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Abstract:

We consider the semilinear heat equation ut = Δu + u p both in ℝN and in a bounded domain with homogeneous Dirichlet boundary conditions, with 1 < p < ps where ps is the Sobolev exponent. This problem has solutions with finite-time blowup; that is, for large enough initial data there exists T < ∞ such that u is a classical solution for 0 < t < T, while it becomes unbounded as t ↗ T. In order to understand the situation for t > T, we consider a natural approximation by reaction problems with global solution and pass to the limit. As is well-known, the limit solution undergoes complete blowup: after it blows up at t = T, the continuation is identically infinite for all t > T. We perform here a deeper analysis of how complete blowup occurs. Actually, the singularity set of a solution that blows up as t ↗ T propagates instantaneously at time t = T to cover the whole space, producing a catastrophic discontinuity between t = T- and t = T +. This is called the "avalanche." We describe its formation as a layer appearing in the limit of the natural approximate problems. After a suitable scaling, the initial structure of the layer is given by the solution of a limit problem, described by a simple ordinary differential equation. As t proceeds past T, the solutions of the approximate problems have a traveling wave behavior with a speed that we compute. The situation in the inner region depends on the type of approximation: a conical pattern is formed with time when we approximate the power up by a flat truncation at level n, while for tangent truncations we get an exponential increase in time and a diffusive spatial pattern. © 2003 Wiley Periodicals, Inc.

Registro:

Documento: Artículo
Título:Thermal avalanche for blowup solutions of semilinear heat equations
Autor:Quirós, F.; Rossi, J.D.; Vázquez, J.L.
Filiación:Universidad Autonoma de Madrid, Madrid, Spain
Universidad de Buenos Aires, Buenos Aires, Argentina
Universidad Autonoma de Madrid, Departamento de Matemáticas, 28049 Madrid, Spain
Universidad de Buenos Aires, Departamento de Matemática, 1428 Buenos Aires, Argentina
Año:2004
Volumen:57
Número:1
Página de inicio:0059
Página de fin:0098
Título revista:Communications on Pure and Applied Mathematics
Título revista abreviado:Commun. Pure Appl. Math.
ISSN:00103640
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00103640_v57_n1_p0059_Quiros

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Citas:

---------- APA ----------
Quirós, F., Rossi, J.D. & Vázquez, J.L. (2004) . Thermal avalanche for blowup solutions of semilinear heat equations. Communications on Pure and Applied Mathematics, 57(1), 0059-0098.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00103640_v57_n1_p0059_Quiros [ ]
---------- CHICAGO ----------
Quirós, F., Rossi, J.D., Vázquez, J.L. "Thermal avalanche for blowup solutions of semilinear heat equations" . Communications on Pure and Applied Mathematics 57, no. 1 (2004) : 0059-0098.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00103640_v57_n1_p0059_Quiros [ ]
---------- MLA ----------
Quirós, F., Rossi, J.D., Vázquez, J.L. "Thermal avalanche for blowup solutions of semilinear heat equations" . Communications on Pure and Applied Mathematics, vol. 57, no. 1, 2004, pp. 0059-0098.
Recuperado de https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00103640_v57_n1_p0059_Quiros [ ]
---------- VANCOUVER ----------
Quirós, F., Rossi, J.D., Vázquez, J.L. Thermal avalanche for blowup solutions of semilinear heat equations. Commun. Pure Appl. Math. 2004;57(1):0059-0098.
Available from: https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00103640_v57_n1_p0059_Quiros [ ]