Poincaré's inequality and global solutions of a nonlinear parabolic equation (Q1288767)

The authors study positive solutions of the equation \(u_t-\Delta u=u^p-\mu| \nabla u| ^q\), \(t\geq 0\) in a (possibly unbounded) domain \(\Omega\subset\mathbb R^N\), where \(q\geq p>1\) and \(\mu>0\). They show the following: If \(q>p\) or \(\mu\) is sufficiently large, then all solutions are global provided the Poincaré inequality is valid in \(\Omega\). Moreover, all solutions are bounded and decay exponentially to zero if \(\mu\) is large. Conversely, if \(\Omega\) contains arbitrarily large balls, then there always exist unbounded solutions. The authors also obtain various qualitative properties of the solutions.