Regularity for Semilinear Neutral Hyperbolic Equations with Cosine Families

Seong-Ho Cho,Jin-Mun Jeong

doi:10.3390/math8071157

Abstract

The purpose of this paper is to obtain the regularity for solutions of semilinear neutral hyperbolic equations with the nonlinear convolution. The principal operator is the infinitesimal generator of a cosine and sine families. In order to show a variation of constant formula for solutions, we make of using the nature of cosine and sine families.

Highlights

IntroductionThis paper is to establish the regularity of solutions of the following abstract semilinear neutral hyperbolic equation in a Banach space X:
This paper is to establish the regularity of solutions of the following abstract semilinear neutral hyperbolic equation in a Banach space X: ( d dt [ w ( t ) +w (0) = x0, g(t, w(t))] = Aw(t) + F (t, w) + f (t), w (0) = y0 . 0 < t ≤ T, (1)The principal operator A is the infinitesimal generator of a cosine family C (t)(t ∈ R)
This paper investigates the regularity for solutions of semilinear neutral hyperbolic equations with the nonlinear convolution

Summary

Introduction

This paper is to establish the regularity of solutions of the following abstract semilinear neutral hyperbolic equation in a Banach space X:. The regularity of solutions of parabolic type equations under some general conditions of the nonlinear terms is considered in [11,12], which is reasonable for application in nonlinear systems. It is worth giving several examples of the use of such and other classes of differential equations in engineering and scientific tasks, for instance, almost automorphic mild solutions of hyperbolic evolution equations with stepanov-like almost automorphic forcing term have studied in [13], and the local well-posedness to the Cauchy problem of the 2D compressible. An example as application of our results in the last section is given

Preliminaries

Semilinear Neutral Equations

Conclusions