Asymptotic simplification for a reaction-diffusion problem with a nonlinear boundary condition
We study non-negative solutions of the porous medium equation with a source and a nonlinear flux boundary condition, ut = (um)xx + up in (0, ∞) × (0, T); -(um)x (0, t) = uq(0, t) for t ∈ (0, T); u(x, 0) = u0(x) in (0, ∞), where m > 1, p, q > 0 are parameters. For every fixed m we prove...
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| Autores principales: | , , |
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| Formato: | JOUR |
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| Acceso en línea: | http://hdl.handle.net/20.500.12110/paper_02724960_v67_n1_p69_DePablo |
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| Sumario: | We study non-negative solutions of the porous medium equation with a source and a nonlinear flux boundary condition, ut = (um)xx + up in (0, ∞) × (0, T); -(um)x (0, t) = uq(0, t) for t ∈ (0, T); u(x, 0) = u0(x) in (0, ∞), where m > 1, p, q > 0 are parameters. For every fixed m we prove that there are two critical curves in the (p, q)-plane: (i) the critical existence curve, separating the region where every solution is global from the region where there exist blowing-up solutions, and (ii) the Fujita curve, separating a region of parameters in which all solutions blow up from a region where both global in time solutions and blowing-up solutions exist. In the case of blow up we find the blow-up rates, the blow-up sets and the blow-up profiles, showing that there is a phenomenon of asymptotic simplification. If 2q < p + m the asymptotics are governed by the source term. On the other hand, if 2q > p + m the evolution close to blow up is ruled by the boundary flux. If 2q = p + m both terms are of the same order. |
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