Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions
Abstract This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: {(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾. $$\textstyle\begin{cases} (h(u) )_{t} =\nabla\cdot (|\nabla u|^{p-2}\nabla u )+f(u)...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-04-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-018-1665-3 |
| _version_ | 1818040620274417664 |
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| author | Juntang Ding |
| author_facet | Juntang Ding |
| author_sort | Juntang Ding |
| collection | DOAJ |
| description | Abstract This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: {(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾. $$\textstyle\begin{cases} (h(u) )_{t} =\nabla\cdot (|\nabla u|^{p-2}\nabla u )+f(u) &\mbox{in } D\times(0,t^{*}), \\ \frac{\partial u}{\partial n}=g(u) &\mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \mbox{in } \overline{D}. \end{cases} $$ Here p>2 $p>2$, the spatial region D in RN $\mathbb{R}^{N}$ ( N≥2 $N\geq2$) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique. |
| first_indexed | 2024-12-10T08:17:25Z |
| format | Article |
| id | doaj.art-c7dc44ade22647d39199a34fc59e5aa7 |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-10T08:17:25Z |
| publishDate | 2018-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-c7dc44ade22647d39199a34fc59e5aa72022-12-22T01:56:26ZengSpringerOpenJournal of Inequalities and Applications1029-242X2018-04-012018111410.1186/s13660-018-1665-3Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditionsJuntang Ding0School of Mathematical Sciences, Shanxi UniversityAbstract This paper is devoted to studying the global existence and blow-up results for the following p-Laplacian parabolic problems: {(h(u))t=∇⋅(|∇u|p−2∇u)+f(u)in D×(0,t∗),∂u∂n=g(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾. $$\textstyle\begin{cases} (h(u) )_{t} =\nabla\cdot (|\nabla u|^{p-2}\nabla u )+f(u) &\mbox{in } D\times(0,t^{*}), \\ \frac{\partial u}{\partial n}=g(u) &\mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq0 & \mbox{in } \overline{D}. \end{cases} $$ Here p>2 $p>2$, the spatial region D in RN $\mathbb{R}^{N}$ ( N≥2 $N\geq2$) is bounded, and ∂D is smooth. We set up conditions to ensure that the solution must be a global solution or blows up in some finite time. Moreover, we dedicate upper estimates of the global solution and the blow-up rate. An upper bound for the blow-up time is also specified. Our research relies mainly on constructing some auxiliary functions and using the parabolic maximum principles and the differential inequality technique.http://link.springer.com/article/10.1186/s13660-018-1665-3Blow-upp-Laplacian equationNonlinear boundary condition |
| spellingShingle | Juntang Ding Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions Journal of Inequalities and Applications Blow-up p-Laplacian equation Nonlinear boundary condition |
| title | Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions |
| title_full | Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions |
| title_fullStr | Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions |
| title_full_unstemmed | Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions |
| title_short | Global existence and blow-up results for p-Laplacian parabolic problems under nonlinear boundary conditions |
| title_sort | global existence and blow up results for p laplacian parabolic problems under nonlinear boundary conditions |
| topic | Blow-up p-Laplacian equation Nonlinear boundary condition |
| url | http://link.springer.com/article/10.1186/s13660-018-1665-3 |
| work_keys_str_mv | AT juntangding globalexistenceandblowupresultsforplaplacianparabolicproblemsundernonlinearboundaryconditions |