Abstract
We consider the initial boundary value problem for nonlinear damped wave equations of the form \(u^{\prime \prime }+M({\textstyle \int _{\Omega }}\left \vert (-\Delta )^{s}u\right \vert ^{2}dx)\Delta u+(-\Delta )^{\alpha }u^{\prime }=f,\) with Neumann boundary conditions. We prove global existence of solutions, when s ∈ [1∕2, 1] and α ∈ (0, 1], and we show that the energy of these ones decays exponentially, as t →∞. The uniqueness of solutions is also obtained when α ∈ [1∕2, 1].
Dedicated to Prof. Enrique Fernández-Cara on the occasion of his 60th birthday.
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The author “F. D. Araruna” is partially supported by INCTMat, CAPES, CNPq (Brazil).
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Araruna, F.D., Matias, F.d.O., Oliveira, M.d.L., Souza, S.M.S.e. (2018). Well-Posedness and Asymptotic Behavior for a Nonlinear Wave Equation. In: Doubova, A., González-Burgos, M., Guillén-González, F., Marín Beltrán, M. (eds) Recent Advances in PDEs: Analysis, Numerics and Control. SEMA SIMAI Springer Series, vol 17. Springer, Cham. https://doi.org/10.1007/978-3-319-97613-6_2
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