Coupled System Of Hyperbolic Equations Research Articles

Blow-Up of Solutions with High Energies of a Coupled System of Hyperbolic Equations

We consider an abstract coupled evolution system of second order in time. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We compare our results with those in the literature and show how we improve them.

Group analysis of a hyperbolic Lane–Emden system

In this paper we carry out a complete Noether and Lie group classification of the radial form of a coupled system of hyperbolic equations. From the Noether symmetries we establish the corresponding conserved vectors. We also determine constraints that the nonlinearities should satisfy in order for the scaling symmetries to be Noetherian. This led us to a critical hyperbola for the systems under consideration. An explicit solution is also obtained for a particular choice of the parameters.

Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping

We consider a wave equation with nonlinear acoustic boundary conditions. This is a nonlinearly coupled system of hyperbolic equations modeling an acoustic/structure interaction, with an additional boundary damping term to induce both existence of solutions as well as stability. Using the methods of Lasiecka and Tataru for a wave equation with nonlinear boundary damping, we demonstrate well-posedness and uniform decay rates for solutions in the finite energy space, with the results depending on the relationship between (i) the mass of the structure, (ii) the nonlinear coupling term, and (iii) the size of the nonlinear damping. We also show that solutions (in the linear case) depend continuously on the mass of the structure as it tends to zero, which provides rigorous justification for studying the case where the mass is equal to zero.

On the significance of finite propagation speeds in multicomponent reacting systems

A continuum model of diffusion in a reacting mixture is built around a constitutive relation for momentum exchange by frictional interaction between diffusing species. The resulting continuity and momentum equations incorporate a finite relaxation time for diffusion. Under appropriate conditions, linearization of these equations produces a coupled system of hyperbolic equations for small disturbances of a stationary state. Fick’s law can be recovered from the linear equations by assuming instantaneous relaxation of the flux, provided that the stationary state is uniform. Fick’s law is generally inconsistent with the momentum equation when the stationary state is nonuniform. The stability of uniform stationary solutions predicted by the parabolic system obtained when Fick’s law is used is compared with the stability predicted by the hyperbolic system. When the former predicts stability and has at least one pair of complex-conjugate roots, the latter may predict that the stationary state is unstable. Thus, inclusion of relaxation in the model can lead to qualitatively different predictions of stability.