Quasilinear hyperbolic systems with dissipation effects

Abstract

We study the Cauchy and the initial boundary value problems for a class of quasilinear hyperbolic systems with dissipation effects, such as frictional damping and relaxation. This kind of systems includes Euler equations as a special example. We aim to prove the existence and non-existence of global classical solutions, and obtain the asymptotic behavior and convergence rate of the solutions to some hyperbolic conservation laws. Precisely, we study the following problems: 1. The existence of global smooth solutions to general quasilinear hyperbolic systems with dissipations provided that the Cº-norm of the initial data is sufficiently small by applying the maximum principle and the generalized Lax transformation; 2. The existence and non-existence of global smooth solutions to Euler equations with damping and spherical symmetry; 3. The existence and non-existence of global smooth solutions to p-system with relaxation; 4. The asymptotic behavior of solutions to the Cauchy problem of p-system with relaxation, and the nonlinear stability of hyperbolic waves when the limits of the initial data at x = ±∞ are not at equilibrium; 5. The asymptotic behavior of solutions to the initial boundary value problem of a nonlinear hyperbolic conservation laws with relaxation and the convergence to stationary solutions, the rarefaction waves or the superposition of these nonlinear waves respectively; 6. The P-convergence rate for p-system with relaxation.