奇异半线性反应扩散方程初值问题的一些注记(英文)

Some Remarks on Cauchy Problem of Singular Semilinear Reaction-diffusion Equation

考虑下述奇异半线性反应扩散方程初值问题: u t-1t△u=ur+f(x), t>0,x∈RNlimt→0+u(t,x)=0, x∈RN其中r>0,△=∑Ni=1 2 x2i,f(x)非负且f(x)∈L∞(RN).首先利用增算子不动点定理,重新证明了IVP在(0,+∞)上至少存在一个非负解,并给出了IVP解的迭代逼近序列.其次获得了一个有关IVP(1)正解的无限增长性的结果.最后,证明了当r>1时,去掉条件1r-1 n2,IVP的正解u(t)同样会产生爆破.研究结果表明情形limt→+∞u(t,x)=+∞不会出现.

This paper deals with the following Cauchy problem of singular reaction-diffusion equationut-1tΔu=u^r+f(x),t>0, x∈R^N(lim)t→0^+ u(t, x)=0, x∈R^N (1) where r>0, Δ=∑Ni=1~2x^2_i, f(x) is nonnegative and f(x)∈L~∞(R^N). First we reprove that IVP(1) has at least one nonnegative solution on (0, +∞), which is obtained in our reference. It is remarkable that the method used here is fixed point theorem of increasing operator, which is very different from the literature and our proof is brief. At the same time, an iterative sequence of approximation solution, which convergence the solution of IVP(1), is also given. Second a result about the infinite growing up of positive solution is obtained. Finally we prove that if r>1, the positive solution u(t) of IVP(1) will blow up, which gets rid of the condition 1r-1n2 of our reference. Consequently, this result shows that the case (lim)t→+∞u(t, x)=+∞ does not appear in the literature.