An abstract semilinear hyperbolic Volterra integrodifferential equation

Abstract We consider semilinear integrodifferential equations of the form u′(t) + A(t) u(t) = ∝0t g(t, s, u(s)) ds + f(t), u(0) = u0. For each t ⩾ 0, the operator A(t) is assumed to be the negative generator of a strongly continuous semigroup in a Banach space X. The domain D(A(t)) of A(t) is allowed to vary with t. Thus our models are Volterra integrodifferential equations of “hyperbolic type.” These types of equations arise naturally in the study of viscoelasticity. Our main results are the proofs of existence, uniqueness, continuation and continuous dependence of the solutions.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  H. Fattorini,et al.  Ordinary differential equations in linear topological space. II. , 1969 .

[3]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[4]  J. Nohel,et al.  Energy methods for nonlinear hyperbolic volterra integrodifferential equations , 1979 .

[5]  J. Goldstein.  Semigroups and second-order differential equations , 1969 .

[6]  G. F. Webb,et al.  Abstract volterra integrodifferential equations and a class of reaction-diffusion equations , 1979 .

[7]  W. Edelstein.  Existence of solutions to the displacement problem for quasistatic viscoelasticity , 1966 .

[8]  Y. Choe.  Linear evolution equations of hyperbolic type in a locally convex space , 1981 .

[9]  Constantine M. Dafermos,et al.  An abstract Volterra equation with applications to linear viscoelasticity , 1970 .

[10]  Tosio Kato,et al.  Linear evolution equations of ``hyperbolic'' type, II , 1973 .

[11]  W. Edelstein,et al.  Uniqueness theorems in the linear dynamic theory of anisotropic viscoelastic solids , 1964 .

[12]  K. Friedrichs.  The identity of weak and strong extensions of differential operators , 1944 .

[13]  R. C. MacCamy,et al.  An integro-differential equation with application in heat flow , 1977 .

[14]  C. Travis,et al.  An abstract second order semi-linear Volterra integrodi er-ential equation , 1979 .

[15]  Jerome A. Goldstein,et al.  Time dependent hyperbolic equations , 1969 .

[16]  Kurt Friedrichs,et al.  Symmetric positive linear differential equations , 1958 .

[17]  Tosio Kato,et al.  Integration of the equation of evolution in a Banach space , 1953 .

[18]  I. Tadjbakhsh,et al.  Uniqueness in the linear theory of viscoelasticity , 1965 .

[19]  Tosio Kato,et al.  On linear differential equations in banach spaces , 1956 .

[20]  R. MacCamy.  A model for one-dimensional, nonlinear viscoelasticity , 1977 .

[21]  G. Webb.  An abstract semilinear Volterra integrodifferential equation , 1978 .

[22]  C. Travis,et al.  Cosine families and abstract nonlinear second order differential equations , 1978 .