Abstract volterra integrodifferential equations and a class of reaction-diffusion equations

Regularity for solutions of nonlinear second order evolution equations

“…These equations arise naturally in the study of viscoelasticity in Edelstein and Gurtin [7]. Our formulation of (SE) is a direct attempt to generalize some results of Webb [3] and Heard [8], who studied problems similar to (SE) in the case when A = A(t) does not depend on t. By using the useful integral inequalities, we will show that there exists a solution for the class of nonlinear second order evolution equations by a similar method to that for the linear heat equations of [6]. Section 2 gives some basic results on existence, uniqueness, and a representation formula of solutions for the given equation (SE).…”

“…Here A is the operator associated with a sesquilinear form defined on V × V and satisfying Gårding's inequality, where V is another Hilbert space such that V ⊂ H ⊂ V * (the dual space of V ). The nonlinear term f (·, x), which is a Lipschitz continuous operator with respect to x from V to H , is a semilinear version of the quasilinear one considered in [1][2][3]. Precise assumptions are given in the next section.…”

Integrodifferential equations with analytic semigroups

“…Webb [15] has also considered (1.3) and has assumed that f maps R × X 1 into X α and for each t ∈ R there exists a positive constant C(t) such that…”