Such equations have physical application; for example, they arise in problems concerned with heat flow in materials with memory. Linear versions of (1.2) are treated in [6], [ll], [24], [25], and [26] ; the nonlinear version is analysed in [14], [15], and [20] ; similar equations are also treated by [12] and [13]. If we replace u,(x, t) by u,,(x, t) we obtain an equation arising in the theory of viscoelasticity [4] and [5]. There is a growing recent literature concerning Volterra equations in abstract spaces. In [lo] Friedman and Shinbrot study a linear version of (1.1); they investigate the existence, boundedness and asymptotic behavior of solutions. R. Miller [23] associates a linear semigroup with solutions and he obtains information concerning the continuous dependence of solutions on initial data. Barbu [l] considers Volterra equations in Hilbert space; Crandall, Londen, and Nohel [2] and Crandall and Nohel [3] apply the theory of multivalued accretive operators to study such equations; Webb in [36] studies a version with A 3 0 and examines the semilinear equation in [33] and [34]. F or other related work the reader is referred to [7], [8], [30], [31], [32]. In this study we are motivated to apply the techniques of the theory of abstract quasi-linear equations of parabolic type found in Friedman [9] and Sobolevskii [29] to the class of semi-

Semilinear integrodifferential equations in Banach space

An improperly posed problem is studied for a linear partial integro-differential equation of convolution type on the semi-axis. The problem originates from a generalised process of heat conduction in materials with fading memory, where the temperature of the material has to be determined for prescribed homogeneous boundary conditions and for a given final temperature distribution. By using eigenfunctions of the n -dimensional Laplacian, the problem is reduced to a family of equivalent initial-value problems for ordinary integro-differential equations; the latter are treated by the method of factorisation in a suitable function algebra. Sufficient conditions for the existence and uniqueness of solutions to the original problem are obtained in terms of the solvability conditions of the reduced problems. The whole analysis is performed simultaneously in a broad variety of spaces consisting of functions with an exponential growth rate (in the time variable) at infinity. One of the main advantages in the present approach is that solutions, if they exist, can always be computed explicitly.

On linear theory of heat conduction in materials with memory. Existence and uniqueness theorems for the final value problem

In this paper we establish the restrictions imposed by the Second law of Thermodynamics on the relaxation functions which arise in the theory of heat conduction with finite wave speeds. We show that (i) the initial values of the energy relaxation function and the heat flux relaxation function are non-negative, (ii) the initial slope of the heat flux relaxation function is non-positive, and (iii) the equilibrium conductivity is non-negative. These results have important implications with regard to the behavior of waves and the uniqueness of solutions.

Thermodynamic restrictions on the relaxation functions of the theory of heat conduction with finite wave speeds

Wave propagation and uniqueness theorems for elastic materials with internal state variables

On the spatial decay of solutions in the nonlinear theory of heat conduction

On uniqueness and wave propagation in nonlinear heat conductors with memory

A uniqueness theorem for the general theory of heat conduction with finite wave speeds

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Robert P. Herrmann

Papers

On uniqueness and wave propagation in nonlinear heat conductors with memory

A uniqueness theorem for the general theory of heat conduction with finite wave speeds

A theorem on the uniqueness of solutions in nonlinear heat conduction