Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion
We establish the existence and uniqueness of weak solutions to the parabolic system with nonstandard growth condition and cross diffusion, $$\displaylines{ \partial_tu-\text{div}a(x,t,\nabla u)) =\text{div}|F|^{p(x,t)-2}F),\cr \partial_tv-\text{div}a(x,t,\nabla v))=\delta\Delta u, }$$ where...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2020-12-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2020/123/abstr.html |
| _version_ | 1818677315979182080 |
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| author | Gurusamy Arumugam Andre H. Erhardt |
| author_facet | Gurusamy Arumugam Andre H. Erhardt |
| author_sort | Gurusamy Arumugam |
| collection | DOAJ |
| description | We establish the existence and uniqueness of weak solutions to the parabolic system with
nonstandard growth condition and cross diffusion,
$$\displaylines{
\partial_tu-\text{div}a(x,t,\nabla u))
=\text{div}|F|^{p(x,t)-2}F),\cr
\partial_tv-\text{div}a(x,t,\nabla v))=\delta\Delta u,
}$$
where $\delta\ge0$ and $\partial_tu,~\partial_tv$ denote the partial derivative of u
and v with respect to the time variable t, while $\nabla u$ and $\nabla v$ denote
the one with respect to the spatial variable x. Moreover, the vector field $a(x,t,\cdot)$
satisfies certain nonstandard p(x,t) growth, monotonicity and coercivity conditions. |
| first_indexed | 2024-12-17T08:57:26Z |
| format | Article |
| id | doaj.art-2a22d86f96ef4341b5947bc41fbd2bde |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-17T08:57:26Z |
| publishDate | 2020-12-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-2a22d86f96ef4341b5947bc41fbd2bde2022-12-21T21:55:54ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912020-12-012020123,113Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusionGurusamy Arumugam0Andre H. Erhardt1 Indian Inst. of Technology Gandhinagar, Gujarat, India Univ. of Oslo, Oslo, Norway We establish the existence and uniqueness of weak solutions to the parabolic system with nonstandard growth condition and cross diffusion, $$\displaylines{ \partial_tu-\text{div}a(x,t,\nabla u)) =\text{div}|F|^{p(x,t)-2}F),\cr \partial_tv-\text{div}a(x,t,\nabla v))=\delta\Delta u, }$$ where $\delta\ge0$ and $\partial_tu,~\partial_tv$ denote the partial derivative of u and v with respect to the time variable t, while $\nabla u$ and $\nabla v$ denote the one with respect to the spatial variable x. Moreover, the vector field $a(x,t,\cdot)$ satisfies certain nonstandard p(x,t) growth, monotonicity and coercivity conditions.http://ejde.math.txstate.edu/Volumes/2020/123/abstr.htmlnonlinear parabolic problemnonstandard growthcross diffusion |
| spellingShingle | Gurusamy Arumugam Andre H. Erhardt Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion Electronic Journal of Differential Equations nonlinear parabolic problem nonstandard growth cross diffusion |
| title | Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion |
| title_full | Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion |
| title_fullStr | Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion |
| title_full_unstemmed | Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion |
| title_short | Existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion |
| title_sort | existence and uniqueness of weak solutions to parabolic problems with nonstandard growth and cross diffusion |
| topic | nonlinear parabolic problem nonstandard growth cross diffusion |
| url | http://ejde.math.txstate.edu/Volumes/2020/123/abstr.html |
| work_keys_str_mv | AT gurusamyarumugam existenceanduniquenessofweaksolutionstoparabolicproblemswithnonstandardgrowthandcrossdiffusion AT andreherhardt existenceanduniquenessofweaksolutionstoparabolicproblemswithnonstandardgrowthandcrossdiffusion |