Opuscula Math. 40, no. 6 (2020), 725-736
https://doi.org/10.7494/OpMath.2020.40.6.725
Opuscula Mathematica
Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains
Abstract. We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)\lt \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.
Keywords: energy decay, global existence, semilinear wave equation, noncylindrical domains.
Mathematics Subject Classification: 35B35, 35L70.
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- Mitsuhiro Nakao
- Faculty of Mathematics, Kyushu University, Moto-oka 744, Fukuoka 819-0395, Japan
- Communicated by Marek Galewski.
- Received: 2020-03-20.
- Revised: 2020-11-07.
- Accepted: 2020-11-07.
- Published online: 2020-12-01.
