Asymptotic behavior of solutions for a new general class of parabolic Kirchhoff type equation with variable exponent sources

Abstract

In this paper, we study the following problem:\begin{align*} \begin{cases} \displaystyle u_t -M\left(\int_{\Omega}\frac{\left|\nabla u\right|^{p(x)}}{p(x)}\diff x\right)\Delta_{p(x)}u = \left|u\right|^{m(x)-2}u,& (x,t)\in \Omega\times (0,T),\\ u(x,t)=0,&(x,t)\in \partial\Omega\times (0,T),\\ u(x,0)=u_0(x),&x\in \Omega, \end{cases}\end{align*}where $\Omega\subset\mathbb{R}^N$ is a bounded domain with smooth boundary $\partial\Omega$, the functions $p,m:\bar{\Omega}\to\mathbb{R}$ and the Kirchhoff function $M:[0,\infty)\to [0,\infty)$ are specified below. We give a new class of general Kirchhoff function which covers both non-degenerate and degenerate cases, and investigate its effects on the existence and non-existence of global weak solutions to \eqref{1.1} in case the initial energy is subcritical. Moreover, we also give the decay estimate for the energy functional in the former case and an upper bound for the maximal existence time in the latter case. This is a continue working on Nhan et al., [Nonlinear Anal. Real World Appl., 56 (2020), 103155] and also an affirmative answer for the comments by Guo et al., [J. Differential Equation., 340 (2022), 45-82.] 2010 MSC: 35A01, 35B44, 35K55