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

Nakao Mitsuhiro

Opuscula Mathematica

2020

Vol. 40, no. 6

725--736

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain [formula]. We are interested in finite energy solution. We derive an exponential decay of the energy in the case Ω (t) is bounded in [formula] and the estimate [formula] in the case Ω (t) is unbounded. Existence and uniqueness of finite energy solution are also proved.

Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity

Nakao M.

Opuscula Mathematica

2014

Vol. 34, no. 3

569--590

We prove the existence and uniqueness of a global decaying solution to the initial boundary value problem for the quasilinear wave equation with Kelvin-Voigt dissipation and a derivative nonlinearity. To derive the required estimates of the solutions we employ a ‘loan’ method and use a difference inequality on the energy.