Existence and nonexistence of global solutions of quasilinear parabolic equations

Abstract

We consider nonnegative solutions to the Cauchy problem for the quasilinear parabolic equations ut=Δum+K(x)up where x∈RN, 1≤m<p and K(x)≥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 pm,σ*=pm,σ*(N) such that if p≤pm,σ*, then every nontrivial nonnegative solution is not global in time; whereas if p>pm,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|-' as |x|→∞ in some cone D′ and u0(x)=0 in the complement of D′.