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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 Hölder 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
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