Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level
Abstract In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation ut−△ut−△u−div(|∇u|2q∇u)=up $$ u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}\bigl(| \nabla u|^{2q}\nabla u\bigr)=u^{p} $$ which was studied by Peng et al. (Appl. Math....
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-10-01
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| Series: | Boundary Value Problems |
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| Online Access: | http://link.springer.com/article/10.1186/s13661-018-1079-7 |
| _version_ | 1819210408579301376 |
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| author | Gongwei Liu Ruimin Zhao |
| author_facet | Gongwei Liu Ruimin Zhao |
| author_sort | Gongwei Liu |
| collection | DOAJ |
| description | Abstract In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation ut−△ut−△u−div(|∇u|2q∇u)=up $$ u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}\bigl(| \nabla u|^{2q}\nabla u\bigr)=u^{p} $$ which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016), where the blow-up phenomena and the lifespan for the initial energy J(u0)<0 $J(u_{0})<0$ were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J(u0)<0 $J(u_{0})<0$. |
| first_indexed | 2024-12-23T06:10:42Z |
| format | Article |
| id | doaj.art-d933bb387876460c9b8ee8ead65d8c6f |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-23T06:10:42Z |
| publishDate | 2018-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-d933bb387876460c9b8ee8ead65d8c6f2022-12-21T17:57:27ZengSpringerOpenBoundary Value Problems1687-27702018-10-012018111010.1186/s13661-018-1079-7Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy levelGongwei Liu0Ruimin Zhao1College of Science, Henan University of TechnologyCollege of Science, Henan University of TechnologyAbstract In this paper, we continue to study the initial boundary value problem of the quasi-linear pseudo-parabolic equation ut−△ut−△u−div(|∇u|2q∇u)=up $$ u_{t}-\triangle u_{t}-\triangle u-\operatorname{div}\bigl(| \nabla u|^{2q}\nabla u\bigr)=u^{p} $$ which was studied by Peng et al. (Appl. Math. Lett. 56:17–22, 2016), where the blow-up phenomena and the lifespan for the initial energy J(u0)<0 $J(u_{0})<0$ were obtained. We establish the finite time blow-up of the solution for the initial data at arbitrary energy level and the lifespan of the blow-up solution. Furthermore, as a product, we obtain the blow-up rate and refine the lifespan when J(u0)<0 $J(u_{0})<0$.http://link.springer.com/article/10.1186/s13661-018-1079-7Blow-upLifespanBlow-up rateQuasi-linear pseudo-parabolic equation |
| spellingShingle | Gongwei Liu Ruimin Zhao Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level Boundary Value Problems Blow-up Lifespan Blow-up rate Quasi-linear pseudo-parabolic equation |
| title | Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level |
| title_full | Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level |
| title_fullStr | Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level |
| title_full_unstemmed | Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level |
| title_short | Blow-up phenomena and lifespan for a quasi-linear pseudo-parabolic equation at arbitrary initial energy level |
| title_sort | blow up phenomena and lifespan for a quasi linear pseudo parabolic equation at arbitrary initial energy level |
| topic | Blow-up Lifespan Blow-up rate Quasi-linear pseudo-parabolic equation |
| url | http://link.springer.com/article/10.1186/s13661-018-1079-7 |
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