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

Mitsuhiro Nakao

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.
• Revised: 2020-11-07.
• Accepted: 2020-11-07.
• Published online: 2020-12-01.