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Critical Fujita exponent for a fast diffusive equation with variable coefficients
Zhongping Li,Chunlai Mu,Wanjuan Du 대한수학회 2013 대한수학회보 Vol.50 No.1
Abstract. In this paper, we consider the positive solution to a Cauchy problem in RN of the fast diffusive equation: [수식], with nontrivial, nonnegative initial data. Here [수식}. We prove that [수식] is the critical Fujita exponent. That is, if 1 < q ≤ qc, then every positive solution blows up in finite time, but for q > qc, there exist both global and non-global solutions to the problem.
CRITICAL FUJITA EXPONENT FOR A FAST DIFFUSIVE EQUATION WITH VARIABLE COEFFICIENTS
Li, Zhongping,Mu, Chunlai,Du, Wanjuan Korean Mathematical Society 2013 대한수학회보 Vol.50 No.1
In this paper, we consider the positive solution to a Cauchy problem in $\mathbb{B}^N$ of the fast diffusive equation: ${\mid}x{\mid}^mu_t={div}(\mid{\nabla}u{\mid}^{p-2}{\nabla}u)+{\mid}x{\mid}^nu^q$, with nontrivial, nonnegative initial data. Here $\frac{2N+m}{N+m+1}$ < $p$ < 2, $q$ > 1 and 0 < $m{\leq}n$ < $qm+N(q-1)$. We prove that $q_c=p-1{\frac{p+n}{N+m}}$ is the critical Fujita exponent. That is, if 1 < $q{\leq}q_c$, then every positive solution blows up in finite time, but for $q$ > $q_c$, there exist both global and non-global solutions to the problem.