Global Stability of a Fractional Partial Differential Equation

0 b(t− s)utxx(s, x) ds+ (g(ux(t, x))x, t > 0, x ∈ (0, 1) with boundary conditions (1.2) u(t, 0) = u(t, 1) = 0, t > 0, and initial values (1.3) u(0, x) = u(x), ut(0, x) = u(x). This problem is motivated by the theory of viscoelastic materials, cf., Pego [10], Ball et al. [1]. The nonlinearity g behaves typically like a power at infinity, g(ξ) ∼ sign (ξ)|ξ|m, |ξ| → ∞, and we assume that it vanishes solely at zero so that zero is the unique stationary state of (1.1). The convolution term represents a fractional derivative with respect to t; explicitly, let