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

We consider the nonlinear heat equation with a nonlinear gradient term: $\partial_t u =\Delta u+\mu|\nabla u|^q+|u|^{p-1}u,\; \mu>0,\; q=2p/(p+1),\; p>3,\; t\in (0,T),\; x\in \R^N.$ 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.