Existence and nonexistence of global solutions of quasilinear parabolic equations

Dedicated to Professor Kunihiko Kajitani on his sixtieth birthday

Abstract

We consider nonnegative solutions to the Cauchy problem for the quasi-linear parabolic equations u_t=Δ u^m+K(x)u^p where x∈ \bm{R}^N, 1≤ m<p and K(x)≥q 0 has the following properties: K(x)∼|x|^σ (-∞≤σ<∞) as |x|→∞ in some cone D and K(x)=0 in the complement of D, where for σ=-∞ we define that K(x) has a compact support. We find a critical exponent p_{m, σ}^*=p_{m, σ}^*(N) such that if p≤ p_{m, σ}^*, then every nontrivial nonnegative solution is not global in time, whereas if p>p_{m, σ}^* then there exits a global solution. We also find a second critical exponent, which is another critical exponent on the growth order α of the initial data u₀(x) such that u₀(x)∼|x|^-α as |x|→∞ in some cone D^{'} and u₀(x)=0 in the complement of D^{'}.