Reinhard Racke

Smoothing properties, decay, and global existence of solutions to nonlinear coupled systems of thermoelastic type

We consider a nonlinear coupled system of evolution equations, the simplest of which models a thermoelastic plate. Smoothing and decay properties of solutions are investigated as well as the local well-posedness and the global existence of solutions. For the system of standard thermoelasticity it is proved that there is no similar smoothing effect.

Multidimensional contact problems in thermoelasticity

We consider dynamical and quasi-static thermoelastic contact problems in

Generalized fourier transforms and global small solutions to kirchhoff equations

hbox{{bbb R}}^n

Chapter 4 – Thermoelasticity

modeling the evolution of temperature and displacement in an elastic body that may come into contact with a rigid foundation. The existence of solutions to these dynamical and quasi-static nonlinear problems and the exponential stability are investigated using a penalty method. Interior smoothing effects in the quasi-static case are also discussed.

DECAY RATES FOR SEMILINEAR VISCOELASTIC SYSTEMS IN WEIGHTED SPACES

It is proved that the inverse of the generalized Fourier transform associated to V an appropriate compactly supported potential, maps into the space of rapidly decreasing functions. This is used for the study of wave equations with non-local nonlinearities of the type being an exterior domain in R3 with V = 0, assuming Dirichlet boundary conditions for u. For a class of smooth data we obtain global existence of small solutions, as well as a partial characterization of the asymptotic behavior as t→∞ . The existence of global large solutions generated by eigenvectors corresponding to negative eigenvalues of is also investigated.

Global existence and decay of solutions of the Cauchy problem in thermoelasticity with second sound

This work presents an extensive discussion of initial boundary value problems in thermoelasticity. First, the classical hyperbolic–parabolic model with Fourier’s law of heat conduction is considered giving results on linear systems, nonlinear systems, in one or more space dimensions, in particular discussing the asymptotic behavior of solutions as time tends to infinity. Second, recent developments for hyperbolic models using Cattaneo’s law for heat conduction are described, including the comparison to classical thermoelasticity.

On (non-)exponential decay in generalized thermoelasticity with two temperatures

We consider a class of second-order hyperbolic systems which describe viscoelastic materials, and we extend the recent results by Dharmawardane and Conti et al. More precisely, for all initial data (u0, u1)∈(Hs+1(ℝN) ∩ L1, γ(ℝN))×(Hs(ℝN) ∩ L1, γ(ℝN)) with γ∈[0, 1], we derive faster decay estimates for both dissipative structure or regularity-loss type models. To this end, we first transform our problem into Fourier space and then, by using the pointwise estimate derived by Dharmawardane et al., combined with a device to treat the Fourier transform in the low frequency region, we derive optimal decay results for the solutions to our problem. Finally, we use these decay estimates for the linear problem, combined with the weighted energy method introduced by Todorova and Yordanov, and tackle a semilinear problem.

Transmission Problems in (Thermo)Viscoelasticity with Kelvin-Voigt Damping : Nonexponential, Strong, and Polynomial Stability

We consider the one-dimensional Cauchy problem in non-linear thermoelasticity with second sound, where the heat conduction is modelled by Cattaneo’s law. After presenting decay estimates for solutions to the linearized problem, including refined estimates for data in weighted Lebesgue-spaces, we prove a global existence theorem for small data together with improved decay estimates, in particular for derivatives of the solutions.

Gleichungen erster Ordnung und Typeinteilung

We study solutions for the one-dimensional problem of the Green-Lindsay and the Lord-Shulman theories with two temperatures. First, existence and uniqueness of weakly regular solutions are obtained. Second, we prove the exponential stability in the Green-Lindsay model, but the non-exponential stability for the Lord-Shulman model.

Der Cauchysche Integralsatz und Potenzreihenentwicklung

We investigate transmission problems between a (thermo)viscoelastic system with Kelvin--Voigt damping, and a purely elastic system. It is shown that neither the elastic damping by Kelvin--Voigt mechanisms nor the dissipative effect of the temperature in one material can assure the exponential stability of the total system when it is coupled through transmission to a purely elastic system. The approach shows the lack of exponential stability using Weyls theorem on perturbations of the essential spectrum. Instead, strong stability can be shown using the principle of unique continuation. To prove polynomial stability we provide an extended version of the characterizations in [A. Borichev and Y. Tomilov, Math. Ann., 347 (2009), pp. 455--478]. Observations on the lack of compacity of the inverse of the arising semigroup generators are included too. The results apply to thermoviscoelastic systems, to purely elastic systems as well as to the scalar case consisting of wave equations.