.10).

Proof

By taking u ∈ N , i.e., I ( u ) = 0 and u ≠ 0 , we have

(2.12) ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 = ∫ 0 L σ ( u x ) u x d x .

Here, by similar arguments to those in the proof of Lemma 2.5 (iii) in [22], we can observe that N is non-empty. From Assumption ( H ) and (2.12), we have

(2.13) ∫ 0 L G ( u x ) d x ≤ 1 p + 1 ∫ 0 L σ ( u x ) u x d x = 1 p + 1 ( ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 ) .

Substituting (2.13) into J ( u ) , we know

(2.14) J ( u ) ≥ p − 1 2 ( p + 1 ) ( ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 ) .

Next, we estimate the term ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 in (2.14). By Assumption ( H ) and (2.3), we note that (2.12) becomes

‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 ≤ C 0 ‖ u x ‖ p + 1 p + 1 ≤ C 0 R p + 1 p + 1 ‖ u x x ‖ 2 p + 1 ≤ C 0 R p + 1 p + 1 ( ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 ) p + 1 2 ,

which gives

(2.15) ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 ≥ 1 C 0 R p + 1 p + 1 2 p − 1 .

By substituting (2.15) into (2.14), we have

(2.16) J ( u ) ≥ p − 1 2 ( p + 1 ) 1 C 0 R p + 1 p + 1 2 p − 1 ,

i.e., J ( u ) ≥ d 1 , which means d ≥ d 1 .□

3 Continuous dependence of global solution on initial data

This section studies the continuous dependence of the global solution to problem (1.1) on the initial data. In order to obtain the conclusion regarding the continuous dependence that reflects the dissipative characteristics of the model, it is essential to establish the following energy decay estimate, which is independent of the initial data in the stable manifold.

Lemma 3.1

(Energy decay estimate independent of initial data in stable manifold) Let ω > 0 , γ ≥ 0 , and Assumption ( H ) hold. For any 0 < ϱ < 1 , if 0 < E ( u 0 , u 1 ) ≤ ϱ d 1 and u 0 ∈ W , then one has

(3.1) E ( u ( t ) , u t ( t ) ) ≤ K e − λ t , t > 0 ,

where

(3.2) λ ≔ 2 ε min 1 − ϱ p − 1 2 , 1 − γ 2 θ 2 2 α 2 > 0 ,

(3.3) K ≔ α 2 ϱ d 1 α 1 > 0 ,

(3.4) α 1 ≔ 1 − ε max { ∣ R 2 2 − ω ∣ , 1 } ( p + 1 ) p − 1 ,

(3.5) α 2 ≔ 1 + ε max { R 2 2 + ω , 1 } ( p + 1 ) p − 1 ,

θ > 0 is chosen such that

(3.6) 1 − γ 2 θ 2 2 > 0 ,

and ε > 0 is selected to satisfy

(3.7) ω R 2 2 + γ − ε 2 − ϱ p − 1 2 + R 2 2 2 θ 2 ≥ 0

and α 1 > 0 .

Proof

Theorems 3.2 and 4.2 and Lemma 4.3 in [16] ensure the existence of the global solution u . Then, we define

V ( t ) ≔ E ( u ( t ) , u t ( t ) ) + ε ( u , u t ) + ε ω 2 ‖ u x ‖ 2 2 ,

where the constant ε > 0 will be chosen later. Here, by applying Hölder’s inequality, Young’s inequality, and (2.2), we know

(3.8) V ( t ) ≤ E ( u ( t ) , u t ( t ) ) + ε ‖ u ‖ 2 ‖ u t ‖ 2 + ε ω 2 ‖ u x ‖ 2 2 ≤ E ( u ( t ) , u t ( t ) ) + ε 2 ‖ u ‖ 2 2 + ε 2 ‖ u t ‖ 2 2 + ε ω 2 ‖ u x ‖ 2 2 ≤ E ( u ( t ) , u t ( t ) ) + ε 2 R 2 2 ‖ u x ‖ 2 2 + ε 2 ‖ u t ‖ 2 2 + ε ω 2 ‖ u x ‖ 2 2 ≤ E ( u ( t ) , u t ( t ) ) + ε max { R 2 2 + ω , 1 } 1 2 ‖ u x ‖ 2 2 + 1 2 ‖ u t ‖ 2 2 ≤ E ( u ( t ) , u t ( t ) ) + ε max { R 2 2 + ω , 1 } 1 2 ‖ u x ‖ 2 2 + 1 2 ‖ u x x ‖ 2 2 + 1 2 ‖ u t ‖ 2 2

and

(3.9) V ( t ) ≥ E ( u ( t ) , u t ( t ) ) − ε ‖ u ‖ 2 ‖ u t ‖ 2 + ε ω 2 ‖ u x ‖ 2 2 ≥ E ( u ( t ) , u t ( t ) ) − ε 2 R 2 2 ‖ u x ‖ 2 2 − ε 2 ‖ u t ‖ 2 2 + ε ω 2 ‖ u x ‖ 2 2 ≥ E ( u ( t ) , u t ( t ) ) − ε 2 R 2 2 − ε ω 2 ‖ u x ‖ 2 2 − ε 2 ‖ u t ‖ 2 2 − ε 2 ‖ u x x ‖ 2 2 ≥ E ( u ( t ) , u t ( t ) ) − ε max { ∣ R 2 2 − ω ∣ , 1 } 1 2 ‖ u x ‖ 2 2 + 1 2 ‖ u x x ‖ 2 2 + 1 2 ‖ u t ‖ 2 2 .

From Lemma 4.1 in [16], we see u ( t ) ∈ W for any time t ∈ ( 0 , + ∞ ) . Based on this fact, we can deduce

(3.10) ‖ u x ‖ 2 2 + ‖ u x x ‖ 2 2 ≥ ∫ 0 L σ ( u x ) u x d x .

By Assumption ( H ) , (3.10) turns to

(3.11) ‖ u x ‖ 2 2 + ‖ u x x ‖ 2 2 ≥ ∫ 0 L σ ( u x ) u x d x ≥ ( p + 1 ) ∫ 0 L G ( u x ) d x .

Substituting (3.11) into E ( u ( t ) , u t ( t ) ) , we obtain

(3.12) 0 ≤ p − 1 2 ( p + 1 ) ‖ u x ‖ 2 2 + p − 1 2 ( p + 1 ) ‖ u x x ‖ 2 2 + 1 2 ‖ u t ‖ 2 2 ≤ E ( u ( t ) , u t ( t ) ) ,

which means that (3.8) and (3.9) turn to

(3.13) V ( t ) ≤ E ( u ( t ) , u t ( t ) ) + ε max { R 2 2 + ω , 1 } ( p + 1 ) p − 1 E ( u ( t ) , u t ( t ) )

and

(3.14) V ( t ) ≥ E ( u ( t ) , u t ( t ) ) − ε max { ∣ R 2 2 − ω ∣ , 1 } ( p + 1 ) p − 1 E ( u ( t ) , u t ( t ) ) ,

respectively. According to (3.13) and (3.14), we know

(3.15) α 1 E ( u ( t ) , u t ( t ) ) ≤ V ( t ) ≤ α 2 E ( u ( t ) , u t ( t ) ) ,

where α 1 and α 2 are defined by (3.4) and (3.5), respectively. In (3.15), we choose ε > 0 to make

1 > ε max { ∣ R 2 2 − ω ∣ , 1 } ( p + 1 ) p − 1 ,

i.e., α 1 > 0 . By differentiating V ( t ) with respect to time t , we obtain

(3.16) V ′ ( t ) = d d t E ( u ( t ) , u t ( t ) ) + ε ‖ u t ‖ 2 2 + ε ∫ 0 L u t t u d x + ε ω ∫ 0 L u x u x t d x .

In (3.16), by similar arguments to those used to prove the energy identity (14) in Theorem 2.4 of [9], we can derive

(3.17) d d t E ( u ( t ) , u t ( t ) ) = − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 .

Subsequently, we treat the term ε ∫ 0 L u t t u d x + ε ω ∫ 0 L u x u x t d x in (3.16). By multiplying both sides of the equation in (1.1) by u and then integrating over [ 0 , L ] , we obtain

(3.18) ∫ 0 L u t t u d x + ω ∫ 0 L u x u x t d x = − ‖ u x x ‖ 2 2 − ‖ u x ‖ 2 2 − γ ∫ 0 L u u t d x + ∫ 0 L σ ( u x ) u x d x .

From (3.17) and (3.18), we note that (3.16) becomes

(3.19) V ′ ( t ) ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 − γ ε ∫ 0 L u u t d x + ε ∫ 0 L σ ( u x ) u x d x .

Using Hölder’s inequality and Sobolev’s inequality (2.2), we derive

(3.20) V ′ ( t ) ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 + ε γ ∫ 0 L ∣ u t ∣ ∣ u ∣ d x + ε ∫ 0 L σ ( u x ) u x d x ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 + ε γ ‖ u t ‖ 2 ‖ u ‖ 2 + ε ∫ 0 L σ ( u x ) u x d x ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 + ε γ ‖ u t ‖ 2 R 2 ‖ u x ‖ 2 + ε ∫ 0 L σ ( u x ) u x d x .

Then, we deal with ε ∫ 0 L σ ( u x ) u x d x in (3.20). According to Assumption ( H ) , we have

(3.21) ∫ 0 L σ ( u x ) u x d x ≤ ∫ 0 L ∣ σ ( u x ) u x ∣ d x ≤ C 0 ∫ 0 L ∣ u x ∣ p + 1 d x = C 0 ‖ u x ‖ p + 1 p + 1 .

Using (2.3), (3.21) turns to

(3.22) ∫ 0 L σ ( u x ) u x d x ≤ C 0 R p + 1 p + 1 ‖ u x x ‖ 2 p + 1 = C 0 R p + 1 p + 1 ‖ u x x ‖ 2 p − 1 ‖ u x x ‖ 2 2 .

Then, we estimate the term ‖ u x x ‖ 2 p − 1 in (3.22). By (3.11), we have

(3.23) J ( u ( t ) ) = 1 2 ‖ u x x ( t ) ‖ 2 2 + 1 2 ‖ u x ( t ) ‖ 2 2 − ∫ 0 L G ( u x ( t ) ) d x ≥ 1 2 ‖ u x x ( t ) ‖ 2 2 + 1 2 ‖ u x ( t ) ‖ 2 2 − 1 p + 1 ‖ u x x ( t ) ‖ 2 2 − 1 p + 1 ‖ u x ( t ) ‖ 2 2 = p − 1 2 ( p + 1 ) ‖ u x x ( t ) ‖ 2 2 + p − 1 2 ( p + 1 ) ‖ u x ( t ) ‖ 2 2 ≥ p − 1 2 ( p + 1 ) ‖ u x x ( t ) ‖ 2 2 .

Due to (3.17), i.e.,

d d t E ( u ( t ) , u t ( t ) ) ≤ 0 ,

we have

E ( u ( t ) , u t ( t ) ) ≤ E ( u 0 , u 1 ) .

Thus, we know that (3.23) gives

(3.24) p − 1 2 ( p + 1 ) ‖ u x x ( t ) ‖ 2 2 ≤ J ( u ( t ) ) ≤ E ( u ( t ) , u t ( t ) ) ≤ E ( u 0 , u 1 ) ,

i.e.,

(3.25) ‖ u x x ( t ) ‖ 2 p − 1 ≤ 2 ( p + 1 ) p − 1 E ( u 0 , u 1 ) p − 1 2 ,

for any t > 0 , which means that (3.22) gives

(3.26) ∫ 0 L σ ( u x ) u x d x ≤ C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 E ( u 0 , u 1 ) p − 1 2 ‖ u x x ‖ 2 2 .

Since 0 < E ( u 0 , u 1 ) ≤ ϱ d 1 is assumed, we know that (3.26) becomes

(3.27) ∫ 0 L σ ( u x ) u x d x ≤ ϱ p − 1 2 ‖ u x x ‖ 2 2 .

Substituting (3.27) into (3.20), we have

(3.28) V ′ ( t ) ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 + ε γ ‖ u t ‖ 2 R 2 ‖ u x ‖ 2 + ε ϱ p − 1 2 ‖ u x x ‖ 2 2 .

For θ > 0 , by Young’s inequality and (2.2), (3.28) becomes

(3.29) V ′ ( t ) ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 + ε γ θ R 2 ‖ u t ‖ 2 θ ‖ u x ‖ 2 + ε ϱ p − 1 2 ‖ u x x ‖ 2 2 ≤ − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε ‖ u x x ‖ 2 2 − ε ‖ u x ‖ 2 2 + ε R 2 2 2 θ 2 ‖ u t ‖ 2 2 + ε γ 2 θ 2 2 ‖ u x ‖ 2 2 + ε ϱ p − 1 2 ‖ u x x ‖ 2 2 = − ω ‖ u x t ‖ 2 2 − γ ‖ u t ‖ 2 2 + ε ‖ u t ‖ 2 2 − ε 1 − ϱ p − 1 2 ‖ u x x ‖ 2 2 − ε 1 − γ 2 θ 2 2 ‖ u x ‖ 2 2 − ε 1 − ϱ p − 1 2 ‖ u t ‖ 2 2 + ε 1 − ϱ p − 1 2 ‖ u t ‖ 2 2 + ε R 2 2 2 θ 2 ‖ u t ‖ 2 2 ≤ − ω R 2 2 + γ − ε 2 − ϱ p − 1 2 + R 2 2 2 θ 2 ‖ u t ‖ 2 2 − ε 1 − ϱ p − 1 2 ‖ u x x ‖ 2 2 − ε 1 − γ 2 θ 2 2 ‖ u x ‖ 2 2 − ε 1 − ϱ p − 1 2 ‖ u t ‖ 2 2 .

Here, due to 0 < ϱ < 1 , we know 1 − ϱ p − 1 2 > 0 . In (3.29), we choose θ > 0 such that (3.6) holds. Then, we choose ε > 0 to satisfy (3.7) and α 1 > 0 . Based on these facts and Assumption ( H ) , (3.29) turns to

(3.30) V ′ ( t ) ≤ − 2 ε min 1 − ϱ p − 1 2 , 1 − γ 2 θ 2 2 1 2 ‖ u x x ‖ 2 2 + 1 2 ‖ u x ‖ 2 2 + 1 2 ‖ u t ‖ 2 2 ≤ − 2 ε min 1 − ϱ p − 1 2 , 1 − γ 2 θ 2 2 E ( u ( t ) , u t ( t ) ) .

From (3.15), i.e., V ( t ) ≤ α 2 E ( u ( t ) , u t ( t ) ) , (3.30) becomes

(3.31) V ′ ( t ) ≤ − λ V ( t ) ,

where λ is defined by (3.2). Subsequently, using Gronwall’s inequality, we note that (3.31) gives

(3.32) V ( t ) ≤ e − λ t V ( 0 ) .

According to (3.15), (3.32), and the condition 0 < E ( u 0 , u 1 ) ≤ ϱ d 1 with 0 < ϱ < 1 , we obtain

(3.33) E ( u ( t ) , u t ( t ) ) ≤ α 2 E ( u 0 , u 1 ) α 1 e − λ t ≤ K e − λ t ,

i.e., (3.1), where K is defined by (3.3).□

Next, using Lemma 3.1, we present the results concerning the continuous dependence of the global solution on initial data, which reflect the dissipative properties of the model equation with the damping terms and the nonlinear strain term.

Theorem 3.1

(Continuous dependence of global solution on initial data) Let ω > 0 , γ ≥ 0 , and Assumption ( H ) hold. For any 0 < ϱ < 1 , suppose 0 < E ( u 0 , u 1 ) ≤ ϱ d 1 , u 0 ∈ W , 0 < E ( v 0 , v 1 ) ≤ ϱ d 1 , and v 0 ∈ W . Suppose that u and v are the global solutions to problem (1.1) with the initial data u 0 , u 1 , and v 0 , v 1 , respectively. Taking any 0 < β < 1 , one has

(3.34) D ( u ( t ) , v ( t ) ) ≤ C 1 ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) β e − C 3 t , t > 0 ,

where

C 1 ≔ 1 + C 4 e C 4 λ ( p − 1 ) λ ( p − 1 ) β 4 ( p + 1 ) K p − 1 1 − β > 0 ,

(3.35) C 2 ≔ C 0 2 κ ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ > 0 ,

C 3 ≔ λ ( 1 − β ) > 0 ,

(3.36) C 4 ≔ 4 C 0 2 ω R 6 2 R 3 ( p − 1 ) 2 ( p − 1 ) 2 2 3 2 ( p + 1 ) p − 1 K p − 1 ,

0 < δ < 1 is chosen such that

(3.37) κ ≔ λ ( 1 − δ ) − δ M > 0 ,

(3.38) M ≔ C 0 2 3 2 2 ω R 6 2 L + 2 R 3 ( p − 1 ) 3 ( p − 1 ) 2 ( p + 1 ) ϱ d 1 p − 1 3 ( p − 1 ) 2 2 3 ,

λ and K are defined in (3.2) and (3.3), respectively.

Proof

From Theorems 3.2, 4.2, and Lemma 4.3 in [16], the existence of the global solutions u and v can be ensured. To facilitate the analysis for the stability of the global solution, we use the notation η ≔ u − v in the following proof. In the case of η x t ∈ L 2 ( [ 0 , T ] ; H 0 1 ( 0 , L ) ) for any T ∈ ( 0 , + ∞ ) , by testing both sides of the equation in (1.1) with the initial data u 0 and u 1 by η t , we obtain

(3.39) ∫ 0 L u t t η t d x + ∫ 0 L u x x η x x t d x + ∫ 0 L u x η x t d x + ω ∫ 0 L u x t η x t d x + γ ∫ 0 L u t η t d x − ∫ 0 L σ ( u x ) η x t d x = 0 .

And testing both sides of the equation in (1.1) with the initial data v 0 and v 1 by η t , we have

(3.40) ∫ 0 L v t t η t d x + ∫ 0 L v x x η x x t d x + ∫ 0 L v x η x t d x + ω ∫ 0 L v x t η x t d x + γ ∫ 0 L v t η t d x − ∫ 0 L σ ( v x ) η x t d x = 0 .

By subtracting (3.40) from (3.39), we have

(3.41) ∫ 0 L η t t η t d x + ∫ 0 L η x x η x x t d x + ∫ 0 L η x η x t d x = − ω ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + ∫ 0 L ( σ ( u x ) − σ ( v x ) ) η x t d x ,

i.e.,

(3.42) d d t D ( u ( t ) , v ( t ) ) = − ω ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + ∫ 0 L ( σ ( u x ) − σ ( v x ) ) η x t d x .

For the case that η x t ∈ L 2 ( [ 0 , T ] ; H 0 1 ( 0 , L ) ) does not hold for some or any T ∈ ( 0 , + ∞ ) , by employing similar arguments to those used to prove (131) or the energy identity (14) in [9], we can see that (3.42) also holds in this case. From Assumption ( H ) , we can observe that (3.42) turns to

(3.43) d d t D ( u ( t ) , v ( t ) ) ≤ − ω ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x ,

i.e.,

(3.44) D ( u ( t ) , v ( t ) ) ≤ D ( u ( 0 ) , v ( 0 ) ) − ω ∫ 0 t ‖ η x t ‖ 2 2 d τ − γ ∫ 0 t ‖ η t ‖ 2 2 d τ + ∫ 0 t ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x d τ .

Next, in order to complete the proof, we first extend the strategy outlined in [9,14], which was originally applied to the local solution, to derive a growth estimate for D ( u ( t ) , v ( t ) ) of the global solution. Building upon this conclusion, we apply the strategy of utilizing the specific energy decay estimate provided in Lemma 3.1 to derive a more accurate result, i.e., the decay estimate for D ( u ( t ) , v ( t ) ) . Accordingly, we divide the following proof into two distinct steps, i.e., Step I (global estimate for D ( u ( t ) , v ( t ) ) ) and Step II (decay estimate for D ( u ( t ) , v ( t ) ) ).

Step I: Global estimate for D ( u ( t ) , v ( t ) ) .

We deal with the term ∫ 0 t ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x d τ in (3.44) as follows. By applying Hölder’s inequality and Young’s inequality, we obtain

(3.45) ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x ≤ C 0 ∫ 0 L ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 3 d x 1 3 ∫ 0 L ∣ u x − v x ∣ 6 d x 1 6 ∫ 0 L ∣ η x t ∣ 2 d x 1 2 = C 0 ∫ 0 L ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 3 d x 1 3 ‖ η x ‖ 6 ‖ η x t ‖ 2 = C 0 ( 2 ω ) 1 2 ∫ 0 L ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 3 d x 1 3 ‖ η x ‖ 6 ( 2 ω ) 1 2 ‖ η x t ‖ 2 ≤ C 0 2 4 ω ∫ 0 L ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 3 d x 2 3 ‖ η x ‖ 6 2 + ω ‖ η x t ‖ 2 2 .

Next, we treat the term ∫ 0 L ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 3 d x in (3.45). For any k 1 ≥ 0 , k 2 ≥ 0 , k 3 ≥ 0 , we have

( k 1 + k 2 + k 3 ) 3 ≤ ( 3 max { k 1 , k 2 , k 3 } ) 3 = 3 3 max { k 1 3 , k 2 3 , k 3 3 } ≤ 3 3 ( k 1 3 + k 2 3 + k 3 3 ) .

From the aforementioned observation, we have

(3.46) ∫ 0 L ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 3 d x ≤ 3 3 ∫ 0 L ( 1 + ∣ u x ∣ 3 ( p − 1 ) + ∣ v x ∣ 3 ( p − 1 ) ) d x .

According to (3.46), (3.45) turns to

(3.47) ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x ≤ C 0 2 3 2 4 ω ∫ 0 L ( 1 + ∣ u x ∣ 3 ( p − 1 ) + ∣ v x ∣ 3 ( p − 1 ) ) d x 2 3 ‖ η x ‖ 6 2 + ω ‖ η x t ‖ 2 2 = C 0 2 3 2 4 ω ( L + ‖ u x ‖ 3 ( p − 1 ) 3 ( p − 1 ) + ‖ v x ‖ 3 ( p − 1 ) 3 ( p − 1 ) ) 2 3 ‖ η x ‖ 6 2 + ω ‖ η x t ‖ 2 2 .

Using (2.3), we know that (3.47) becomes

(3.48) ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x ≤ C 0 2 3 2 4 ω R 6 2 ( L + R 3 ( p − 1 ) 3 ( p − 1 ) ‖ u x x ‖ 2 3 ( p − 1 ) + R 3 ( p − 1 ) 3 ( p − 1 ) ‖ v x x ‖ 2 3 ( p − 1 ) ) 2 3 ‖ η x x ‖ 2 2 + ω ‖ η x t ‖ 2 2 .

From (3.25) and the condition 0 < E ( u 0 , u 1 ) ≤ ϱ d 1 , we have

(3.49) ‖ u x x ‖ 2 2 ≤ 2 ( p + 1 ) ϱ d 1 p − 1 .

By applying a similar process to that used in deriving (3.49), we obtain

(3.50) ‖ v x x ‖ 2 2 ≤ 2 ( p + 1 ) ϱ d 1 p − 1 .

Substituting (3.49) and (3.50) into (3.48), we have

(3.51) ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x ≤ C 0 2 3 2 4 ω R 6 2 L + 2 R 3 ( p − 1 ) 3 ( p − 1 ) 2 ( p + 1 ) ϱ d 1 p − 1 3 ( p − 1 ) 2 2 3 ‖ η x x ‖ 2 2 + ω ‖ η x t ‖ 2 2 ≤ M D ( u ( t ) , v ( t ) ) + ω ‖ η x t ‖ 2 2 ,

where M is defined by (3.38). Due to (3.51), we know

(3.52) − ∫ 0 t ω ‖ η x t ‖ 2 2 d τ + ∫ 0 t ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x d τ ≤ M ∫ 0 t D ( u ( τ ) , v ( τ ) ) d τ .

Subsequently, substituting (3.52) into (3.44), we have

(3.53) D ( u ( t ) , v ( t ) ) ≤ D ( u ( 0 ) , v ( 0 ) ) + M ∫ 0 t D ( u ( τ ) , v ( τ ) ) d τ .

Then, by applying Gronwall’s inequality, we derive

(3.54) D ( u ( t ) , v ( t ) ) ≤ D ( u ( 0 ) , v ( 0 ) ) e M t .

Next, we shall utilize the energy decay estimate obtained in Lemma 3.1, which is independent of the initial data in the stable manifold, to enhance (3.54). This enhancement will yield an improved estimate that effectively captures the dissipative properties of the model equation in (1.1).

Step II: Decay estimate for D ( u ( t ) , v ( t ) ) .

We can rewrite the term ∫ 0 t ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x d τ in (3.44) as follows:

(3.55) ∫ 0 t ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x d τ = ∫ 0 t ∫ 0 L C 0 ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x d τ ︸ ≔ A 1 + ∫ 0 t ∫ 0 L C 0 ∣ η x ∣ ∣ η x t ∣ d x d τ ︸ ≔ A 2 .

By the similar process of dealing with (3.48), we can treat A 1 as

(3.56) A 1 ≤ ∫ 0 t C 0 2 2 2 2 ω R 6 2 ( R 3 ( p − 1 ) 3 ( p − 1 ) ‖ u x x ‖ 2 3 ( p − 1 ) + R 3 ( p − 1 ) 3 ( p − 1 ) ‖ v x x ‖ 2 3 ( p − 1 ) ) 2 3 ‖ η x x ‖ 2 2 + ω 2 ‖ η x t ‖ 2 2 d τ = ∫ 0 t 2 C 0 2 ω R 6 2 R 3 ( p − 1 ) 2 ( p − 1 ) ( ‖ u x x ‖ 2 3 ( p − 1 ) + ‖ v x x ‖ 2 3 ( p − 1 ) ) 2 3 ‖ η x x ‖ 2 2 + ω 2 ‖ η x t ‖ 2 2 d τ .

Due to (3.23) and Lemma 3.1, we know

(3.57) 1 2 ‖ u t ‖ 2 2 + p − 1 2 ( p + 1 ) ‖ u x x ‖ 2 2 + p − 1 2 ( p + 1 ) ‖ u x ‖ 2 2 ≤ E ( u ( t ) , u t ( t ) ) ≤ K e − λ t .

By the similar process of deriving (3.57), we have

(3.58) 1 2 ‖ v t ‖ 2 2 + p − 1 2 ( p + 1 ) ‖ v x x ‖ 2 2 + p − 1 2 ( p + 1 ) ‖ v x ‖ 2 2 ≤ E ( v ( t ) , v t ( t ) ) ≤ K e − λ t .

From (3.57) and (3.58), we have

(3.59) ‖ u x x ‖ 2 3 ( p − 1 ) ≤ 2 ( p + 1 ) p − 1 K 3 2 ( p − 1 ) e − 3 2 λ ( p − 1 ) t

and

(3.60) ‖ v x x ‖ 2 3 ( p − 1 ) ≤ 2 ( p + 1 ) p − 1 K 3 2 ( p − 1 ) e − 3 2 λ ( p − 1 ) t ,

which means

(3.61) ( ‖ u x x ‖ 2 3 ( p − 1 ) + ‖ v x x ‖ 2 3 ( p − 1 ) ) 2 3 ≤ 2 2 3 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) t .

By substituting (3.61) into (3.56), we obtain

(3.62) A 1 ≤ ∫ 0 t 2 C 0 2 ω R 6 2 R 3 ( p − 1 ) 2 ( p − 1 ) 2 2 3 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) τ ‖ η x x ‖ 2 2 d τ + ∫ 0 t ω 2 ‖ η x t ‖ 2 2 d τ .

Next, we deal with A 2 in (3.55). According to (3.54), we know

(3.63) ‖ η x ‖ 2 ≤ ( 2 D ( u ( 0 ) , v ( 0 ) ) e M t ) 1 2 .

Take 0 < δ < 1 satisfying (3.37). Then, (3.63) becomes

(3.64) ‖ η x ‖ 2 δ ≤ ( 2 D ( u ( 0 ) , v ( 0 ) ) ) δ 2 e δ M t 2 .

Meanwhile, combining (3.57) and (3.58), we also have

(3.65) ‖ η x ‖ 2 ≤ ‖ u x ‖ 2 + ‖ v x ‖ 2 ≤ 2 2 ( p + 1 ) K p − 1 1 2 e − λ 2 t ,

i.e.,

(3.66) ‖ η x ‖ 2 1 − δ ≤ 2 1 − δ 2 ( p + 1 ) K p − 1 1 − δ 2 e − λ ( 1 − δ ) 2 t .

Using Cauchy-Schwarz inequality, we have

(3.67) A 2 ≤ ∫ 0 t C 0 ‖ η x ‖ 2 ‖ η x t ‖ 2 d τ ≤ ∫ 0 t C 0 2 2 ω ‖ η x ‖ 2 2 d τ + ∫ 0 t ω 2 ‖ η x t ‖ 2 2 d τ = ∫ 0 t C 0 2 2 ω ‖ η x ‖ 2 2 δ ‖ η x ‖ 2 2 ( 1 − δ ) d τ + ∫ 0 t ω 2 ‖ η x t ‖ 2 2 d τ .

From (3.64) and (3.66), we know that (3.67) becomes

(3.68) A 2 ≤ ∫ 0 t C 0 2 ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ D ( u ( 0 ) , v ( 0 ) ) δ e − ( λ ( 1 − δ ) − δ M ) τ d τ + ∫ 0 t ω 2 ‖ η x t ‖ 2 2 d τ .

Substituting (3.62) and (3.68) into (3.55), we have

(3.69) ∫ 0 t ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ u x − v x ∣ ∣ η x t ∣ d x d τ ≤ ∫ 0 t 2 C 0 2 ω R 6 2 R 3 ( p − 1 ) 2 ( p − 1 ) 2 2 3 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) τ ‖ η x x ‖ 2 2 d τ + ω ∫ 0 t ‖ η x t ‖ 2 2 d τ + ∫ 0 t C 0 2 ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ D ( u ( 0 ) , v ( 0 ) ) δ e − ( λ ( 1 − δ ) − δ M ) τ d τ ≤ ∫ 0 t 4 C 0 2 ω R 6 2 R 3 ( p − 1 ) 2 ( p − 1 ) 2 2 3 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) τ D ( u ( τ ) , v ( τ ) ) d τ + ω ∫ 0 t ‖ η x t ‖ 2 2 d τ + ∫ 0 t C 0 2 ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ D ( u ( 0 ) , v ( 0 ) ) δ e − ( λ ( 1 − δ ) − δ M ) τ d τ = ∫ 0 t C 4 e − λ ( p − 1 ) τ D ( u ( τ ) , v ( τ ) ) d τ + ω ∫ 0 t ‖ η x t ‖ 2 2 d τ + ∫ 0 t C 0 2 ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ D ( u ( 0 ) , v ( 0 ) ) δ e − κ τ d τ ,

where C 4 and κ are defined in (3.36) and (3.37), respectively. Then, by substituting (3.69) into (3.44), we obtain

(3.70) D ( u ( t ) , v ( t ) ) ≤ D ( u ( 0 ) , v ( 0 ) ) + ∫ 0 t C 4 e − λ ( p − 1 ) τ D ( u ( τ ) , v ( τ ) ) d τ + D ( u ( 0 ) , v ( 0 ) ) δ ∫ 0 t C 0 2 ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ e − κ τ d τ .

Here, by direct calculation, we derive

(3.71) ∫ 0 t C 0 2 ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ e − κ τ d τ = C 0 2 κ ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ − C 0 2 κ ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ e − κ t ≤ C 0 2 κ ω 2 2 − 2 δ ( p + 1 ) K p − 1 1 − δ .

Subsequently, (3.71) implies that (3.70) turns to

(3.72) D ( u ( t ) , v ( t ) ) ≤ D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ + ∫ 0 t C 4 e − λ ( p − 1 ) τ D ( u ( τ ) , v ( τ ) ) d τ ,

i.e.,

(3.73) e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) ≤ e − λ ( p − 1 ) t ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) + C 4 e − λ ( p − 1 ) t ∫ 0 t e − λ ( p − 1 ) τ D ( u ( τ ) , v ( τ ) ) d τ ,

where C 2 is defined by (3.35). Next, we define

(3.74) F ( t ) ≔ ∫ 0 t e − λ ( p − 1 ) τ D ( u ( τ ) , v ( τ ) ) d τ .

Thus, we can rewrite (3.73) as

(3.75) d d t F ( t ) ≤ e − λ ( p − 1 ) t ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) + C 4 e − λ ( p − 1 ) t F ( t ) .

By applying Gronwall’s inequality, we note that (3.75) gives

F ( t ) ≤ ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) e C 4 ∫ 0 t e − λ ( p − 1 ) τ d τ ∫ 0 t e − λ ( p − 1 ) τ d τ = ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) e C 4 λ ( p − 1 ) e − C 4 e − λ ( p − 1 ) t λ ( p − 1 ) 1 − e − λ ( p − 1 ) t λ ( p − 1 ) ≤ ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) e C 4 λ ( p − 1 ) λ ( p − 1 ) ,

which implies that (3.72) becomes

(3.76) D ( u ( t ) , v ( t ) ) ≤ ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) 1 + C 4 e C 4 λ ( p − 1 ) λ ( p − 1 ) .

According to (3.76), for any 0 < β < 1 , we have

(3.77) D ( u ( t ) , v ( t ) ) = D ( u ( t ) , v ( t ) ) β D ( u ( t ) , v ( t ) ) 1 − β ≤ ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) β 1 + C 4 e C 4 λ ( p − 1 ) λ ( p − 1 ) β D ( u ( t ) , v ( t ) ) 1 − β .

Here, by applying Young’s inequality, we know

(3.78) D ( u ( t ) , v ( t ) ) 1 − β = 1 2 ‖ u t − v t ‖ 2 2 + 1 2 ‖ u x x − v x x ‖ 2 2 + 1 2 ‖ u x − v x ‖ 2 2 1 − β ≤ 1 2 ( ‖ u t ‖ 2 + ‖ v t ‖ 2 ) 2 + 1 2 ( ‖ u x x ‖ 2 + ‖ v x x ‖ 2 ) 2 + 1 2 ( ‖ u x ‖ 2 + ‖ v x ‖ 2 ) 2 1 − β = 1 2 ‖ u t ‖ 2 2 + ‖ u t ‖ 2 ‖ v t ‖ 2 + 1 2 ‖ v t ‖ 2 2 + 1 2 ‖ u x x ‖ 2 2 + ‖ u x x ‖ 2 ‖ v x x ‖ 2 + 1 2 ‖ v x x ‖ 2 2 + 1 2 ‖ u x ‖ 2 2 + ‖ u x ‖ 2 ‖ v x ‖ 2 + 1 2 ‖ v x ‖ 2 2 1 − β ≤ ( ‖ u t ‖ 2 2 + ‖ v t ‖ 2 2 + ‖ u x x ‖ 2 2 + ‖ v x x ‖ 2 2 + ‖ u x ‖ 2 2 + ‖ v x ‖ 2 2 ) 1 − β .

From (3.57) and (3.58), we have

(3.79) p − 1 2 ( p + 1 ) ( ‖ u t ‖ 2 2 + ‖ u x x ‖ 2 2 + ‖ u x ‖ 2 2 ) ≤ K e − λ t

and

(3.80) p − 1 2 ( p + 1 ) ( ‖ v t ‖ 2 2 + ‖ v x x ‖ 2 2 + ‖ v x ‖ 2 2 ) ≤ K e − λ t ,

respectively. By substituting (3.79) and (3.80) into (3.78), we obtain

D ( u ( t ) , v ( t ) ) 1 − β ≤ 4 ( p + 1 ) K p − 1 1 − β e − λ ( 1 − β ) t ,

which means that (3.77) gives (3.34).□

Remark 3.1

(Advantage of continuous dependence conclusion in Theorem 3.1 given by utilizing some energy decay estimate) For the damped fourth-order wave equation with the general nonlinear strain term in problem (1.1), we initially apply the strategy outlined in [9,14] to give growth estimate (3.54) of the distance D ( u ( t ) , v ( t ) ) between the global solutions u and v corresponding to the initial data u 0 , u 1 and v 0 , v 1 , respectively, in Step I of the proof of Theorem 3.1, which is also a conclusion of continuous dependence of the global solution on initial data. Building on this initial result, we then apply the present paper’s strategy that involves utilizing the energy dissipation estimate independent of the initial data in the stable manifold given in Lemma 3.1 to derive a stronger one, i.e., the decay estimate (3.34) of the distance D ( u ( t ) , v ( t ) ) , in Step II of the proof of Theorem 3.1, which effectively captures the dissipative characteristics of the model equation, thereby showing a deeper understanding for the solution’s behavior. The presence of the decay function e − C 3 t ( C 3 > 0 ) in (3.34) ensures that the distance D ( u ( t ) , v ( t ) ) between global solutions u and v converges to zero at an accelerating rate over time t . In other words, to ensure that the distance D ( u ( t ) , v ( t ) ) between global solutions becomes sufficiently small, it is only necessary to impose progressively more relaxed conditions on the distance of the initial data D ( u ( 0 ) , v ( 0 ) ) as time t progresses. In contrast, the growth function e M t ( M > 0 ) renders (3.54) a growth estimate of the distance D ( u ( t ) , v ( t ) ) . Consequently, as time t progresses, in order to ensure that D ( u ( t ) , v ( t ) ) remains small enough, it is necessary to impose increasingly stronger conditions on D ( u ( 0 ) , v ( 0 ) ) . The conclusion (3.34) obtained in Theorem 3.1 not only is more precise but also effectively captures the dissipative nature of the damped fourth-order wave equation with the general nonlinear strain term in problem (1.1). More precisely, due to the presence of damping terms ω u x x t and γ u t in the equation of (1.1), the global solutions u and v corresponding to different initial data u 0 , u 1 and v 0 , v 1 , respectively, exhibit decay characteristics. A natural conjecture is that as time t progresses, the distance between solutions u and v should gradually diminish, which is indeed corroborated by conclusion (3.34). In contrast, the growth estimate (3.54) fails to reflect this phenomenon. To more effectively highlight the advantages of conclusion (3.34), which is derived by utilizing the energy dissipation estimate independent of the initial data in stable manifold, we shall present a diagrammatic sketch in Figure 1 to provide a clear comparison between conclusion (3.54) obtained without applying the energy decay characteristics and conclusion (3.34). In Figure 1, the red growth curve is used to depict the growth estimate function

D ( u ( 0 ) , v ( 0 ) ) e M t ,

for the distance D ( u ( t ) , v ( t ) ) between the global solutions u and v in (3.54). Here, to enhance the clarity of the observation, we assume D ( u ( 0 ) , v ( 0 ) ) = M = 1 2 . And we use the blue decay curve to indicate the decay estimate function

C 1 ( D ( u ( 0 ) , v ( 0 ) ) + C 2 D ( u ( 0 ) , v ( 0 ) ) δ ) β e − C 3 t ,

for the distance D ( u ( t ) , v ( t ) ) between u and v in (3.34). Similarly, we assume D ( u ( 0 ) , v ( 0 ) ) = 1 2 , C 1 = C 2 = 1 , and C 3 = β = δ = 1 2 to facilitate observation.

$Figure 1 Estimate functions for the distance D ( u ( t ) , v ( t ) ) {\mathbb{D}}\left(u\left(t),v\left(t)) between global solutions u u and v v in conclusions (3.34) and (3.54).$

Figure 1

Estimate functions for the distance D ( u ( t ) , v ( t ) ) between global solutions u and v in conclusions (3.34) and (3.54).

Next, with the help of Lemma 3.1, we provide another version of the stability of the global solution, i.e., its continuous dependence on the initial data, using another strategy different from Theorem 3.1, which can also reflect the dissipative characteristics of the model equation with the damping and the nonlinear strain term.

Theorem 3.2

(Another version of continuous dependence of global solution on initial data) Let ω > 0 , γ ≥ 0 , and Assumption ( H ) hold. Taking any 0 < ϱ < 1 , suppose 0 < E ( u 0 , u 1 ) ≤ ϱ d 1 , u 0 ∈ W , 0 < E ( v 0 , v 1 ) ≤ ϱ d 1 , and v 0 ∈ W . Assume that u and v are the global solutions to (1.1) with the initial data u 0 , u 1 , and v 0 , v 1 , respectively. Then, there holds

(3.81) D ( u ( t ) , v ( t ) ) ≤ C 5 D ( u ( 0 ) , v ( 0 ) ) e − C 6 t , t > 0 ,

where

C 5 ≔ β 2 β 1 e Y > 0 ,

C 6 ≔ 2 ς − ς C 0 − C 0 2 2 ω 1 β 2 > 0 ,

(3.82) β 1 ≔ 1 − ς max { ∣ R 2 2 + γ R 2 2 − ω ∣ , 1 } ,

(3.83) β 2 ≔ 1 + ς max { R 2 2 + ω + γ R 2 2 , 1 } ,

ς > 0 is chosen such that β 1 > 0 and

(3.84) ς − γ 2 − ω 8 R 2 2 < 0 ,

C 0 > 0 in Assumption ( H ) satisfies

(3.85) C 0 2 2 ω + ς C 0 < ς ,

(3.86) Y ≔ M 1 λ ( p − 1 ) + 2 M 2 λ ( p − 1 ) ,

(3.87) M 1 ≔ 8 C 0 ω β 1 R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 ,

(3.88) M 2 ≔ 4 ς C 0 β 1 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 ,

λ and K are defined in (3.2) and (3.3), respectively.

Proof

The existence of the global solutions u and v is guaranteed by Theorems 3.2, 4.2, and Lemma 4.3 in [16]. To facilitate the analysis, we also use the notation η = u − v . We define

P ( t ) ≔ D ( u ( t ) , v ( t ) ) + ς ( u ( t ) − v ( t ) , u t ( t ) − v t ( t ) ) + ς ω 2 ‖ u x ( t ) − v x ( t ) ‖ 2 2 + ς γ 2 ‖ u ( t ) − v ( t ) ‖ 2 2 ,

where the constant ς > 0 will be chosen later. We first prove that the constants β 1 and β 2 , defined by (3.82) and (3.83), respectively, satisfy

(3.89) β 1 D ( u ( t ) , v ( t ) ) ≤ P ( t ) ≤ β 2 D ( u ( t ) , v ( t ) ) ,

where ς > 0 in (3.82) is chosen such that β 1 > 0 . Applying Hölder’s inequality, Young’s inequality, and Sobolev’s inequality (2.2), we derive

(3.90) P ( t ) ≤ D ( u ( t ) , v ( t ) ) + ς ∣ ( η , η t ) ∣ + ς ω 2 ‖ η x ‖ 2 2 + ς γ 2 ‖ η ‖ 2 2 ≤ D ( u ( t ) , v ( t ) ) + ς ‖ η ‖ 2 ‖ η t ‖ 2 + ς ω 2 ‖ η x ‖ 2 2 + ς γ R 2 2 2 ‖ η x ‖ 2 2 ≤ D ( u ( t ) , v ( t ) ) + ς R 2 2 2 ‖ η x ‖ 2 2 + ς 2 ‖ η t ‖ 2 2 + ς ω 2 ‖ η x ‖ 2 2 + ς γ R 2 2 2 ‖ η x ‖ 2 2 = D ( u ( t ) , v ( t ) ) + ς R 2 2 2 + γ R 2 2 2 + ω 2 ‖ η x ‖ 2 2 + ς 2 ‖ η t ‖ 2 2 ≤ D ( u ( t ) , v ( t ) ) + ς max { R 2 2 + ω + γ R 2 2 , 1 } D ( u ( t ) , v ( t ) ) .

Similarly, we have

(3.91) P ( t ) ≥ D ( u ( t ) , v ( t ) ) − ς ∣ ( η , η t ) ∣ + ς ω 2 ‖ η x ‖ 2 2 − ς γ 2 ‖ η ‖ 2 2 ≥ D ( u ( t ) , v ( t ) ) − ς ‖ η ‖ 2 ‖ η t ‖ 2 + ς ω 2 ‖ η x ‖ 2 2 − ς γ R 2 2 2 ‖ η x ‖ 2 2 ≥ D ( u ( t ) , v ( t ) ) − ς R 2 2 2 ‖ η x ‖ 2 2 − ς 2 ‖ η t ‖ 2 2 + ς ω 2 ‖ η x ‖ 2 2 − ς γ R 2 2 2 ‖ η x ‖ 2 2 = D ( u ( t ) , v ( t ) ) − ς R 2 2 2 + γ R 2 2 2 − ω 2 ‖ η x ‖ 2 2 − ς 2 ‖ η t ‖ 2 2 ≥ D ( u ( t ) , v ( t ) ) − ς max { ∣ R 2 2 + γ R 2 2 − ω ∣ , 1 } D ( u ( t ) , v ( t ) ) .

Here, ς > 0 is selected such that

1 − ς max { ∣ R 2 2 + γ R 2 2 − ω ∣ , 1 } > 0 ,

in (3.91). From (3.90) and (3.91), we see that (3.89) holds. By differentiating P ( t ) with respect to t , we have

(3.92) d d t P ( t ) = d d t D ( u ( t ) , v ( t ) ) + ς ‖ η t ‖ 2 2 + ς ∫ 0 L η t t η d x + ς ω ∫ 0 L η x η x t d x + ς γ ∫ 0 L η η t d x .

Here, we can use (3.43) to estimate d d t D ( u ( t ) , v ( t ) ) in (3.92). Subsequently, we deal with the terms

ς ∫ 0 L η t t η d x + ς ω ∫ 0 L η x η x t d x + ς γ ∫ 0 L η η t d x ,

in (3.92). Testing both sides of the equation in (1.1) with the initial data u 0 and u 1 by η and integrating over [ 0 , L ] , we obtain

(3.93) ∫ 0 L u t t η d x + ∫ 0 L u x x η x x d x + ∫ 0 L u x η x d x + ω ∫ 0 L u x t η x d x + γ ∫ 0 L u t η d x − ∫ 0 L σ ( u x ) η x d x = 0 .

Then, by testing both sides of the equation in (1.1) with the initial data v 0 and v 1 by η and integrating over [ 0 , L ] , we know

(3.94) ∫ 0 L v t t η d x + ∫ 0 L v x x η x x d x + ∫ 0 L v x η x d x + ω ∫ 0 L v x t η x d x + γ ∫ 0 L v t η d x − ∫ 0 L σ ( v x ) η x d x = 0 .

By subtracting (3.94) from (3.93) and using Assumption ( H ) , we have

(3.95) ∫ 0 L η t t η d x + ω ∫ 0 L η x t η x d x + γ ∫ 0 L η t η d x = − ‖ η x x ‖ 2 2 − ‖ η x ‖ 2 2 + ∫ 0 L ( σ ( u x ) − σ ( v x ) ) η x d x ≤ − ‖ η x x ‖ 2 2 − ‖ η x ‖ 2 2 + ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x ∣ d x .

Subsequently, by substituting (3.43) and (3.95) into (3.92), we derive

(3.96) d d t P ( t ) ≤ − ω ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x + ς ‖ η t ‖ 2 2 − ς ‖ η x x ‖ 2 2 − ς ‖ η x ‖ 2 2 + ς ∫ 0 L C 0 ( 1 + ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x ∣ d x = − ω ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + C 0 ∫ 0 L ∣ η x ∣ ∣ η x t ∣ d x + C 0 ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x + ς ‖ η t ‖ 2 2 − ς ‖ η x x ‖ 2 2 − ς ‖ η x ‖ 2 2 + ς C 0 ‖ η x ‖ 2 2 + ς ∫ 0 L C 0 ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x ∣ d x .

Next, we estimate the fourth and ninth terms on the right-hand side of (3.96). For the fourth term, by applying Hölder’s inequality, Young’s inequality, and (2.3), we have

(3.97) ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x ≤ ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) 2 p + 2 p − 1 d x p − 1 2 p + 2 ∫ 0 L ∣ η x ∣ p + 1 d x 1 p + 1 ∫ 0 L ∣ η x t ∣ 2 d x 1 2 ≤ ∫ 0 L 2 p + 3 p − 1 ( ∣ u x ∣ 2 p + 2 + ∣ v x ∣ 2 p + 2 ) d x p − 1 2 p + 2 ‖ η x ‖ p + 1 ‖ η x t ‖ 2 = 2 p + 3 2 p + 2 ( ‖ u x ‖ 2 p + 2 2 p + 2 + ‖ v x ‖ 2 p + 2 2 p + 2 ) p − 1 2 p + 2 ‖ η x ‖ p + 1 ‖ η x t ‖ 2 ≤ 2 1 2 ω 1 2 2 p + 3 2 p + 2 R p + 1 R 2 p + 2 p − 1 ( ‖ u x x ‖ 2 2 p + 2 + ‖ v x x ‖ 2 2 p + 2 ) p − 1 2 p + 2 ‖ η x x ‖ 2 ω 2 1 2 ‖ η x t ‖ 2 ≤ 1 ω 2 p + 3 p + 1 R p + 1 2 R 2 p + 2 2 ( p − 1 ) ( ‖ u x x ‖ 2 2 p + 2 + ‖ v x x ‖ 2 2 p + 2 ) p − 1 p + 1 ‖ η x x ‖ 2 2 + ω 4 ‖ η x t ‖ 2 2 ≤ 2 ω 2 p + 3 p + 1 R p + 1 2 R 2 p + 2 2 ( p − 1 ) ( ‖ u x x ‖ 2 2 p + 2 + ‖ v x x ‖ 2 2 p + 2 ) p − 1 p + 1 D ( u ( t ) , v ( t ) ) + ω 4 ‖ η x t ‖ 2 2 .

Then, substituting (3.59) and (3.60) into (3.97), we have

(3.98) ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x t ∣ d x ≤ 2 ω 2 p + 3 p + 1 R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 p − 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) + ω 4 ‖ η x t ‖ 2 2 .

Next, we treat the ninth term on the right-hand side of (3.96). By Hölder’s inequality and (2.3), we obtain

(3.99) ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x ∣ d x ≤ ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) p + 1 p − 1 d x p − 1 p + 1 ∫ 0 L ∣ η x ∣ p + 1 d x 1 p + 1 ∫ 0 L ∣ η x ∣ p + 1 d x 1 p + 1 ≤ ∫ 0 L 2 2 p − 1 ( ∣ u x ∣ p + 1 + ∣ v x ∣ p + 1 ) d x p − 1 p + 1 ‖ η x ‖ p + 1 ‖ η x ‖ p + 1 = 2 2 p + 1 ( ‖ u x ‖ p + 1 p + 1 + ‖ v x ‖ p + 1 p + 1 ) p − 1 p + 1 ‖ η x ‖ p + 1 2 ≤ 2 2 p + 1 R p + 1 p + 1 ( ‖ u x x ‖ 2 p + 1 + ‖ v x x ‖ 2 p + 1 ) p − 1 p + 1 ‖ η x x ‖ 2 2 ≤ 2 p + 3 p + 1 R p + 1 p + 1 ( ‖ u x x ‖ 2 p + 1 + ‖ v x x ‖ 2 p + 1 ) p − 1 p + 1 D ( u ( t ) , v ( t ) ) .

Substituting (3.59) and (3.60) into (3.99), we have

(3.100) ∫ 0 L ( ∣ u x ∣ p − 1 + ∣ v x ∣ p − 1 ) ∣ η x ∣ ∣ η x ∣ d x ≤ 2 p + 3 p + 1 R p + 1 p + 1 2 p − 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) .

Subsequently, by substituting (3.98) and (3.100) into (3.96), we derive

(3.101) d d t P ( t ) ≤ − 3 ω 4 ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + C 0 ∫ 0 L ∣ η x ∣ ∣ η x t ∣ d x + ς ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς ‖ η x x ‖ 2 2 − ς ‖ η x ‖ 2 2 + ς C 0 ‖ η x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) .

Using Hölder’s inequality, Young’s inequality and (2.2), (3.101) turns to

(3.102) d d t P ( t ) ≤ − 3 ω 4 ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + C 0 ‖ η x ‖ 2 ‖ η x t ‖ 2 + ς ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 × e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς ‖ η x x ‖ 2 2 − ς ‖ η x ‖ 2 2 + ς C 0 ‖ η x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) = − 3 ω 4 ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + C 0 1 ω 1 2 ‖ η x ‖ 2 ω 1 2 ‖ η x t ‖ 2 + ς ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 × e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς ‖ η x x ‖ 2 2 − ς ‖ η x ‖ 2 2 + ς C 0 ‖ η x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) ≤ − 3 ω 4 ‖ η x t ‖ 2 2 − γ ‖ η t ‖ 2 2 + C 0 2 1 2 ω ‖ η x ‖ 2 2 + ω 2 ‖ η x t ‖ 2 2 + ς ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 × e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς ‖ η x x ‖ 2 2 − ς ‖ η x ‖ 2 2 + ς C 0 ‖ η x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 × e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) ≤ − ω 4 R 2 2 + γ − 2 ς + ς C 0 + C 0 2 2 ω ‖ η t ‖ 2 2 − ς − ς C 0 − C 0 2 2 ω ‖ η x ‖ 2 2 − ς − ς C 0 − C 0 2 2 ω ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς ‖ η x x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) ≤ − ω 4 R 2 2 + γ − 2 ς + ς C 0 + C 0 2 2 ω ‖ η t ‖ 2 2 − ς − ς C 0 − C 0 2 2 ω ‖ η x ‖ 2 2 − ς − ς C 0 − C 0 2 2 ω ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς − ς C 0 − C 0 2 2 ω ‖ η x x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) .

In (3.102), ς > 0 is chosen such that (3.84) and β 1 > 0 hold. And C 0 > 0 satisfies (3.85). Based on these preparations and (3.89), we know that (3.102) gives

(3.103) d d t P ( t ) ≤ − ς − ς C 0 − C 0 2 2 ω ‖ η x ‖ 2 2 − ς − ς C 0 − C 0 2 2 ω ‖ η t ‖ 2 2 + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) − ς − ς C 0 − C 0 2 2 ω ‖ η x x ‖ 2 2 + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) = − 2 ς − ς C 0 − C 0 2 2 ω D ( u ( t ) , v ( t ) ) + 8 C 0 ω R p + 1 2 R 2 p + 2 2 ( p − 1 ) 2 ( p + 1 ) p − 1 K p − 1 × e − λ ( p − 1 ) t D ( u ( t ) , v ( t ) ) + 4 ς C 0 R p + 1 p + 1 2 ( p + 1 ) p − 1 K p − 1 2 e − λ ( p − 1 ) 2 t D ( u ( t ) , v ( t ) ) ≤ − 2 ς − ς C 0 − C 0 2 2 ω β 2 P ( t ) + Q ( t ) P ( t ) ,

where

Q ( t ) ≔ M 1 e − λ ( p − 1 ) t + M 2 e − λ ( p − 1 ) 2 t ,

M 1 and M 2 are defined by (3.87) and (3.88), respectively. We note that (3.103) yields

d d t e − ∫ 0 t Q ( τ ) d τ + ς − ς C 0 − C 0 2 2 ω 2 β 2 t P ( t ) ≤ 0 ,

which means

e − ∫ 0 t Q ( τ ) d τ + ς − ς C 0 − C 0 2 2 ω 2 β 2 t P ( t ) ≤ P ( 0 ) ,

i.e.,

(3.104) P ( t ) ≤ P ( 0 ) e ∫ 0 t Q ( τ ) d τ − ς − ς C 0 − C 0 2 2 ω 2 β 2 t .

Here,

∫ 0 t Q ( τ ) d τ = M 1 λ ( p − 1 ) ( 1 − e − λ ( p − 1 ) t ) + 2 M 2 λ ( p − 1 ) 1 − e − λ ( p − 1 ) 2 t ≤ Y ,

and Y is defined in (3.86). Based on (3.89), we note that (3.104) implies

β 1 D ( u ( t ) , v ( t ) ) ≤ β 2 e Y D ( u ( 0 ) , v ( 0 ) ) e − ς − ς C 0 − C 0 2 2 ω 2 β 2 t ,

which gives (3.81).□

Acknowledgments

The authors would like to appreciate the referees for taking their valuable time to provide constructive and insightful comments and suggestions, which markedly enhanced the quality of this article.

Funding information: Authors state no funding involved.
Author contributions: All authors contributed to the research and read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Data availability statement: This research did not involve the use of any data.

References

[1] G. Andrews and J. M. Ball, Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity, J. Differential Equations 44 (1982), 306–341, https://doi.org/10.1016/0022-0396(82)90019-5. Search in Google Scholar

[2] L. J. An, Loss of hyperbolicity in elastic-plastic material at finite strains, SIAM J. Appl. Math. 53 (1993), 621–654, https://doi.org/10.1137/015303. Search in Google Scholar

[3] L. J. An and A. Peirce, A weakly nonlinear analysis of elastoplastic-microstructure models, SIAM J. Appl. Math. 55 (1995), 136–155, https://doi.org/10.1137/S0036139993255327. Search in Google Scholar

[4] G. W. Chen and Z. J. Yang, Existence and non-existence of global solutions for a class of non-linear wave equations, Math. Methods Appl. Sci. 23 (2000), 615–631, https://doi.org/10.1002/(SICI)1099-1476(20000510)23:73.3.CO;2-5. Search in Google Scholar

[5] Y. X. Chen, V. D. Rădulescu, and R. Z. Xu, High energy blowup and blowup time for a class of semilinear parabolic equations with singular potential on manifolds with conical singularities, Commun. Math. Sci. 21 (2023), 25–63, https://dx.doi.org/10.4310/CMS.2023.v21.n1.a2. Search in Google Scholar

[6] Y. X. Chen, Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity, Commun. Anal. Mech. 15 (2023), 658–694, https://doi.org/10.3934/cam.2023033. Search in Google Scholar

[7] I. Chueshov and I. Lasiecka, On global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity, Commun. Partial Differ. Equ. 36 (2011), 67–99, https://doi.org/10.1080/03605302.2010.484472. Search in Google Scholar

[8] J. A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal. 63 (2005), e331–e343, https://doi.org/10.1016/j.na.2005.02.108. Search in Google Scholar

[9] M. M. Freitas, M. L. Santos, and J. A. Langa, Porous elastic system with nonlinear damping and sources terms, J. Differential Equations 264 (2018), 2970–3051, https://doi.org/10.1016/j.jde.2017.11.006. Search in Google Scholar

[10] T. F. Feng, Y. Dong, K. L. Zhang, and Y. Zhu, Global existence and blow-up to coupled fourth-order parabolic systems arising from modeling epitaxial thin film growth, Commun. Anal. Mech. 17 (2025), 263–289, https://doi.org/10.3934/cam.2025011. Search in Google Scholar

[11] F. Gazzola and M. Squassina, Global solutions and finite-time blow up for damped semilinear wave equations, Ann. Inst. Henri Poincaré Anal. Non Linéaire 23 (2006), 185–207, https://doi.org/10.1016/j.anihpc.2005.02.007. Search in Google Scholar

[12] J. B. Han, R. Z. Xu, and C. Yang, Continuous dependence on initial data and high energy blowup time estimate for porous elastic system, Commun. Anal. Mech. 15 (2023), 214–244, https://doi.org/10.3934/cam.2023012. Search in Google Scholar

[13] J. B. Han, K. Y. Wang, R. Z. Xu, and C. Yang, Global quantitative stability of wave equations with strong and weak dampings, J. Differential Equations 390 (2024), 228–344, https://doi.org/10.1016/j.jde.2024.01.033. Search in Google Scholar

[14] M. A. Jorge Silva and T. F. Ma, On a viscoelastic plate equation with history setting and perturbation of p-Laplacian type, IMA J. Appl. Math. 78 (2013), 1130–1146, https://doi.org/10.1093/imamat/hxs011. Search in Google Scholar

[15] J. E. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics, 10. Society for Industrial and Applied Mathematics, SIAM, Philadelphia, PA, 1989, https://doi.org/10.1137/1.9781611970821.Search in Google Scholar

[16] W. Lian, V. D. Rădulescu, R. Z. Xu, Y. B. Yang, and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var. 14 (2021), 589–611, https://doi.org/10.1515/acv-2019-0039. Search in Google Scholar

[17] W. Lian, J. Wang, and R. Z. Xu, Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations 269 (2020), 4914–4959, https://doi.org/10.1016/j.jde.2020.03.047. Search in Google Scholar

[18] Y. B. Luo, R. Z. Xu, and C. Yang, Global well-posedness for a class of semilinear hyperbolic equations with singular potentials on manifolds with conical singularities, Calc. Var. Partial Differential Equations 61 (2022), 210, https://doi.org/10.1007/s00526-022-02316-2. Search in Google Scholar

[19] Q. Lin and R. Z. Xu, Global well-posedness of the variable-order fractional wave equation with variable exponent nonlinearity, J. Lond. Math. Soc. 111 (2025), no. 2, e70091, https://doi.org/10.1112/jlms.70091. Search in Google Scholar

[20] R. L. Pego, Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability, Arch. Ration. Mech. Anal. 97 (1987), 353–394, https://doi.org/10.1007/BF00280411. Search in Google Scholar

[21] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math. 22 (1975), 273–303, https://doi.org/10.1007/BF02761595. Search in Google Scholar

[22] Y. Pang, X. T. Qiu, R. Z. Xu, and Y. B. Yang, The Cauchy problem for general nonlinear wave equations with doubly dispersive, Commun. Anal. Mech. 16 (2024), 416–430, https://doi.org/10.3934/cam.2024019. Search in Google Scholar

[23] R. Racke and C. Y. Shang, Global attractors for nonlinear beam equations, Proc. Roy. Soc. Edinburgh Sect. A 142 (2012), 1087–1107, https://doi.org/10.1017/S030821051000168X Search in Google Scholar

[24] D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Ration. Mech. Anal. 30 (1968), 148–172, https://doi.org/10.1007/BF00250942. Search in Google Scholar

[25] Y. J. Wang and Y. F. Wang, On the initial-boundary problem for fourth order wave equations with damping, strain and source terms, J. Math. Anal. Appl. 405 (2013), 116–127, https://doi.org/10.1016/j.jmaa.2013.03.060. Search in Google Scholar

[26] X. C. Wang and R. Z. Xu, Global existence and finite-time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal. 10 (2021), 261–288, https://doi.org/10.1515/anona-2020-0141.Search in Google Scholar

[27] R. Z. Xu and J. Su, Global existence and finite-time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal. 264 (2013), 2732–2763, https://doi.org/10.1016/j.jfa.2013.03.010. Search in Google Scholar

[28] R. Z. Xu, W. Lian, and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math. 63 (2020), 321–356, https://doi.org/10.1007/s11425-017-9280-x. Search in Google Scholar

[29] H. Y. Xu, Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials, Commun. Anal. Mech. 15 (2023), 132–161, https://doi.org/10.3934/cam.2023008. Search in Google Scholar

[30] Z. J. Yang, Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations 187 (2003), 520–540, https://doi.org/10.1016/S0022-0396(02)00042-6. Search in Google Scholar

[31] Y. B. Yang and R. Z. Xu, Nonlinear wave equation with both strongly and weakly damped terms: supercritical initial energy finite-time blow up, Commun. Pure Appl. Anal. 18 (2019), 1351–1358, https://doi.org/10.3934/cpaa.2019065.Search in Google Scholar

[32] C. Yang and Y. B. Yang, Long-time behavior for fourth-order wave equations with strain term and nonlinear weak damping term, Discrete Contin. Dyn. Syst. Ser. S 14 (2021), 4643–4658, https://doi.org/10.3934/dcdss.2021110. Search in Google Scholar

[33] C. Yang, V. D. Rădulescu, R. Z. Xu, and M. Y. Zhang, Global well-posedness analysis for the nonlinear extensible beam equations in a class of modified Woinowsky-Krieger models, Adv. Nonlinear Stud. 22 (2022), 436–468, https://doi.org/10.1515/ans-2022-0024. Search in Google Scholar

Received: 2025-04-30

Revised: 2025-07-06

Accepted: 2025-07-10

Published Online: 2025-08-30

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

Articles in the same Issue

https://doi.org/10.1515/anona-2025-0104

Keywords for this article

continuous dependence; fourth-order wave equation; general nonlinear strain term; strong damping; weak damping; energy decay estimate

Creative Commons

BY 4.0