Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations

Abstract

A complete characterization of the long-time behaviour of nonnegative solutions to the problem $$ u_t -\Delta u + u + \sum b_j u^{r_j} - \sum a_i u^{p_i} = 0 , \ u=u(x,t) , \ x \in \rn , \ t>0 $$ $$ 1 <r_j < p_i \leq {N\over N-2}, \ a_i, \ b_j > 0 $$ is given. For any nonnegative compactly supported datum $ \bar u \not\equiv 0 $, there is a constant $ \alpha_c >0 $ such that the solution converges to a ground state for $ u(0)= \alpha_c \bar u $, blows up at a finite time if $ u(0) = \alpha \bar u ,$ $ \alpha > \alpha_c $ and tends to zero provided $ u(0) = \alpha \bar u $ with $ 0 \leq \alpha < \alpha_c $. The proof is based on a combination of the concentrated compactness, comparison theorems and some recent results on uniqueness for the corresponding stationary equation.