Thermal avalanche for blowup solutions of semilinear heat equations
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 e...
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todo:paper_00103640_v57_n1_p0059_Quiros2023-10-03T14:09:02Z Thermal avalanche for blowup solutions of semilinear heat equations Quirós, F. Rossi, J.D. Vázquez, J.L. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00103640_v57_n1_p0059_Quiros |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| description |
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. |
| format |
JOUR |
| author |
Quirós, F. Rossi, J.D. Vázquez, J.L. |
| spellingShingle |
Quirós, F. Rossi, J.D. Vázquez, J.L. Thermal avalanche for blowup solutions of semilinear heat equations |
| author_facet |
Quirós, F. Rossi, J.D. Vázquez, J.L. |
| author_sort |
Quirós, F. |
| title |
Thermal avalanche for blowup solutions of semilinear heat equations |
| title_short |
Thermal avalanche for blowup solutions of semilinear heat equations |
| title_full |
Thermal avalanche for blowup solutions of semilinear heat equations |
| title_fullStr |
Thermal avalanche for blowup solutions of semilinear heat equations |
| title_full_unstemmed |
Thermal avalanche for blowup solutions of semilinear heat equations |
| title_sort |
thermal avalanche for blowup solutions of semilinear heat equations |
| url |
http://hdl.handle.net/20.500.12110/paper_00103640_v57_n1_p0059_Quiros |
| work_keys_str_mv |
AT quirosf thermalavalancheforblowupsolutionsofsemilinearheatequations AT rossijd thermalavalancheforblowupsolutionsofsemilinearheatequations AT vazquezjl thermalavalancheforblowupsolutionsofsemilinearheatequations |
| _version_ |
1807320111242018816 |