A note on the dispersive-dissipative wave equation having strong damping and nonlinear sources
Year: 2024
Type: article
Abstract: In this paper, we delve into an initial-boundary value problem of the dispersive-dissipative wave equation having strong damping, $ u_{tt}-\Delta u-\Delta u_{t}+\Delta^{2} u-\Delta u_{tt} = a|u|^{p-1} u-b|u|^{q-1} u \quad \text{in}\ \Omega\times(0, T_{\max}), $where $ \Omega $ is an open bounded domain in $ \mathbb{R}^{n} $ with smooth boundary; $ T_{\max} $ is the maximal existence time; $ a, b>0 $, $ p, q>1 $ are constants. Through the combination of the contraction mapping principle and the Faedo-Galerkin method, we achieve the local existence and uniqueness of the weak solution. For $ p>q $ or $ p = q $ and $ a>b $, we obtain the existence of the global weak solution, and the energy decay estimate. Moreover, based on the auxiliary function approach, we demonstrate that the weak solutions blow up in finite time under distinct conditions related to the initial energy. Furthermore, we succeed in obtaining lower and upper bounds for blow-up time (life span). For $ p
DOI: 10.3934/cpaa.2024080
Authors: Mengyao Liu, Bingchen Liu
Institution: China University of Petroleum, Beijing
Language: English
Subfield: Control and Systems Engineering
Field: Engineering
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4403950730