Asymptotic behavior of a semilinear non-autonomous wave equation with distributed delay and analytic nonlinearity

Abstract

This paper is concerned with a semilinear non-autonomous wave equation with distributed delay and analytic nonlinearity. The distributed delay is represented by an integral operator integrating on the delay interval, which considers a segment of the past dynamic information. Firstly we show the global boundedness of solutions and then prove that every globally bounded solution has a relative compact range in the phase space. By the Łojasiewicz–Simon inequality, we prove the convergence to an equilibrium of globally bounded solutions and then show the convergence rate dependence on the Łojasiewicz exponent and the decay conditions on non-autonomous term. The existence and uniqueness of a global strong solution is finally proved. Note that the result of existence and uniqueness of the solution does not need any restrictions on the coefficients between the damping and the delay. The presence of an integral kernel makes distributed delays more difficult to be analyzed compared to its pointwise counterpart.