Long-Time Behavior of Solutions to Von Karman Equations with Variable Sources

Abstract

The interest of this paper is to deal with long-time behavior of the solutions to the following Von Karman equation involving variable sources and clamped boundary conditions:

$$\begin{aligned} \quad u_{tt}+\Delta ^{2} u+a|u_t|^{m(x)-2}u_{t}=[u,F(u)]+b|u|^{p(x)-2}u,\quad \Delta ^{2}F(u)=-[u,u]. \end{aligned}$$

First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships among initial energy value, the term $\int _{\Omega }\frac{1}{p(x)}|u|^{p(x)}\mathrm{d}x$ and the Airy stress functions, which ensure that the energy functional are nonnegative with respect to time variable. And then, some energy estimates and Komornik inequality is used to prove a uniform estimate of decay rates of the solution which provides an estimation of long-time behavior of solutions. As we know, such results are seldom seen for the variable exponent case.