CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS

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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