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

Abstract

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.

Keywords:

• Optimal decay rate
• Neumann boundary condition
• evolution equation
• frictional dissipation
• wave equation with memory
• decay estimates

AMS Subject Classifications:

• 35L53
• 35L05
• 35B40
• 45K05

Disclosure statement

The authors declare that they have no competing interests.

Additional information

Funding

The work was supported by the National Natural Science Foundation of China [grant number 11371095].