The critical exponent for fast diffusion equation with nonlocal source

Boundary Value Problems

Linghua Kong¹  Qing Tian²  Chunxiao Yang²  Yingxue Wu² 

[1] School of Science, Dalian Ocean University;School of Science, Xi’an University of Architecture and Technology;

关键词: Critical exponents;    Fast diffusion;    Nonlocal;    Global solutions;    Blow-up;   

DOI  :  10.1186/s13661-019-1282-1

来源: DOAJ

【 摘 要 】

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 $p>p_{c}$.

【 授权许可】

Unknown