Opuscula Math. 40, no. 6 (2020), 725-736

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.
  • Received: 2020-03-20.
  • Revised: 2020-11-07.
  • Accepted: 2020-11-07.
  • Published online: 2020-12-01.
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Cite this article as:
Mitsuhiro Nakao, Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains, Opuscula Math. 40, no. 6 (2020), 725-736, https://doi.org/10.7494/OpMath.2020.40.6.725

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