| In this Ph.D. thesis, we firstly discuss the existence, stability and instability of traveling waves for first order quasilinear hyperbolic systems. Then we study the global existence of classical solutions to a kind of partially dissipative systems. Finally, we establish the global existence and the blow-up mechanism of classical solutions to the Goursat problem for quasilinear hyperbolic systems.In Chapter1, we briefly give the history and the present situation on the Cauchy problem for quasilinear hyperbolic systems, and a survey of our results in this thesis.In Chapter2. we study the theory on traveling waves for homogeneous linearly degenerate quasilinear hyperbolic systems. We first give the formulas to represent all the traveling waves, which show that these waves can range and only range on characteristic trajectories of the system. Then, we prove the stability of the leftmost (rightmost) traveling waves. At last, we give examples to illustrate the instabilities of the intermediate traveling waves.In Chapter3, we consider a kind of partially dissipative systems, one part of which possesses the strict dissipation and the others are weakly linearly degenerate. Under certain suitable hypotheses on the intersection between the two parts, we obtain the global existence of classical solutions to this kind of systems.In Chapter4, we consider the Goursat problem for first-order quasilinear hyper-bolic systems. Under the assumptions that the system is weakly linearly degenerate and the boundary functions given on the characteristic boundaries possess small C1norms and certain decaying properties, we obtain the existence of global C1solutions. When the system is not weakly linearly degenerate and the boundary functions given on the characteristic boundaries possess small C1norms and certain decaying properties, we obtain the blow-up mechanism and the asymptotic behavior of the life-span of C1solutions to the Goursat problem. |
PDF Full Text Request