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

Article type: Research Article

Authors: Bartier, Jean-Philippe

Affiliations: Ceremade (UMR CNRS no. 7534), Université Paris IX-Dauphine, Place de Lattre de Tassigny, 75775 Paris cédex 16, France E-mail: [email protected]

Abstract: We study the equation ∂tu−Δu=up−μ |∇u|qur 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.

Keywords: nonlinear parabolic equations, gradient term, finite time blow-up, global existence, inradius, Poincaré inequality

Journal: Asymptotic Analysis, vol. 46, no. 3-4, pp. 325-347, 2006