Blow-up and continuous dependence of solutions for a class of Love-type damped wave equations with p-Laplacian and memory term

Introduction

In this paper, we consider the following initial boundary value problem for the Love-type damped wave equation with the p-Laplacian and a memory term

1.1

$$\begin{array}{cc}& u_{\mathit{tt}}-u_{\mathit{ttxx}}-u_{\mathit{txx}}-{\displaystyle \frac{\partial }{\partial x}}\left[\left(1,+,\mu ,(x,t),{\left|u_x\right|}^{p-2}\right),u_x\right]\hfill \\ \hfill & +{\int }_0^{t}g(t-s){\displaystyle \frac{\partial }{\partial x}}\left[\overline{\mu },(x,s),u_x,(x,s)\right]ds\hfill \\ \hfill & =f(x,t,u,u_t,u_x,u_{\mathit{tx}}),0<x<1,0<t<T,\hfill \end{array}$$

1.2

$$\begin{array}{cc}& u(0,t)=u(1,t)=0,\hfill \end{array}$$

1.3

$$\begin{array}{cc}& u(x,0)={\tilde{u}}_0(x),u_t(x,0)={\tilde{u}}_1(x),\hfill \end{array}$$where $p\ge 3$ is the given constant and $\mu ,$$\overline{\mu },$f,  g,  ${\tilde{u}}_0,$${\tilde{u}}_1$ are given functions satisfying certain conditions that will be specified later.

It has long been known that waves are created when a vibrating source disturbs a medium. Among those, the model of waves generated by the longitudinal vibration of rods has been a topic of interest to researchers, engineers, mathematicians, and scientists alike. In this case, one can immediately mention Love-type equations. For instance, we refer to the models examined in [1] and [2], which are described by the following fourth-order partial differential equation

1.4

$$\begin{array}{c}\hfill u_{\mathit{tt}}-c_0^2{u}_{\mathit{xx}}-{\nu }^2{K}^2{u}_{\mathit{ttxx}}=0,\end{array}$$where $u=u(x,t)$ is the displacement, $c_0=\sqrt{E/{\rho }_0}$ is the wave speed, E is the Young modulus of the material, ${\rho }_0$ is the mass density, $\nu$ is a constant, K is a cross-section radius of gyration about the central line. The equation (1.4) has been extensively studied, with numerous attempts made to understand it. For example, Davies [3] and Thomson [4] have successfully obtained the existence and uniqueness of an exact solution for...