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The problem utt = a(0)a(ux)x + a(t — r)a(ux)x dr + /, 0 < x < 1, t > 0, Jo u(0, t) = m(1, t) = 0, u(x, 0) = u0(x), u,(x, 0) = u,{x) is considered. The essential hypotheses are that a(t) = a„ + A(t), a„ > 0, A £ L'(0, ), (— l)kaa)(t) > 0, k = 0, 1, 2, a(0) = 0, a'(£) > e > 0. It is shown that the problem has a unique classical solution for all t if the data are sufficiently small and, if / is suitably restricted, this solution tends to zero as t tends to infinity. It is shown that the problem provides a special model for elastic materials which exhibit a memory effect.

A model for one-dimensional, nonlinear viscoelasticity