The Large Time Behavior Of Solutions For Hyperbolic And Hyperbolic-parabolic Systems

Within the progress of development in the world, the change of different physical variables will restrict or promote each other. PDEs used to reflect this relationship. The researchs in PDEs help people to understand the physical phenomema or the laws in nature, and to guide the production and practices. So people paid attention to and researched in the study of PDEs all along.For evolution equation, we can consider the large time behavior of solutions i.e. the tendency of the solutions as soon as the smooth solutions exist globally. Here we study the large time behavior of solutions to quasilinear hyperbolic systems in one dimension and hyperbolic-parabolic systems in multi-dimensions. Quasilinear hyperbolic systems reflect the restriction between different materials in gas dynamics, shallow water theory, combustion theory, nonliear elasticity, acoustics, classial fluid dynamics and petroleum resevoir engineering. Classical solutions to these systems will describe the wave propagation in fluid dynamics or continuum mechanics. We will know more about the character of the systems by studing the large time behavior of the classical solutions. The equations would be hyperbolic-parabolic systems if we consider viscidity. Naviable-Stokes equation is a typical example for hyperbolic-parabolic sytems. It models the progress of some physical variables like mass, momentum and energy in the field of acoustics, shallow water theory. This equation has broad applications in water conservancy, too. It is very important to study these PDEs for the theorem in mathematics and the applications in realism.Because the physical equations come from practices in the world, the behavior of the solutions must coincide with the physical phenomena. PDEs are worthiness when they describe the laws of the movememt in physics. On the one hand, solutions to hyperbolic systems reflect the wave propagation. On the other hand, waves propagation with finite speed and follow Huygens' principle. It is nature that we expect the behavior of solutions to these systems would coincide with these physical phenomena.The method of Green function has been used to study pointwise estimates of solutions to all kinds of equations widely. In particular, we can get the sharp estimates of the solution by this method if the smooth solutions exist globally. In this paper, using the method of Green fuction, we consider pointwise estimates of solutions to quasilinear hyperbolic systems in one dimension and the large time behavior of solution to compressible isentropic Navier-Stokes equaation in even dimensions.The well-posedness of the solutions for one order quasilinear hyperbolic system in one dimensionhas been researched completely. In general, we just need to get a priori estimates of the suitably norms about the solution by using characteristic, because the well-posedness of solution in local has been proved. In addition, the regularity of solution would increases as the regularity of initial data and the systems increases. So we also can study the higher order derivations of the solutions. It is well known that the existence of classical solutions to Cauchy problem for quasilinear hyperbolic systems linked closely to the nonlinearity of the system. If the systems satisfy linearly degenerate or weakly linearly degenerate, the classical solutions to Cauchy problem for quasilinear hyperbolic system will exist as the initial data with compact support is small enough. Otherwise, the classical solutions shall blowup in finite time. So we study the systems which satisfy (weakly) linearly degenerate when we consider the large time behavior of solutions.For the isentropic Navier-Stokes equation in even dimensions, we can prove that it is well-posed to the perturbation in a neighborhood of any non-vacuum constant state using the energy method and Fourier analysis. Furthermore, we have obtained the large time behavior of solutions in odd but larger than one space dimensions. Fortunately, it coincides with weak Huygens' principle. Because the affect of Huygens' pri...