The critical exponent for fast diffusion equation with nonlocal source

Abstract

Abstract This paper considers the Cauchy problem for fast diffusion equation with nonlocal source ut=Δum+(∫Rnuq(x,t)dx)p−1qur+1 $u_{t}=\Delta u^{m}+ (\int_{\mathbb{R}^{n}}u^{q}(x,t)\,dx )^{\frac{p-1}{q}}u^{r+1}$, which was raised in [Galaktionov et al. in Nonlinear Anal. 34:1005–1027, 1998]. We give the critical Fujita exponent pc=m+2q−n(1−m)−nqrn(q−1) $p_{c}=m+\frac{2q-n(1-m)-nqr}{n(q-1)}$, namely, any solution of the problem blows up in finite time whenever 1pc $p>p_{c}$.

Keywords

• Critical exponents
• Fast diffusion
• Nonlocal
• Global solutions
• Blow-up