SDDEs limits solutions to sublinear reaction-diffusion SPDEs
We start by introducing a new definition of solutions to heat-based SPDEs driven by space-time white noise: SDDEs (stochastic differential-difference equations) limits solutions. In contrast to the standard direct definition of SPDEs solutions; this new notion, which builds on and refines our SDDEs...
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| Format: | Article |
| Language: | English |
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Texas State University
2003-11-01
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| Series: | Electronic Journal of Differential Equations |
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| Online Access: | http://ejde.math.txstate.edu/Volumes/2003/111/abstr.html |
| _version_ | 1830211084110266368 |
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| author | Hassan Allouba |
| author_facet | Hassan Allouba |
| author_sort | Hassan Allouba |
| collection | DOAJ |
| description | We start by introducing a new definition of solutions to heat-based SPDEs driven by space-time white noise: SDDEs (stochastic differential-difference equations) limits solutions. In contrast to the standard direct definition of SPDEs solutions; this new notion, which builds on and refines our SDDEs approach to SPDEs from earlier work, is entirely based on the approximating SDDEs. It is applicable to, and gives a multiscale view of, a variety of SPDEs. We extend this approach in related work to other heat-based SPDEs (Burgers, Allen-Cahn, and others) and to the difficult case of SPDEs with multi-dimensional spacial variable. We focus here on one-spacial-dimensional reaction-diffusion SPDEs; and we prove the existence of a SDDEs limit solution to these equations under less-than-Lipschitz conditions on the drift and the diffusion coefficients, thus extending our earlier SDDEs work to the nonzero drift case. The regularity of this solution is obtained as a by-product of the existence estimates. The uniqueness in law of our SPDEs follows, for a large class of such drifts/diffusions, as a simple extension of our recent Allen-Cahn uniqueness result. We also examine briefly, through order parameters $epsilon_1$ and $epsilon_2$ multiplied by the Laplacian and the noise, the effect of letting $epsilon_1,epsilon_2o 0$ at different speeds. More precisely, it is shown that the ratio $epsilon_2/epsilon_1^{1/4}$ determines the behavior as $epsilon_1,epsilon_2o 0$. |
| first_indexed | 2024-12-18T05:37:25Z |
| format | Article |
| id | doaj.art-2b9e1bb1a2b54304964984875813ddfe |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-18T05:37:25Z |
| publishDate | 2003-11-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-2b9e1bb1a2b54304964984875813ddfe2022-12-21T21:19:15ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912003-11-012003111121SDDEs limits solutions to sublinear reaction-diffusion SPDEsHassan AlloubaWe start by introducing a new definition of solutions to heat-based SPDEs driven by space-time white noise: SDDEs (stochastic differential-difference equations) limits solutions. In contrast to the standard direct definition of SPDEs solutions; this new notion, which builds on and refines our SDDEs approach to SPDEs from earlier work, is entirely based on the approximating SDDEs. It is applicable to, and gives a multiscale view of, a variety of SPDEs. We extend this approach in related work to other heat-based SPDEs (Burgers, Allen-Cahn, and others) and to the difficult case of SPDEs with multi-dimensional spacial variable. We focus here on one-spacial-dimensional reaction-diffusion SPDEs; and we prove the existence of a SDDEs limit solution to these equations under less-than-Lipschitz conditions on the drift and the diffusion coefficients, thus extending our earlier SDDEs work to the nonzero drift case. The regularity of this solution is obtained as a by-product of the existence estimates. The uniqueness in law of our SPDEs follows, for a large class of such drifts/diffusions, as a simple extension of our recent Allen-Cahn uniqueness result. We also examine briefly, through order parameters $epsilon_1$ and $epsilon_2$ multiplied by the Laplacian and the noise, the effect of letting $epsilon_1,epsilon_2o 0$ at different speeds. More precisely, it is shown that the ratio $epsilon_2/epsilon_1^{1/4}$ determines the behavior as $epsilon_1,epsilon_2o 0$.http://ejde.math.txstate.edu/Volumes/2003/111/abstr.htmlReaction-diffusion SPDESDDESDDE limits solutionsmultiscale. |
| spellingShingle | Hassan Allouba SDDEs limits solutions to sublinear reaction-diffusion SPDEs Electronic Journal of Differential Equations Reaction-diffusion SPDE SDDE SDDE limits solutions multiscale. |
| title | SDDEs limits solutions to sublinear reaction-diffusion SPDEs |
| title_full | SDDEs limits solutions to sublinear reaction-diffusion SPDEs |
| title_fullStr | SDDEs limits solutions to sublinear reaction-diffusion SPDEs |
| title_full_unstemmed | SDDEs limits solutions to sublinear reaction-diffusion SPDEs |
| title_short | SDDEs limits solutions to sublinear reaction-diffusion SPDEs |
| title_sort | sddes limits solutions to sublinear reaction diffusion spdes |
| topic | Reaction-diffusion SPDE SDDE SDDE limits solutions multiscale. |
| url | http://ejde.math.txstate.edu/Volumes/2003/111/abstr.html |
| work_keys_str_mv | AT hassanallouba sddeslimitssolutionstosublinearreactiondiffusionspdes |