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On decay and blow-up of solutions for a nonlinear Petrovsky system with conical degeneration
- Source :
- Boundary Value Problems, Vol 2020, Iss 1, Pp 1-26 (2020)
- Publication Year :
- 2020
- Publisher :
- SpringerOpen, 2020.
-
Abstract
- 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 $E(0)< d$ .
Details
- Language :
- English
- ISSN :
- 16872770
- Volume :
- 2020
- Issue :
- 1
- Database :
- Directory of Open Access Journals
- Journal :
- Boundary Value Problems
- Publication Type :
- Academic Journal
- Accession number :
- edsdoj.68dfb3aec36249aaae2a1aadf77d8ab0
- Document Type :
- article
- Full Text :
- https://doi.org/10.1186/s13661-020-01438-w