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Abstract:

We study the continuation after blow-up of solutions u(x, t) of the heat equation, ut = uxx, with a nonlinear flux condition at the boundary, -ux(0, t) = f(u(0, t)). We prove that such a continuation is trivial (i.e., identically infinite for all t > T, where T is the blow-up time) in the following sense: if we replace f(u) by a sequence of functions fn(u) that do not cause blow-up and fn converges to f uniformly on compact sets, then the corresponding solutions un satisfy limn→∞ un × (x, t) = +∞ for every x > 0 and every t > T. This is called complete blow-up and happens in the present problem for all positive and continuous functions f for which solutions blow up. An interesting phenomenon related to complete blow-up is the thermal avalanche: in the cases in which there is single-point blow-up as t ↗ T the singularity at the origin propagates instantaneously at time t = T to cover the whole space. We describe the formation of the avalanche in the case f(u) = up, p > 1, as a boundary layer which appears in the limit of the approximate problems by choosing a suitable scaling and passing to self-similar variables. We then show that the layer is described by the solution of a limit problem. We also describe the asymptotic behaviour for the approximate problems as t goes to infinity. We also consider the nonlinear diffusion equation ut = (um)xx, m > 0, with nonlinear boundary condition -(um)x(0, t) = f(u(0, t)). We prove that blow-up solutions have a nontrivial continuation if and only if m < 1, independently of f. The thermal avalanche when m > 1 and f is a power is also described in this case, as well as the asymptotic behaviour for the approximations for all m.

Registro:

Documento: Artículo
Título:Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions
Autor:Quirós, F.; Rossi, J.D.; Vazquez, J.L.
Filiación:Departamento de Matemáticas, U. Autónoma de Madrid, Madrid, 28049, Spain
Departamento de Matemática, F.C.E y N.. UBA, (1428) Buenos Aires, Argentina
Palabras clave:Avalanche; Blow-up; Boundary layer; Heat equation; Nonlinear boundary condition
Año:2002
Volumen:27
Número:1-2
Página de inicio:395
Página de fin:424
DOI: http://dx.doi.org/10.1081/PDE-120002792
Título revista:Communications in Partial Differential Equations
Título revista abreviado:Commun. Partial Differ. Equ.
ISSN:03605302
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03605302_v27_n1-2_p395_Quiros

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Citas:

---------- APA ----------
Quirós, F., Rossi, J.D. & Vazquez, J.L. (2002) . Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions. Communications in Partial Differential Equations, 27(1-2), 395-424.
http://dx.doi.org/10.1081/PDE-120002792
---------- CHICAGO ----------
Quirós, F., Rossi, J.D., Vazquez, J.L. "Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions" . Communications in Partial Differential Equations 27, no. 1-2 (2002) : 395-424.
http://dx.doi.org/10.1081/PDE-120002792
---------- MLA ----------
Quirós, F., Rossi, J.D., Vazquez, J.L. "Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions" . Communications in Partial Differential Equations, vol. 27, no. 1-2, 2002, pp. 395-424.
http://dx.doi.org/10.1081/PDE-120002792
---------- VANCOUVER ----------
Quirós, F., Rossi, J.D., Vazquez, J.L. Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions. Commun. Partial Differ. Equ. 2002;27(1-2):395-424.
http://dx.doi.org/10.1081/PDE-120002792