Ronald Grimmer

Wellposedness and wave propagation for a class of integrodifferential equations in Banach space

In recent years there has been an increasing interest in the mathematical treatment of problems which originate in the theory of viscoelastic materials. In such problems the constitutive relations for these materials incorporate the history of the strain or the strain rate, and, in the linear theory, lead to linear Volterra integrodifferential equations with infinite memory in a Banach space X. The purpose of this paper is to examine the problems of wellposedness and wave propagation for one such equation. In particular, we are concerned with equations of the form x’(t)=/4 x(r)+J’ [ F(t-s)x(s)ds -al

Nonoscillatory solutions of higher order delay equations

where f is a continuous real valued function for f > 0 and x E R such that f(t, x) is nondecreasing in x for fixed t, and xf(t, x) > 0 if x # 0. The delay function g(t) is continuous and satisfies g(t) to in that it satisfies for r>, t, x(t)x”‘(t) > 0 for i = 0, l,..., I, and (-1)“’ ‘x(t)x”‘(t) < 0, i = 1 + 1, I + 2 ,..., n.

Singular relaxation moduli and smoothing in three-dimensional viscoelasticity

We develop a semigroup setting for linear viscoelasticity in threedimensional space with tensor-valued relaxation modulus and give a criterion on the relaxation kernel for differentiability and analyticity of the solutions. The method is also extended to a simple problem in thermoviscoelasticity.

Integral equations as evolution equations

Abstract We study a class of time-dependent linear integrodifferential equations (VE) with the evolution equation approach. We determine the generators of a time-dependent evolution equation (DE) which is equivalent to the given integrodifferential equation. Under very general assumptions we prove the well-posedness and continuity of (VE) from the stability of (DE). The related question of convergence of a family of approximate solutions is examined. As an application, we include an example of hyperbolic integro-partial-differential equation to illustrate the theory.

On the wellposedness of constitutive laws involving dissipation potentials

We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic. As a technical tool we generalize the notion of an Orlicz space to a cone “normed” by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.

Nonoscillatory solutions of forced second-order linear equations

The number of nonoscillatory solutions of a forced second order linear differential equation is studied under the hypothesis that the homogeneous equation is oscillatory. The main technique involves expressing a general solution of the forced equation in terms of two parameters, given a pair of independent solutions of the homogeneous equation (see (2.4) below).

Nonlinear volterra integrodifferential equations in a Banach space

We study the Cauchy problem associated with the Volterra integrodifferential equation u\left( t \right) \in Au\left( t \right) + \int {_0^1 B\left( {t - s} \right)u\left( s \right)ds + f\left( t \right),} u\left( 0 \right) = u_0 \in D\left( A \right), whereA is anm-dissipative non-linear operator (or more generally, anm-D(ω) operator), defined onD(A) ⊂X, whereX is a real reflexive Banach space. We show that ifB is of the formB=FA+K, whereF, K :X →D(Ds), whereDs is the differentiation operator, withF bounded linear andK andDsK Lipschitz continuous, then the Cauchy problem is well-posed. In addition we obtain an approximation result for the Cauchy problem.

Energy Estimates for Symmetric Hyperbolic Integro-Differential Equations

Publisher Summary This chapter discusses the Cauchy problem associated with the linear integro-differential equation where the partial differential equation is symmetric equation. The chapter discusses the symmetric hyperbolic differential equations by the use of a-priori L 2 estimates (so-called energy estimates). These estimates lead to the existence of unique weak solution. The estimates provide regularity results as well as show that that the solutions propagate with finite speed. The chapter proves energy estimates for which are analogous to the estimates found for symmetric hyperbolic differential equations. The chapter presents two examples associated with continuum mechanics for the materials with memory. The first example discusses the generalized linear theory with the electromagnetic theory for inhomogeneous anisotropic stable media with memory. The chapter discusses the constitutive relations proposed by Volterra.