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- Publisher
- Springer Journals
- Copyright
- Copyright © The Editorial Office of AMAS & Springer-Verlag GmbH Germany 2020
- ISSN
- 0168-9673
- eISSN
- 1618-3932
- DOI
- 10.1007/s10255-020-0984-6
- Publisher site
- See Article on Publisher Site
Abstract
In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients utt+Δ2u+k1(t)∣ut∣m−2ut=k2(t)∣u∣p−2u,x∈Ω,t>0,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${u_{tt}} + {{\rm{\Delta }}^2}u + {k_1}\left(t \right){\left| {{u_t}} \right|^{m - 2}}{u_t} = {k_2}\left(t \right){\left| u \right|^{p - 2}}u,\,\,\,\,\,\,\,\,\,\,x \in {\rm{\Omega,}}\,\,\,{\rm{t > 0,}}$$\end{document} where Ω is a bounded domain in Rn. Under certain conditions on k1(t), k2(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.
Journal
Acta Mathematicae Applicatae Sinica – Springer Journals
Published: Dec 27, 2020