Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating
We consider a non-local initial boundary-value problem for the equation $$ u_t=Delta u+lambda f(u)/Big(int_{Omega}f(u),dxBig)^2 ,quad x in Omega subset mathbb{R}^2 ,,;t>0, $$ where $u$ represents a temperature and $f$ is a positive and decreasing function. It is shown that for the radially symmet...
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| Format: | Article |
| Language: | English |
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Texas State University
2002-02-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2002/11/abstr.html |
| _version_ | 1819058591686983680 |
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| author | Dimitrios E. Tzanetis |
| author_facet | Dimitrios E. Tzanetis |
| author_sort | Dimitrios E. Tzanetis |
| collection | DOAJ |
| description | We consider a non-local initial boundary-value problem for the equation $$ u_t=Delta u+lambda f(u)/Big(int_{Omega}f(u),dxBig)^2 ,quad x in Omega subset mathbb{R}^2 ,,;t>0, $$ where $u$ represents a temperature and $f$ is a positive and decreasing function. It is shown that for the radially symmetric case, if $int_{0}^{infty}f(s),ds <infty $ then there exists a critical value $lambda^{ast}>0$ such that for $lambda>lambda^{ast}$ there is no stationary solution and $u$ blows up, whereas for $lambda<lambda^{ast}$ there exists at least one stationary solution. Moreover, for the Dirichlet problem with $-s,f'(s)<f(s)$ there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if $lambda<lambda^{ast}$ then there are at least two solutions, while if $lambda=lambda^{ast}$ at least one solution. Stability and blow-up of these solutions are examined in this article. |
| first_indexed | 2024-12-21T13:57:38Z |
| format | Article |
| id | doaj.art-938052aa2e7c4302a7fe94a713576d12 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-21T13:57:38Z |
| publishDate | 2002-02-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-938052aa2e7c4302a7fe94a713576d122022-12-21T19:01:28ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912002-02-01200211126Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heatingDimitrios E. TzanetisWe consider a non-local initial boundary-value problem for the equation $$ u_t=Delta u+lambda f(u)/Big(int_{Omega}f(u),dxBig)^2 ,quad x in Omega subset mathbb{R}^2 ,,;t>0, $$ where $u$ represents a temperature and $f$ is a positive and decreasing function. It is shown that for the radially symmetric case, if $int_{0}^{infty}f(s),ds <infty $ then there exists a critical value $lambda^{ast}>0$ such that for $lambda>lambda^{ast}$ there is no stationary solution and $u$ blows up, whereas for $lambda<lambda^{ast}$ there exists at least one stationary solution. Moreover, for the Dirichlet problem with $-s,f'(s)<f(s)$ there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if $lambda<lambda^{ast}$ then there are at least two solutions, while if $lambda=lambda^{ast}$ at least one solution. Stability and blow-up of these solutions are examined in this article.http://ejde.math.txstate.edu/Volumes/2002/11/abstr.htmlNonlocal parabolic equationsblow-upglobal existencesteady states. |
| spellingShingle | Dimitrios E. Tzanetis Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating Electronic Journal of Differential Equations Nonlocal parabolic equations blow-up global existence steady states. |
| title | Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating |
| title_full | Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating |
| title_fullStr | Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating |
| title_full_unstemmed | Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating |
| title_short | Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating |
| title_sort | blow up of radially symmetric solutions of a non local problem modelling ohmic heating |
| topic | Nonlocal parabolic equations blow-up global existence steady states. |
| url | http://ejde.math.txstate.edu/Volumes/2002/11/abstr.html |
| work_keys_str_mv | AT dimitriosetzanetis blowupofradiallysymmetricsolutionsofanonlocalproblemmodellingohmicheating |