The decay rate for a fractional differential equation ✩

We consider the fractional differential equation ut t (t , x) = t 0 k(t −s)usxx(s, x) ds + uxx(t , x), t > 0, x ∈ (0, 1), with Dirichlet boundary conditions and initial values. This problem, with a particular kernel, may be looked at as an internally damped wave equation with (a strong) damping of order less than one. It is proved that the solution of this problem with a weakly singular kernel decays exponentially to zero.  2004 Elsevier Inc. All rights reserved