On the influence of damping in hyperbolic equations with parabolic degeneracy

Abstract

This paper examines the effect of damping on a nonstrictly hyperbolic 2×2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

“…Recently, the authors in [7] proved convergence rates for the half space problem. Moreover, the authors in [3] and [4] considered phase transition and the effect of damping respectively. When (V) = − with being a given positive constant and ℎ(V) = 1, system (10) reduces to the well-known -system with linear damping.…”

“…[1][2][3][4] and references therein). To improve the model, one may generalize the history dependence of q by modifying (4) or, as was done in [2], by introducing a suitable nonlinear dependence in (7),…”