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This note is concerned with the damped wave equation ε2∂ttu+∂tu-Δu+f(u)=g depending on a small parameter ℇ and with the corresponding parabolic equation ∂tu-Δu+f(u)=g obtained in the singular limit ℇ→0. The existence of a family ℳℇ of exponential attractors which is Hölder continuous with respect to ℇ is proved. [ABSTRACT FROM AUTHOR]

We establish some decay properties of the semigroup generated by a linear integro-differential equation in a Hilbert space, which is an abstract version of the equation ut(t)-βΔu(t)-∫0∞k(s)Δu(t-s) ds=0 describing hereditary heat conduction. [ABSTRACT FROM AUTHOR]

We consider a phase‐field model based on hereditary constitutive equations for the internal energy and the heat flux and on the assumption that the spatial average of the order parameter χ is conserved. This model consists of a parabolic integrodifferential equation for the (relative) temperature ϑ coupled with a nonlinear fourth‐order evolution equation for χ. We first show that the obtained system is indeed a nonautonomous dynamical system, provided that the phase space accounts for the past history of ϑ and appropriate boundary conditions are given. Then we establish the existence of an absorbing set, uniformly with respect to a certain class of heat source terms. Finally, we prove that, under suitable assumptions, our dissipative dynamical system possesses a uniform attractor of finite Hausdorff and fractal dimensions. [ABSTRACT FROM AUTHOR]

A strongly damped semilinear wave equation with linear memory is considered in a history space setting. Namely, the evolution of the past history of the displacement vector is contained in the dynamical system. Existence, uniqueness and continuous dependence results are discussed. Under proper assumptions on the memory kernel, the existence of uniform absorbing sets is achieved. Moreover, when the source term is translation compact in a suitable functional space, the system is shown to possess a uniform attractor. [ABSTRACT FROM AUTHOR]