Liouville-type theorem for one-dimensional porous medium systems with sources

Abstract

In this paper, we are concerned with the one-dimensional porous medium system with sources \begin{align*} \begin{cases}u_t-( u^m)_{xx} =a_{11}u^{p}+a_{12} u^rv^{r+m}, (x,t)\in J\times I\subset\mathbb{R}\times \mathbb{R}\\ v_t-(v^m)_{xx} =a_{21} u^{r+m}v^{r}+a_{22}v^{p},\;(x,t)\in J\times I\subset \mathbb{R}\times \mathbb{R}, \end{cases} \end{align*} where $p=2r+m$, $m>1$, $r>0$. Under the conditions $a_{12}\geq 0, a_{21}\geq 0$, $a_{11}>0$, and $a_{22}>0$, we prove that the system does not possess any nontrivial nonnegative weak solution.