On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration
Abstract This paper deals with a class of Petrovsky system with nonlinear damping w t t + Δ B 2 w − k 2 Δ B w t + a w t | w t | m − 2 = b w | w | p − 2 $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{alig...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2020-08-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-020-01438-w |
| _version_ | 1811327736391663616 |
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| author | Jiali Yu Yadong Shang Huafei Di |
| author_facet | Jiali Yu Yadong Shang Huafei Di |
| author_sort | Jiali Yu |
| collection | DOAJ |
| description | Abstract This paper deals with a class of Petrovsky system with nonlinear damping w t t + Δ B 2 w − k 2 Δ B w t + a w t | w t | m − 2 = b w | w | p − 2 $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ on a manifold with conical singularity, where Δ B $\Delta _{\mathbb{B}}$ is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary x 1 = 0 $x_{1}=0$ . We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy E ( 0 ) < d $E(0)< d$ . |
| first_indexed | 2024-04-13T15:13:18Z |
| format | Article |
| id | doaj.art-68dfb3aec36249aaae2a1aadf77d8ab0 |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-04-13T15:13:18Z |
| publishDate | 2020-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-68dfb3aec36249aaae2a1aadf77d8ab02022-12-22T02:41:57ZengSpringerOpenBoundary Value Problems1687-27702020-08-012020112610.1186/s13661-020-01438-wOn decay and blow-up of solutions for a nonlinear Petrovsky system with conical degenerationJiali Yu0Yadong Shang1Huafei Di2School of Science, Dalian Jiaotong UniversitySchool of Mathematics and Information Science, Guangzhou UniversitySchool of Mathematics and Information Science, Guangzhou UniversityAbstract This paper deals with a class of Petrovsky system with nonlinear damping w t t + Δ B 2 w − k 2 Δ B w t + a w t | w t | m − 2 = b w | w | p − 2 $$\begin{aligned} w_{tt}+\Delta _{\mathbb{B}}^{2}w-k_{2} \Delta _{\mathbb{B}}w_{t}+aw_{t} \vert w_{t} \vert ^{m-2}=bw \vert w \vert ^{p-2} \end{aligned}$$ on a manifold with conical singularity, where Δ B $\Delta _{\mathbb{B}}$ is a Fuchsian-type Laplace operator with totally characteristic degeneracy on the boundary x 1 = 0 $x_{1}=0$ . We first prove the global existence of solutions under conditions without relation between m and p, and establish an exponential decay rate. Furthermore, we obtain a finite time blow-up result for local solutions with low initial energy E ( 0 ) < d $E(0)< d$ .http://link.springer.com/article/10.1186/s13661-020-01438-wPetrovsky systemCone Sobolev spacesGlobal existenceDecay rateBlow-up |
| spellingShingle | Jiali Yu Yadong Shang Huafei Di On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration Boundary Value Problems Petrovsky system Cone Sobolev spaces Global existence Decay rate Blow-up |
| title | On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration |
| title_full | On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration |
| title_fullStr | On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration |
| title_full_unstemmed | On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration |
| title_short | On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration |
| title_sort | on decay and blow up of solutions for a nonlinear petrovsky system with conical degeneration |
| topic | Petrovsky system Cone Sobolev spaces Global existence Decay rate Blow-up |
| url | http://link.springer.com/article/10.1186/s13661-020-01438-w |
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