Stability of global and exponential attractors for a three‐dimensional conserved phase‐field system with memory

Abstract

AbstractWe consider a conserved phase‐field system on a tri‐dimensional bounded domain. The heat conduction is characterized by memory effects depending on the past history of the (relative) temperature ϑ, which is represented through a convolution integral whose relaxation kernel k is a summable and decreasing function. Therefore, the system consists of a linear integrodifferential equation for ϑ, which is coupled with a viscous Cahn–Hilliard type equation governing the order parameter χ. The latter equation contains a nonmonotone nonlinearity ϕ and the viscosity effects are taken into account by a term −αΔ∂tχ, for some α⩾0. Rescaling the kernel k with a relaxation time ε>0, we formulate a Cauchy–Neumann problem depending on ε and α. Assuming a suitable decay of k, we prove the existence of a family of exponential attractors {ℰα,ε} for our problem, whose basin of attraction can be extended to the whole phase–space in the viscous case (i.e. when α>0). Moreover, we prove that the symmetric Hausdorff distance of ℰα,ε from a proper lifting of ℰα,0 tends to 0 in an explicitly controlled way, for any fixed α⩾0. In addition, the upper semicontinuity of the family of global attractors {𝒜α,ε} as ε→0 is achieved for any fixed α>0. Copyright © 2009 John Wiley & Sons, Ltd.