Regularity for solutions of nonlinear second order evolution equations

Abstract

This paper deals with the existence of solutions for the class of nonlinear second order evolution equations. The regularity and a variation of solutions of the given equations are also given. As particular cases of our general formulation, some results for Volterra integrodifferential equations of the hyperbolic type are given.

Regular Problems for Semilinear Hyperbolic Type Equations

“…Section 3 is devoted to the study of the regularity of solutions of the linear wave equations in Gelfand triple spaces. Subsequently, our construction of a local solution of the nonlinear equation (SE) is essentially based on [7]. We will show the energy inequalities for our problem (SE) with the aid of estimate of L 2 -type of the solutions.…”

“…In case where A = A(t), by Theorem 3.1 in [7] (or Theorem 6.1.3 in [15]), the problems (SE) has a unique local solution on some interval [0, T c ) for some T c ≤ T . Also when A(t) depends on t, similar results hold when the equation (SE) has a fundamental solution, see Remark 6.1.1 of [15].…”