Exponential Decay in Thermoelastic Systems with Internal Distributed Delay

In this paper, we are concerned with the following problem  autt(x, t)− duxx(x, t) + βθx(x, t) = 0, in (0, L)× (0,∞) bθt(x, t)− k1θxx(x, t)− ∫ τ 2 τ 1 k2(s)θxx(x, t− s)ds+ βuxt(x, t) = 0, in (0, L)× (0,∞) u(x, 0) = u0(x), ut(x, 0) = u1(x), θ(x, 0) = θ0(x), x ∈ (0, L) θx(x,−t) = f0(x, t), x ∈ (0, L), t ∈ (0, τ 2), (1.1) a thermoelastic system with a distributed delay associated with initial data u0, u1, θ0 and history function f0 in suitable function spaces. Here a, b, d, k1, β, τ 2 are positive constants, τ 1 is a nonnegative constant with τ 1 < τ 2, u is the displacement, θ is the temperature difference from a reference value, and k2 : [τ 1, τ 2] → IR is a bounded function. We consider boundary conditions of the type ux(0, t) = ux(L, t) = θ(0, t) = θ(L, t) = 0 (1.2)