Polynomial-exponential stability and blow-up solutions to a nonlinear damped viscoelastic Petrovsky equation

• A. Peyravi ^[1] ; F. Tahamtani ^[1]
  1. [1] Universidad de Shiraz
• Localización: SeMA Journal: Boletín de la Sociedad Española de Matemática Aplicada, ISSN-e 2254-3902, ISSN 2254-3902, Vol. 77, Nº. 2, 2020, págs. 181-201
• Idioma: inglés
• DOI: 10.1007/s40324-019-00210-0
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• Resumen

  • This work is concerned with the initial boundary value problem for a nonlinear viscoelastic Petrovsky equation utt+Δ2u−∫t0g(t−τ)Δ2u(τ)dτ−Δut−Δutt+ut|ut|m−1=u|u|p−1.

    We prove that the solution energy has polynomial rate of decay, even if the kernel g decays exponentially provided m>1 while decay rates is exponentially in the case of weak damping. The unbounded properties of solutions in two cases m=1 and p>m≥1 have been also investigated. For the first case, we prove the blow-up of solutions with different ranges of initial energy. For the second case, we prove blow-up of solutions under some restrictions on g when the initial energy is negative or non negative at less than potential well depth.