Gradient estimates for a nonlinear parabolic equation with potential under geometric flow
Let (M,g) be an n dimensional complete Riemannian manifold. In this article we prove local Li-Yau type gradient estimates for all positive solutions to the nonlinear parabolic equation $$ (\partial_t - \Delta_g + \mathcal{R}) u( x, t) = - a u( x, t) \log u( x, t) $$ along the generalised geom...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2015-01-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2015/12/abstr.html |
| _version_ | 1811200624653500416 |
|---|---|
| author | Abimbola Abolarinwa |
| author_facet | Abimbola Abolarinwa |
| author_sort | Abimbola Abolarinwa |
| collection | DOAJ |
| description | Let (M,g) be an n dimensional complete Riemannian manifold.
In this article we prove local Li-Yau type gradient estimates for all positive
solutions to the nonlinear parabolic equation
$$
(\partial_t - \Delta_g + \mathcal{R}) u( x, t) = - a u( x, t) \log u( x, t)
$$
along the generalised geometric flow on M. Here
$\mathcal{R} = \mathcal{R} (x, t)$ is a smooth potential function and a
is an arbitrary constant. As an application we derive a global estimate and
a space-time Harnack inequality. |
| first_indexed | 2024-04-12T02:06:43Z |
| format | Article |
| id | doaj.art-3e0a555efb39412d861ed1f738165d8c |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-04-12T02:06:43Z |
| publishDate | 2015-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-3e0a555efb39412d861ed1f738165d8c2022-12-22T03:52:31ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912015-01-01201512,111Gradient estimates for a nonlinear parabolic equation with potential under geometric flowAbimbola Abolarinwa0 Univ. of Sussex, Brighton, UK Let (M,g) be an n dimensional complete Riemannian manifold. In this article we prove local Li-Yau type gradient estimates for all positive solutions to the nonlinear parabolic equation $$ (\partial_t - \Delta_g + \mathcal{R}) u( x, t) = - a u( x, t) \log u( x, t) $$ along the generalised geometric flow on M. Here $\mathcal{R} = \mathcal{R} (x, t)$ is a smooth potential function and a is an arbitrary constant. As an application we derive a global estimate and a space-time Harnack inequality.http://ejde.math.txstate.edu/Volumes/2015/12/abstr.htmlGradient estimatesHarnack inequalitiesparabolic equationsgeometric flows |
| spellingShingle | Abimbola Abolarinwa Gradient estimates for a nonlinear parabolic equation with potential under geometric flow Electronic Journal of Differential Equations Gradient estimates Harnack inequalities parabolic equations geometric flows |
| title | Gradient estimates for a nonlinear parabolic equation with potential under geometric flow |
| title_full | Gradient estimates for a nonlinear parabolic equation with potential under geometric flow |
| title_fullStr | Gradient estimates for a nonlinear parabolic equation with potential under geometric flow |
| title_full_unstemmed | Gradient estimates for a nonlinear parabolic equation with potential under geometric flow |
| title_short | Gradient estimates for a nonlinear parabolic equation with potential under geometric flow |
| title_sort | gradient estimates for a nonlinear parabolic equation with potential under geometric flow |
| topic | Gradient estimates Harnack inequalities parabolic equations geometric flows |
| url | http://ejde.math.txstate.edu/Volumes/2015/12/abstr.html |
| work_keys_str_mv | AT abimbolaabolarinwa gradientestimatesforanonlinearparabolicequationwithpotentialundergeometricflow |