Optimal decay rates for semilinear wave equations with memory and Neumann boundary conditions

• Published in: Applicable analysis Vol. 97; no. 15; pp. 2594 - 2609
• Main Authors: Xiao, Ti-Jun, Zhang, Hui
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 18.11.2018; Taylor & Francis Ltd
• Subjects: Boundary conditions; decay estimates; Decay rate; Dirichlet problem; evolution equation; Formulas (mathematics); frictional dissipation; Mathematical analysis; Neumann boundary condition; Optimal decay rate; wave equation with memory; Wave equations
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036811.2017.1377834

We consider a class of semilinear wave equations with memory and Neumann boundary conditions, being subject to frictional dissipation. As can be seen, their solutions have different properties from those in the case of Dirichlet boundary conditions (or Dirichlet-Neumann boundary conditions). We prove for some nonlinear terms with growth exponent that all solutions decay uniformly and at least at the polynomial rate , when the memory kernel decays exponentially or polynomially (with large enough degree r); some other decay rates, depending on both q and r, are also derived, when r is not large enough. Moreover, we show a large class of solutions decaying exactly at the rate , in the case of the memory kernels decaying exponentially.