Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains
We use $Gamma$--convergence to prove existence of stable multiple--layer stationary solutions (stable patterns) to the reaction--diffusion equation. $$ eqalign{ {partial v_varepsilon over partial t} =& varepsilon^2, hbox{div}, (k_1(x)abla v_varepsilon) + k_2(x)(v_varepsilon -alpha)(Beta-v_va...
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| Format: | Article |
| Language: | English |
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Texas State University
1997-12-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/1997/22/abstr.html |
| _version_ | 1811247164231254016 |
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| author | Arnaldo Simal do Nascimento |
| author_facet | Arnaldo Simal do Nascimento |
| author_sort | Arnaldo Simal do Nascimento |
| collection | DOAJ |
| description | We use $Gamma$--convergence to prove existence of stable multiple--layer stationary solutions (stable patterns) to the reaction--diffusion equation. $$ eqalign{ {partial v_varepsilon over partial t} =& varepsilon^2, hbox{div}, (k_1(x)abla v_varepsilon) + k_2(x)(v_varepsilon -alpha)(Beta-v_varepsilon) (v_varepsilon -gamma_varepsilon(x)),,hbox{ in }Omegaimes{Bbb R}^+ cr &v_varepsilon(x,0) = v_0 quad {partial v_varepsilon over partial widehat{n}} = 0,, quadhbox{ for } xin partialOmega,, t >0,.} $$ Given nested simple closed curves in ${Bbb R}^2$, we give sufficient conditions on their curvature so that the reaction--diffusion problem possesses a family of stable patterns. In particular, we extend to two-dimensional domains and to a spatially inhomogeneous source term, a previous result by Yanagida and Miyata. |
| first_indexed | 2024-04-12T15:05:17Z |
| format | Article |
| id | doaj.art-3af05a77b95249cdbbc303a19377a342 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-12T15:05:17Z |
| publishDate | 1997-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-3af05a77b95249cdbbc303a19377a3422022-12-22T03:27:58ZengTexas State UniversityElectronic Journal of Differential Equations1072-66911997-12-01199722117Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domainsArnaldo Simal do NascimentoWe use $Gamma$--convergence to prove existence of stable multiple--layer stationary solutions (stable patterns) to the reaction--diffusion equation. $$ eqalign{ {partial v_varepsilon over partial t} =& varepsilon^2, hbox{div}, (k_1(x)abla v_varepsilon) + k_2(x)(v_varepsilon -alpha)(Beta-v_varepsilon) (v_varepsilon -gamma_varepsilon(x)),,hbox{ in }Omegaimes{Bbb R}^+ cr &v_varepsilon(x,0) = v_0 quad {partial v_varepsilon over partial widehat{n}} = 0,, quadhbox{ for } xin partialOmega,, t >0,.} $$ Given nested simple closed curves in ${Bbb R}^2$, we give sufficient conditions on their curvature so that the reaction--diffusion problem possesses a family of stable patterns. In particular, we extend to two-dimensional domains and to a spatially inhomogeneous source term, a previous result by Yanagida and Miyata.http://ejde.math.txstate.edu/Volumes/1997/22/abstr.htmlDiffusion equationGamma-convergencetransition layersstable equilibria. |
| spellingShingle | Arnaldo Simal do Nascimento Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains Electronic Journal of Differential Equations Diffusion equation Gamma-convergence transition layers stable equilibria. |
| title | Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains |
| title_full | Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains |
| title_fullStr | Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains |
| title_full_unstemmed | Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains |
| title_short | Stable multiple-layer stationary solutions of a semilinear parabolic equation in two-dimensional domains |
| title_sort | stable multiple layer stationary solutions of a semilinear parabolic equation in two dimensional domains |
| topic | Diffusion equation Gamma-convergence transition layers stable equilibria. |
| url | http://ejde.math.txstate.edu/Volumes/1997/22/abstr.html |
| work_keys_str_mv | AT arnaldosimaldonascimento stablemultiplelayerstationarysolutionsofasemilinearparabolicequationintwodimensionaldomains |