The Blow-Up of Solutions for a Class of Semi-linear Equations with p-Laplacian Viscoelastic Term Under Positive Initial Energy

This paper deals with homogeneous Dirichlet boundary value problem to a class of semi-linear equations with p-Laplacian viscoelastic term ∂u∂t-Δu+∫0tg(t-s)Δpu(x,s)ds=uq(x)-2u,x∈Ω,t≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial u}{\partial t}-\Delta u+\int _{0}^{t}g(t-s)\Delta _{p}u(x,s){{\textrm{d}}}s=\left| u\right| ^{q(x)-2}u,\quad x\in \Omega ,\ t\ge 0, \end{aligned}$$\end{document}the bounded domain Ω⊂Rn(n≥3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset R^{n}~(n\ge 3)$$\end{document} with a smooth boundary. We prove that the weak solutions of the above problems blow up in finite time for all 2k<q-<q+<p<2nn-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2k<q^-<q^+<p<\frac{2n}{n-2}$$\end{document} (k is defined in (2.5)), when the initial energy is positive and the function g satisfies suitable conditions. This result generalized and improved the result by Messaoudi (Abstr Appl Anal 2005(2):87–94, 2005).