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This paper is devoted to the analysis of a one-dimensional model for phase transition phenomena in thermoviscoelastic materials. The corresponding parabolic-hyperbolic PDE system features a strongly nonlinear internal energy balance equation, governing the evolution of the absolute temperature , an evolution equation for the phase change parameter , including constraints on the phase variable, and a hyperbolic stress-strain relation for the displacement variable . The main novelty of the model...
The existence of small global (in time) solutions to an abstract evolution equation containing a damping term is proved. The result is then applied to fully nonlinear telegraph equations and to nonlinear equations involving operators with time delay.
The global in time solvability of the one-dimensional nonlinear equations of thermoelasticity, equations of viscoelasticity and nonlinear wave equations in several space dimensions with some boundary dissipation is discussed. The blow up of the solutions which might be possible even for small data is excluded by allowing for a certain dissipative mechanism.
We prove global dynamical stability of steady transonic shock solutions in divergent quasi-one-dimensional nozzles. One of the key improvements compared with previous results is that we assume neither the smallness of the slope of the nozzle nor the weakness of the shock strength. A key ingredient of the proof are the derivation a exponentially decaying energy estimates for a linearized problem.
We prove small data global existence and scattering for quasilinear systems of Klein-Gordon equations with different speeds, in dimension three. As an application, we obtain a robust global stability result for the Euler-Maxwell equations for electrons.
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