Abstract
This paper considers the Cauchy problem for fast diffusion equation with nonlocal source u_{t}=Delta u^{m}+ (int_{mathbb{R}^{n}}u^{q}(x,t),dx )^{frac{p-1}{q}}u^{r+1}, which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent p_{c}=m+frac{2q-n(1-m)-nqr}{n(q-1)}, namely, any solution of the problem blows up in finite time whenever 1< ple p_{c}, and there are both global and non-global solutions if p>p_{c}.
Highlights
We study the following Cauchy problem of fast diffusion parabolic equation with a nonlinear nonlocal source:
The study for the Cauchy problem of nonlocal nonlinear parabolic equation was proposed by Galaktionov et al [1], in which it was proved that the Cauchy problem (1.1) with m = 1 has a critical Fujita exponent, and Wang et al [15] obtained similar results by other methods
This paper shows that the model (1.1) possesses critical Fujita exponent pc = m +
Summary
Namely, any solution of the problem blows up in finite time whenever 1 < p ≤ pc, and there are both global and non-global solutions if p > pc. We study the following Cauchy problem of fast diffusion parabolic equation with a nonlinear nonlocal source: Many important results have appeared on the blow-up problem for a nonlinear parabolic equation with nonlocal source (see [2, 6, 8,9,10,11] and references therein), and for nonlocal nonlinear diffusion equations [12, 13].
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