具有变指数反应项的多孔介质方程的爆破

Blow-up of porous medium equations with variable source power

本文研究下述具有变指数反应项的多孔介质方程解的爆破和整体存在性问题,u_t=?u^m+u^(p(x)),(x,t)∈?×(0,T),其中?为有界域或全空间R^N,p(x)为定义在?上满足条件0<p_=infp(x)≤p(x)≤p+=supp(x)<∞的连续函数.这个方程由于变指数p(x)与定义域?的空间结构之间的相互作用表现出丰富而有趣的动力学特性.粗略地讲,对于全空间R^N上的初值问题,如果p(x)≤1,则方程的解可能不具有唯一性,此时所有非平凡解均整体存在;如果p+>m,此时一定存在爆破解.进一步,当1<p(x)≤m+2/N时,所有非平凡解均爆破;当p(x)>m+2/N时,存在非平凡整体解.当p_<m+2/N时,本文构造的例子表明,对于某些p(x)所有非平凡解均爆破;而对于另外一些p(x),则可能存在整体非平凡解.在有界域上解的性质与全空间又有所不同.此时有p(x)和m及区域性质三个因素相互作用,而仅有一个临界指标p=m表征解的爆破行为.若p+>m,则此时如同全空间情形存在爆破解;若p+<m,则方程所有解均整体存在;又若p(x)>m或者区域足够小,则方程存在整体解.最有意思的是,对于某些满足条件p_<m<p+的p(x),作者发现了对于这类方程特有的有界域上的Fujita现象.

In this paper, we study the blow-up and/or global existence of the following porous medium equation with variable source power,ut = Ωu^m＋ u^（p（x））, in Ω ×（0, T）,where ？ is either a bounded domain or the whole space R^N, and p（x） is a positive and continuous bounded function defined in ？ satisfying 0 p= inf p（x）≤p（x）≤p＋=sup p（x）∞. It is demonstrated that theequation has rich dynamics with interesting phenomena which depend on the interplay of p（x） and the structure of spatial domain Ω.Roughly speaking, for the Cauchy problem, if p（x）≤1, uniqueness of the solutions might not hold, and all nontrivial solutions exist globally. If p＋ m, there are blow-up solutions. If 1 p（x）≤m＋2/N,every nontrivial solution blows up. If p（x） m ＋2/N, there exist nontrivial global solutions. When p m ＋/2N, the examples we constructed suggest that for some p（x） all nontrivial solutions blow up and for others there exist global solutions.The situation is totally different in bounded domains. There is only one critical exponent p = m, while there are three factors, p（x）, m and properties of the domain, interacting with each other. If p＋ m, there are blow-up solutions as in RN. If p＋ m, all solutions exist globally in time. If p（x） m or the domain is small enough,there are global solutions. The most interesting part is that for some p（x） satisfying pmp＋, we find the Fujita phenomenon in the bounded domain.