WSEAS Transactions on Mathematics
Print ISSN: 1109-2769, E-ISSN: 2224-2880
Volume 17, 2018
Global Existence and Exponential Decay of Solutions for a Class of Nonlinear Wave Equations
Author:
Abstract: A initial-boundary value problem for some nonlinear wave equation with damping and source terms $$u_{tt}+ A_{u} + u_{t} + aAu_{t} = b|u|^{[(q-1)]}u$$ in a bounded domain is studied, where $$A = (−∆)^m, m ≥ 1$$ is a nature number $$a ≥ 0, b > 0 $$ and $$ q > 1 $$ are real numbers. The existence of global solutions for this problem is proved by
constructing the stable sets, and show the exponential decay estimate of global solutions as time goes to infinity by
applying the multiplier method. Meanwhile, under the conditions of the nonnegative initial energy and $$ α= 0$$, it is showed that the solution blows up in finite time.
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Keywords: Nonlinear wave equation, Initial boundary value problem, Stable sets, Nonlinear damping and source terms, Exponential decay
Pages: 252-257
WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 17, 2018, Art. #31