Existence of a Stable Blow-up profile for the nonlinear heat equation with a critical power nonlinear gradient term

Abstract

We consider the nonlinear heat equation with a nonlinear gradient term: ${\partial }_{t}u=\Delta u+\mu |\nabla u{|}^q+|u{|}^{p-1}u,\mu >0,q=2p/(p+1),p>3,t\in (0,T),x\in {\mathbb{R}}^N.$ We construct a solution which blows up in finite time $T>0.$ We also give a sharp description of its blow-up profile and show that it is stable with respect to perturbations in initial data. The proof relies on the reduction of the problem to a finite dimensional one, and uses the index theory to conclude. The blow-up profile does not scale as $(T-t{)}^{1/2}|\log (T-t){|}^{1/2},$ like in the standard nonlinear heat equation, i.e. $\mu =0,$ but as $(T-t{)}^{1/2}|\log (T-t){|}^{\beta }$ with $\beta =(p+1)/[2(p-1)]>1/2.$ We also show that $u$ and $\nabla u$ blow up simultaneously and at a single point, and give the final profile. In particular, the final profile is more singular than the case of the standard nonlinear heat equation.