Blow-up Of Solutions To A Class Of Semilinear Heat Equations With Viscoelastic Terms

In this paper,we consider blow-up of solutions to a class of semi-linear heat equations with viscoelastic termswhere ? is a smooth bounded domain of RN with N>2,and the relax-ation function g:R+?R+is a bounded C1(R)function satisfingThe function f(u)? C1(R),and satisfies the following conditions:(?)There exists a constant p>1,such that(?)There exist constants q>1,ak>0,1<pm<pm-1<…<p1<(N+2)/(N-2),N?3,1<pm<Pm-1<…<p1<?,N=1,2,1 ?k?m,such thatThe main results of this paper read as follows:Theorem 1.Suppose that u0 ? V such that E(u0)<1-l/2r0 and thatLet u=u(x,t)be a strong solution to problem(0.1).Then u(x,t)blows up at some finite time T in the sense thatHere l is the positive constant given in(0.2),r0 is a positive constant that satisfies certain specific conditions,V is the corresponding set outside the potential well defined aswhere is the depth of the potential well which is characterized asN is Nehari's manifold defined asand the initial energy E(u0)is given byWe first recall the background and recent development on the study of semilinear heat equations with a visco-elastic terms.Subsequently,some necessary preliminaries are presented.At last,the potential well method and concavity argument are used to prove that the solutions blow up in finite time with positive initial energy.