The critical exponent for fast diffusion equation with nonlocal source
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Abstract This paper considers the Cauchy problem for fast diffusion equation with nonlocal source $u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}$ u t = Δ u m + ( ∫ R n u q ( x , t ) d x ) 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 $p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}$ p c = m + 2 q − n ( 1 − m ) − n q r n ( q − 1 ) , namely, any solution of the problem blows up in finite time whenever $1< p\le p_{c}$ 1 < p ≤ p c , and there are both global and non-global solutions if $p>p_{c}$ p > p c .
2008 ◽
Vol 213
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pp. 95-110
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1996 ◽
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pp. 1349-1365
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2000 ◽
Vol 31
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pp. 1157-1174
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2014 ◽
Vol 90
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pp. 167-183
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