Existence and nonexistence of global solutions of quasilinear parabolic equations

Abstract

We consider nonnegative solutions to the Cauchy problem for the quasilinear parabolic equations $u_t=\Delta u^m+K\left(x\right)u^p$ where $x\in R^N$, and $K\left(x\right)\ge 0$ has the following properties: $K\left(x\right)$- as $\left|x\right|\to \infty$ in some cone $D$ and $K\left(x\right)=0$ in the complement of $D$, where for $\sigma =-\infty$ we define that $K\left(x\right)$ has a compact support. We find a critical exponent $p_{m,\sigma }^{*}=p_{m,\sigma }^{*}\left(N\right)$ such that if $p\le p_{m,\sigma }^{*}$, then every nontrivial nonnegative solution is not global in time; whereas if $p>p_{m,\sigma }^{*}$ then there exits a global solution. We also find a second critical exponent, which is another critical exponent on the growth order $\alpha$ of the initial data $u_0\left(x\right)$ such that $u_0\left(x\right)$-$|x{|}^{-}\text{'}$ as $\left|x\right|\to \infty$ in some cone $D^{\prime }$ and $u_0\left(x\right)=0$ in the complement of $D^{\prime }$.