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Global existence and asymptotic behavior of solutions for a semi-linear wave equation. (English) Zbl 1232.35099
This work is concerned with the Cauchy initial value problem of a damped semilinear wave equation in $n$-dimensional Euclidean space. If we call the first time derivative of the dependent variable $u$ as the velocity and its second time derivative as the acceleration, the linear part of the equation consists of the acceleration, the Laplacian of $u$, the Laplacian of the acceleration and a linear damping term given by the velocity. The non-linear damping is characterised by a smooth non-linear function of the velocity which is assumed to be of the order of the square of the velocity as the velocity tends to zero. Initial conditions for $u$ and the velocity are prescribed functions at $t=0$. In order to attack the non-linear Cauchy problem, the authors first investigate the decay estimate of solutions of the linear damped wave equation by applying a Fourier transform on space variables and then study the solution of the resulting time-dependent ordinary differential equation. Based on this estimate, they define a set of time-weighted Sobolev spaces leading to a Banach space into which the solution functions are placed. Then, they prove the global existence and uniqueness of the solution of the semilinear equation by employing the contraction mapping theorem.
##### MSC:
 35L71 Semilinear second-order hyperbolic equations 35L05 Wave equation (hyperbolic PDE) 35L20 Second order hyperbolic equations, boundary value problems