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Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions.

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Arrieta Algarra, José María y Rodríguez Bernal, Aníbal (2004) Localization on the boundary of blow-up for reaction-diffusion equations with nonlinear boundary conditions. Communications in Partial Differential Equations, 29 (7-8). pp. 1127-1148. ISSN 0360-5302

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URL Oficial: http://www.tandfonline.com/doi/full/10.1081/PDE-200033760


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Resumen

In this work we analyze the existence of solutions that blow-up in finite time for a reaction-diffusion equation ut−Δu=f(x,u) in a smooth domain Ω with nonlinear boundary conditions ∂u∂n=g(x,u). We show that, if locally around some point of the boundary, we have f(x,u)=−βup,β≥0, and g(x,u)=uq, then blow-up in finite time occurs if 2q>p+1 or if 2q=p+1 and β<q. Moreover, if we denote by Tb the blow-up time, we show that a proper continuation of the blow-up solutions are pinned to the value infinity for some time interval [T,τ] with Tb≤T<τ. On the other hand, for the case f(x,u)=−βup, for all x and u, with β>0 and p>1, we show that blow-up occurs only on the boundary.


Tipo de documento:Artículo
Palabras clave:reaction-diffusion; blow-up; nonlinear boundary conditions; heat-equations; parabolic equations; positive solutions; uniqueness; attractors
Materias:Ciencias > Matemáticas > Ecuaciones diferenciales
Código ID:17898
Depositado:23 Ene 2013 08:51
Última Modificación:12 Dic 2018 15:07

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