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On symmetric and nonsymmetric blowup for a weakly quasilinear heat equation. (English) Zbl 0857.35063

Summary: We construct blow-up patterns for the quasilinear heat equation \[ u_t=\nabla\cdot (k(u)\nabla u)+Q(u) \tag{QHE} \] in \(\Omega\times(0,T)\), \(\Omega\) being a bounded open convex set in \(\mathbb{R}^N\) with smooth boundary, with zero Dirichlet boundary condition and nonnegative initial data. The nonlinear coefficients of the equation are assumed to be smooth and positive functions and moreover, \(k(u)\) and \(Q(u)/u^p\) with a fixed \(p>1\) are of slow variation as \(u\to\infty\), so that (QHE) can be treated as a quasilinear perturbation of the well-known semilinear heat equation \[ u_t=\Delta u+u^p.\tag{SHE} \] We prove that the blow-up patterns for the (QHE) and the (SHE) coincide in a structural sense under the extra assumption \(\int^\infty k(f(e^s))ds=\infty\), where \(f(v)\) is a monotone solution of the ODE \(f'(v)=Q(f(v))/v^p\) defined for all \(v\gg1\). If the integral is finite then the (QHE) is shown to admit an infinite number of different blow-up patterns.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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