一个耦合双曲-抛物系统的全局光滑解

Global Smooth Solutions to a Coupled Hyperbolic-Parabolic System

考虑一个源自生物学的耦合双曲-抛物模型的初边值问题.当动能函数为非线性函数以及初始值具有小的L^2能量但其H^2能量可能任意大时,得到了初边值问题光滑解的全局存在性和指数稳定性.而且,如果假定非线性动能函数满足一定的条件,在对初值没任何小条件假定下得到光滑解的全局存在性.通过构造一个新的非负凸熵和做精细的能量估计得到了结果的证明.

The authors investigate the initial-boundary value problem for a coupled hyperbolic-parabolic system arising from biology. When the kinetic function is nonlinear and the initial data are of small L2-norm energy but possibly large H2-norm energy, the authors get both the global existence and the exponential stability of smooth solutions to the initial-boundary value problem. Furthermore, assuming that the nonlinear kinetic function satisfies certain conditions, the authors establish the global existence of smooth solutions without any smallness assumption on the initial data. The proof is obtained by constructing a new nonnegative convex entropy and making delicate energy estimates.