On nonmonotone solutions of an integrodifferential equation in linear viscoelasticity

We consider the integrodifferential equation u(t, x) = ∫0t a (t - s)uxx (s, x )ds with initial and boundary conditions corresponding to the Rayleigh problem. The kernel has the form a(t) a0 + a∞t + ∫0t a1 (s) ds, where a0 ≥ 0, a∞≥ 0, and a1 ∈ Ltoc1 (R+) is of positive type and satisfies the condition ∫0∞ e-∈t|a1 (t)|dt < ∞ for every ∈ > 0. By solving the equation numerically and performing a careful error analysis, we show that the solution u(t, x) need not be nondecreasing in t ≥ 0 for fixed x > 0 if a1 is nonnegative, nonincreasing, and convex. The same result is shown to hold under the assumption that a1 is completely positive. This answers a question that remained unsolved in [J. Prüβ, Math. Ann., 279 (1987), p. 330]. In the case where a1 is convex, piecewise linear, the solution is shown to be almost everywhere equal to a function which is discontinuous across infinitely many parallel lines.