SUMMARYIn this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle-a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established.

We study an initial-boundary-value problem for a quasilinear thermoelastic plate of Kirchhoff & Love-type with parabolic heat conduction due to Fourier, mechanically simply supported and held at the reference temperature on the boundary. For this problem, we show the short-time existence and uniqueness of classical solutions under appropriate regularity and compatibility assumptions on the data. Further, we use barrier techniques to prove the global existence and exponential stability of solutions under a smallness condition on the initial data. It is the first result of this kind established for a quasilinear non-parabolic thermoelastic Kirchhoff & Love plate in multiple dimensions.

We consider a quasilinear nonhomogeneous, anisotropic Maxwell system in a bounded smooth domain of R 3 with a strictly positive conductivity subject to the boundary conditions of a perfect conductor. Under appropriate regularity conditions, adopting a classical L 2 -Sobolev solution framework, a nonlinear energy barrier estimate is established for local-in-time H 3 -solutions to the Maxwell system by a proper combination of higher-order energy and observability-type estimates under a smallness assumption on the initial data. Technical complications due to quasilinearity, anisotropy and the lack of solenoidality, etc., are addressed. Finally, provided the initial data are small, the barrier method is applied to prove that local solutions exist globally and exhibit an exponential decay rate.In (2.1) and (2.2) to follow we impose symmetry assumptions on ε and µ under which (1.1) becomes a symmetric quasilinear hyperbolic system. For such systems in the full space case Ω = R 3 , one has a well-developed local well-posedness theory of H 3 -valued solutions due to Kato [12]. However, the above problem has a characteristic boundary which could lead to a loss of regularity in the normal direction. The available general existence results work in Sobolev spaces of very high order and with weights vanishing at the boundary, see [11,26] as well as [24] for tangential regularity. For absorbing boundary conditions, local existence results in H 3 were given in [22]. However, in our case a local well-posedness theory in H 3 was established only recently in the papers [28,29,30]. We will strongly rely on these results.In our main Theorem 2.2 we show that local solutions are indeed global and exhibit exponential decay rates in H 3 , provided that the initial fields are small and that the conductivity is strictly positive. In Remark 2 of [6], it is explained that certain solutions do not decay to 0 if one drops the simple connectedness of Ω or the magnetic compatibility conditions in (2.5), even for linear ε and µ. It should be emphasized that, while decay rates for the Maxwell system have been studied in a number of works (viz. [1, 6, 8, 9, 13, 14, 20, 21] and references therein), the cited studies only allow for linear permittivity and permeability and partly deal with constant isotropic coefficients. Stabilization results for general hyperbolic systems typically concern damping mechanism acting on all components of the solutions, see [25], whereas in our Maxwell system the dissipation via conductivity only affects the electric field. For Ω = R m the paper [2] allows for partial damping even in the quasilinear case, but its assumptions exclude the Maxwell equations. For the quasilinear Maxwell system we are only aware of a few results on the full space Ω = R 3 that establish global existence and decay for small and smooth solutions, see [16,23,27]. These works rely on dispersive estimates which are not available on bounded domains. On the other hand, it is known that blowup in W 1,∞ or H(curl) can occur in various cases, see [4]...

We consider an initial-boundary-value problem for a thermoelastic Kirchhoff & Love plate, thermally insulated and simply supported on the boundary, incorporating rotational inertia and a quasilinear hypoelastic response, while the heat effects are modeled using the hyperbolic Maxwell-Cattaneo-Vernotte law giving rise to a 'second sound' effect. We study the local wellposedness of the resulting quasilinear mixed-order hyperbolic system in a suitable solution class of smooth functions mapping into Sobolev H k -spaces. Exploiting the sole source of energy dissipation entering the system through the hyperbolic heat flux moment, provided the initial data are small in a lower topology (basic energy level corresponding to weak solutions), we prove a nonlinear stabilizability estimate furnishing global existence & uniqueness and exponential decay of classical solutions.