## Gradient blow-up for multidimensional nonlinear parabolic equations with general boundary conditions.(English)Zbl 1015.35016

Consider the parabolic equation $$u_t-\Delta u=F(x,u,\nabla u)$$ in a smoothly bounded domain $$\Omega$$ in $$\mathbb{R}^N$$, complemented by an initial condition and general Dirichlet boundary conditions. The author shows that for a large class of $$F$$ with superquadratic growth with respect to $$\nabla u$$, gradient blow-up occurs. More precisely, solutions with suitable (or all) initial data remain bounded but their gradient blows up in finite time. Typical examples are $$F=|\nabla u|^p+a(x)u^q$$ or $$F=|\nabla u|^p(1+u^2)^{-m/2}$$, where $$p>2$$, $$a$$ is HĂ¶lder continuous, $$p>q\geq 1$$, $$m\leq p/N$$ and $$m<p-1$$ if $$N=1$$. If $$F=F(\nabla u)$$ then a general condition of the type $$F(\nabla u)\geq|\nabla u|^2h(|\nabla u|)$$ guarantees gradient bow-up for suitable initial data, provided $$h$$ satisfies some technical assumptions and $$\int^\infty_0 1/sh(s) ds<\infty$$. The author also considers nonnegative solutions of initial boundary value problems for the nonlocal equation $$u_t-\Delta u=u^m(\int_\Omega|\nabla u|^2)^r$$, where $$m\geq 1$$, $$r>0$$. These solutions may blow-up in $$L^\infty$$ in finite time but gradient blow-up (for bounded solutions) cannot occur in this case.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 45K05 Integro-partial differential equations 35K55 Nonlinear parabolic equations 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations