Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

• Main Author: Mitsuhiro Nakao
• Format: Article
• Language: English
• Published: AGH Univeristy of Science and Technology Press 2020-12-01
• Series: Opuscula Mathematica
• Subjects: energy decay; global existence; semilinear wave equation; noncylindrical domains
• Online Access: https://www.opuscula.agh.edu.pl/vol40/6/art/opuscula_math_4040.pdf

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.