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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
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