CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS
Zhongping LiCollege of Mathematic and Information China West Normal University Nanchong 637009, P. R. ChinaChunlai MuCollege of Mathematic and Information China West Normal University Nanchong 637009, P. R. ChinaWanjuan DuCollege of Mathematic and Information China West Normal University Nanchong 637009, P. R. China
2013en
ABI
Abstract
In this paper, we consider the positive solution to a Cauchy problem in <TEX>$\mathbb{B}^N$</TEX> of the fast diffusive equation: <TEX>${\mid}x{\mid}^mu_t={div}(\mid{\nabla}u{\mid}^{p-2}{\nabla}u)+{\mid}x{\mid}^nu^q$</TEX>, with nontrivial, nonnegative initial data. Here <TEX>$\frac{2N+m}{N+m+1}$</TEX> < <TEX>$p$</TEX> < 2, <TEX>$q$</TEX> > 1 and 0 < <TEX>$m{\leq}n$</TEX> < <TEX>$qm+N(q-1)$</TEX>. We prove that <TEX>$q_c=p-1{\frac{p+n}{N+m}}$</TEX> is the critical Fujita exponent. That is, if 1 < <TEX>$q{\leq}q_c$</TEX>, then every positive solution blows up in finite time, but for <TEX>$q$</TEX> > <TEX>$q_c$</TEX>, there exist both global and non-global solutions to the problem.
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