Blow-up for a degenerate diffusion problem not in divergence form
We study the behaviour of solutions of the nonlinear diffusion equation in the half-line, ℝ+ = (0, ∞), with a nonlinear boundary condition, {ut = uuxx, (x, t) ∈ ℝ+ (0, T), -ux(0,t) = up(0,t), t ∈ (0,T), u(x,0) = u0(x), x ∈ ℝ+ with p > 0. We describe, in terms of p and the initial datum, when...
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2006
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00222518_v55_n3_p955_Ferreira http://hdl.handle.net/20.500.12110/paper_00222518_v55_n3_p955_Ferreira |
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| Sumario: | We study the behaviour of solutions of the nonlinear diffusion equation in the half-line, ℝ+ = (0, ∞), with a nonlinear boundary condition, {ut = uuxx, (x, t) ∈ ℝ+ (0, T), -ux(0,t) = up(0,t), t ∈ (0,T), u(x,0) = u0(x), x ∈ ℝ+ with p > 0. We describe, in terms of p and the initial datum, when the solution is global in time and when it blows up in finite time. For blowing up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behaviour close to the blow-up time in terms of a self-similar profile. The stationary character of the support is proved both for global solutions and blowing-up solutions. Also we obtain results for the problem in a bounded interval. Indiana University Mathematics Journal ©. |
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