Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions
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 fol...
| Publicado: |
2002
|
|---|---|
| Materias: | |
| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03605302_v27_n1-2_p395_Quiros http://hdl.handle.net/20.500.12110/paper_03605302_v27_n1-2_p395_Quiros |
| Aporte de: |
| id |
paper:paper_03605302_v27_n1-2_p395_Quiros |
|---|---|
| record_format |
dspace |
| spelling |
paper:paper_03605302_v27_n1-2_p395_Quiros2023-06-08T15:34:47Z Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions Avalanche Blow-up Boundary layer Heat equation Nonlinear boundary condition 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. 2002 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03605302_v27_n1-2_p395_Quiros http://hdl.handle.net/20.500.12110/paper_03605302_v27_n1-2_p395_Quiros |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Avalanche Blow-up Boundary layer Heat equation Nonlinear boundary condition |
| spellingShingle |
Avalanche Blow-up Boundary layer Heat equation Nonlinear boundary condition Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| topic_facet |
Avalanche Blow-up Boundary layer Heat equation Nonlinear boundary condition |
| description |
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. |
| title |
Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| title_short |
Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| title_full |
Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| title_fullStr |
Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| title_full_unstemmed |
Complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| title_sort |
complete blow-up and thermal avalanche for heat equations with nonlinear boundary conditions |
| publishDate |
2002 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_03605302_v27_n1-2_p395_Quiros http://hdl.handle.net/20.500.12110/paper_03605302_v27_n1-2_p395_Quiros |
| _version_ |
1768544367081947136 |