Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions

Abstract

We consider nonlinear parabolic equations with gradient-dependent nonlinearities, of the form $u_t-\Delta u=F(u,\nabla u)$. These equations are studied on smoothly bounded domains of ${\mathbb R}^N$, $N\geq 1$, with arbitrary (continuous) Dirichlet boundary data. Under optimal assumptions of (superquadratic) growth of $F$ with respect to $\nabla u$, we show that gradient blow-up occurs for suitably large initial data; i.e., $\nabla u$ blows up in finite time while $u$ remains uniformly bounded. Various extensions and additional results are given. We also consider some equations where the nonlinearity is nonlocal with respect to $\nabla u$, and show that gradient blow-up usually does not occur in this case.