New critical exponents, large time behavior, and life span for a fast diffusive p-Laplacian equation with nonlocal source

Abstract

This paper mainly investigate positive solutions of the Cauchy problem for a fast diffusive p-Laplacian equation with nonlocal source

$$\begin{aligned} u_{t}=\Delta _pu+\left( \,\,\int \limits _{{\mathbb {R}}^N}u^q(y,t)\mathrm{d}y\right) ^{\frac{r-1}{q}}u^{s+1},\quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{aligned}$$

where $N\ge 1$, $\frac{2N}{N+1}<p<2$, $q>1$, $r\ge 1$, $0\le s<\left( 1+\frac{1}{N}\right) p-2$ and $r+s>1$. We obtain the new critical Fujita exponent by virtue of the auxiliary function method and forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.