A hereditary partial differential equation with applications in the theory of simple fluids

The linearized boundary-initial history value problem for simple fluids obeying the Coleman-Noll constitutive equation $$S + pdelta = 2intlimits_0^infty {m(s)(E(t - s} ) - E(t))ds$$ is considered. Here S is the stress tensor, δ the Kronecker delta, p the constitutively indeterminate mean normal stress, E the infinitesimal strain tensor, and m(s) a material function. The shear relaxation modulus G is defined as (i) $$G(s) = intlimits_infty ^s {m(xi )dxi .}$$ In this paper it is shown that if G satisfies the assumptions (i) $$G in C^2 [0,infty ),{text{ }}G(s) to 0{text{ as }}s to infty,$$ (ii) $$( - 1)^k frac{{d^k G(s)}}{{ds^k }} > 0,{text{ }}k = 0,1,$$ (iii) $$G''(s) geqq 0,$$ then the rest state of the fluid is stable in an appropriate “fading memory” norm. The additional assumption (iv) $$ - intlimits_0^infty {G'} (s)s^2 ds < infty$$ yields asymptotic stability.