The Influences Of The Gradient Term To The Blow Up Of The Solution To Semilinear Evolutional Equations

Nonlinear diffusion equations, as an important class of partial diffusion equations, come from a variety of diffusion phenomena appeared widely in nature. They arise from many fields such as physics, chemistry and dynamics of biological groups. In recent years, the study in this direction attracts a large number of mathematicians both in China and aboard, and remarkable progress has been achieved on the local existence of classical solutions, global existence, blow-up as well as estimates on the blow-up time and blow-up set, which enrich enormously the theory of partial differential equations. Until now, the study of diffu-sion equations is still a very active research area.Chipotand Weissler proved a equation whereΩ∈RN with smooth boundary,λ>0, and p, q>1. He proved that when 12p<(N+2)/(N-2), or 10, or q=2p/(p+1),λ>λp=2p(p+1)(p-1)-p-1. then for every u0≥vwithu0≠v, where v is the a stationary solution.then the solutionu(t, u0) of the equation will blow up.Later, Shaohua Chen give a further condition about the blow up of He show when g(x),f(x) satisfy some condition, andφ≤λψwithλ>1, then the solution of equation(2) will blow up in a finite time.In addition to show the blow up is rely on the initial data, Philippe and Souplet show that only under the condition p>q the blow up occurs. They proved that when p>q, if there existλ0=λ0(ψ)>0, andλ>λ0, then the solution of a equation with initial dataφ=λψ(2) will blow up.In addition, many paper show the blow up rates of the solution for the equation (2) under the condition q<2p/(p+1). They show that under some conditions the solution of equation (2) satisfy C1(T-t)1/(p-1)≤‖u(t)‖∞≤C2(T-t)1/(p-1),t→T. Form the two equations we can come to the result that the gradient term has an important influences to the blow up rate.Consider the equation with nonlinear boundary flux ut=Δu+|▽u|r-aepu, M.Chipot, M.Fila and P.Quittner first show the equation without the gradient and the bound-ary flux When p, q>1 and a>0, they show if p<2q-1, then the solution of equation (3) will blow up for large initial data. But when p>2q-1, the solutions of equation (3) are globally bounded. So the p=2q-1 is a critical value for equation (3). If p=2q-1, when aq, then every solution of the equation is globally bounded.When come to the blow up rates, many paper research the reflect diffusion equation with absorption coefficient. The critical value of equation (4) is 2q=p. And the solution of the equation (4) satisfy log C1(T-t)-1/(2q)≤max u(., t)≤log C2(T-1)-1/(2q)Recent, Sining Zheng and Wei Wang research the critical value of the equation They proved that the critical value of equation (5) isμq=p, whereμ=max(r,2). And show that whenμq>p, the solution of equation (5) will blow up,and whenμq