Global behavior of solutions of a reaction–diffusion equation with gradient absorption in unbounded domains

Abstract

We study the equation ∂_tu−Δu=u^p−μ |∇u|^qu^r in a general domain (bounded or not). Many results have been established in the case r=0, and we will generalize some properties. If q≥1 and q+r≥p, we will show that there is a strong connection between the finiteness of the inradius of Ω (or the Poincaré inequality) and the global existence of the solutions. More precisely if the inradius of Ω is finite, then the solutions u are global and if moreover μ is large enough, the solutions decay exponentially to zero. Conversely, if it is infinite, there always exists unbounded solution. Other qualitative results are also obtained.