Decay rates of energy of the 1D damped original nonlinear wave equation |
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Institution: | 1. Antalya Bilim University, Department of Industrial Engineering, Dosemealti, 07190 Antalya, Turkey;2. Azerbaijan State Economic University, Center of Analytical-Information Resources, 194 M. Mukhtarov, AZ1001 Baku, Azerbaijan;3. Istanbul Technical University Maslak, 34467, Turkey;4. University of Alabama, Tuscaloosa, AL 35487, USA |
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Abstract: | We consider the decay rate of energy of the 1D damped original nonlinear wave equation. We first construct a new energy function. Then, employing the perturbed energy method and the generalized Young’s inequality, we prove that, with a general growth assumption on the nonlinear damping force near the origin, the decay rate of energy is governed by a dissipative ordinary differential equation. This allows us to recover the classical exponential, polynomial, or logarithmic decay rate for the linear, polynomial or exponentially degenerating damping force near the origin, respectively. Unlike the linear wave equation, the exponential decay rate constant depends on the initial data, due to the nonlinearity. |
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Keywords: | Original nonlinear wave equation Energy function Nonlinear damping Perturbed energy method Generalized Young’s inequality |
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