Long-time Behavior Of Classical Solutions To First Order Quasilinear Hyperbolic Systems

This thesis concerns the long-time behavior of classical solutions to first order quasi-linear hyperbolic systems. We approach this problem from four different points of view.1. Property of ODE singularity for linearly degenerate quasilinear hyperbolic systems with one space variable:Different from genuinely nonlinear quasilinear hyperbolic systems, the classical solutions to1-D linearly degenerate quasilinear hyperbolic systems with large initial data may form singularities different from the shock formation at the blow-up point. This is the so called ODE singularity conjecture which is still unsolved up to now. Two aspects of this problem are discussed:a general theory and a result for some specific systems.2. Global classical solutions to1-D partially dissipative quasilinear hyperbolic sys-tems:It is well-known that the strict dissipation or the weak linear degeneracy can prevent the formation of singularities. A series of results are given to show that under some suitable hypotheses on the interaction of waves, the combined system, in which a part of equations possess the strict dissipation and the others are weakly linearly degenerate, admit global classical solutions for small initial data.3. Traveling wave solutions and their stability:The1-D linearly degenerate homoge-neous quasilinear hyperbolic systems admit traveling wave solutions. Their representation and stability are analyzed. The result shows that only the rightmost and leftmost traveling waves are definitely stable.4. Global classical solutions to a kind of quasilinear hyperbolic systems with several space variables:The existence of global classical solutions is given to a kind of multidi-mensional linearly degenerate quasilinear hyperbolic systems whose coefficient matrices commute each other.