Well-Posedness and Asymptotic Behavior for a Nonlinear Wave Equation

  • Fágner Dias ArarunaEmail author
  • Frederico de Oliveira Matias
  • Milton de Lacerda Oliveira
  • Shirley Maria Santos e Souza
Part of the SEMA SIMAI Springer Series book series (SEMA SIMAI, volume 17)


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].


Wave equation Well-posedness Asymptotic behavior 

AMS Subject Classifications

35L70 35B40 74K10 



The author “F. D. Araruna” is partially supported by INCTMat, CAPES, CNPq (Brazil).


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Fágner Dias Araruna
    • 1
    Email author
  • Frederico de Oliveira Matias
    • 1
  • Milton de Lacerda Oliveira
    • 1
  • Shirley Maria Santos e Souza
    • 1
  1. 1.Departamento de MatemáticaUniversidade Federal da ParaíbaJoão PessoaBrazil

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