Strong gradient absorption and initial decay prevent blow-up

Consider positive classical solutions of the Cauchy problem for $u_t-Δu=u^p-μ|\nabla u|^q$, with sufficiently smooth initial data $u_0$, in the strong absorption regime, namely $q\ge p>1$ and $μ>0$. It has remained a long standing open problem whether finite time blow-up can occur if $u_0$ decays at space infinity. We here solve this problem, showing that the answer in negative. To this end, we establish a novel, universal Bernstein-type gradient estimate, of independent interest, valid only in the range $q\ge p$. Moreover, the decay assumption on $u_0$ turns out to be essentially optimal. Indeed we show that the solution blows up in finite time whenever $u_0$ has a positive limit at infinity, and we determine the blow-up behavior of $u$ at space infinity under an additional assumption.