Fujita Type Theorems With Bounded Domains Or Bounded Reaction Supports

It is known that in general the study on the critical Fujita admit unbounded domains only. This thesis discusses Fujita phenomena of nonlinear parabolic equations for bounded domains or compactly supported reactions. There are two topics included:The Fujita exponent of nonlinear parabolic systems in bounded domain; The Cauchy problem with sources compactly supported. The topics of blow-up set and blow-up rate etc are involved as well.The thesis composes of four chapters:In Chapter1we summarize the background of the related issues and state the main results of the present thesis.Chapter2deals with the Cauchy problem ut=(um)xx+a(x)up with localized reaction a(x)up, where the reaction a(x)up is compactly supported. As we know, the Fujita exponent of Cauchy problem with source up is pc=m+2, and there is no Fujita type conclusion with bounded domains. The problem studied here is between them, neither the problem with bounded domains, nor general Cauchy problem on the whole space, and its critical Fujita exponent will be shown as pc=m+1. This extends the related result for the slow diffusion situation. It is pointed out that there is no Fujita type conclusion for the high dimension case of this problem. In addition, we prove that the problem admits blow-up rate (T-t)-1/p-1and the blow-up set{0} under suitable initial data.Chapter3studies the Fujita type conclusion for a parabolic system coupled via variable sources up(x) and vg(x).At first consider a region I for global solutions with any initial data, and a coexistence region II of global and non-global solutions. Outside of I and Ⅱ, the existence or not of global solutions is related to the size of the domain:the solutions are global with small initial data if the domain is small to be contained in a small ball, and there is p(x),q(x) such that the solutions are non-global with large initial data if the domain is large to contain a big ball.Chapter4treats the coupled parabolic system with time-weighted sources in a bounded domain:ut=△u+eαtvp, ut=△u+eβtuq in Ω×(0,T) with α,β∈R and p,q>0, subject to null Dirichlet boundary value condition. The critical Fujita curve is determined as (pq)c=1+max{α+βp,β+αq,0} where λ1is the first eigenvalue of the Lapla-cian with null Dirichlet boundary condition, and there is no any additional restriction on α,β,p,q. Next, as an extension, an interesting Fujita phenomenon is observed for another coupled system Ut=△U+mU+Vp,△t=△V+nV+Uq in Ω×(0, T) with pq>1that the critical Fujita curve is represented via the Fujita critical coefficient max{m,n}=λ1namely, any nontrivial solutions blow up in finite time if and only if max{m,n}≥λ1As for the techniques used in this Chapter, it is mentioned that the current studies of critical Fujita curves for coupled systems (especially in critical cases) seem to be heavenly relying upon Jensen's inequality and/or the Kaplan method, for which one has to deal with complicated discussions on the exponents p, q being greater or less than one. Differ-ently, in the framework of this paper, the heat semigroup is introduced to study critical Fujita curves for coupled system problems, where various superlinear and sublinear cases will be treated uniformly by estimates involved. This greatly simplifies the arguments for establishing the Fujita type theorems. Finally, as applications of the framework of the paper, a new and simpler proof is proposed to some previous results of the authors in Chapter3.