Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source

Abstract

Throughout this paper, we consider the equation\[u_t - \Delta u = e^{|\nabla u|}\]with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\overline{\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.