Behaviors Of Solutions For Nonlinear Hyperbolic Conservation Laws And Related Problems

In this thesis, we consider asymptotic behaviors of solutions for the generalized BBM-Burgers equation and the generalized KDV equation with a general boundary data in a half space and the existence of global solutions for the general degradation of the non-homogeneous viscous hyperbolic conservation laws with general boundary condition.For the generalized KDV equation with general boundary data and the smooth and strictly convex flux function, under the condition of small perturbations for the initial-boundary datas, using L2 energy proves that the global solutions of exsit and converge time-asympotically to the rarefaction wave, foreover, we derive an decay rate in one-dimensional half-space.For the non-homogeneous viscous hyperbolic conservation laws with the degenerate viscosity, we use the extension theorem and the maximum principle to study the global existence theorem of solutions for general initial-boundary problem.Under the condition that non-convex-flux and large initial disturbance, using L2 energy proves the asymptotic convergence to a stable wave toward the generalized BBM-Burgers equation in one-dimensional half-space.