In this paper we prove the existence of the unique global classical solution with small initial data to the Cauchy problem of a scalar conservation law with degenerate diffusion by establishing the uniform a priori decay estimates of solutions. In order to compensate the degeneracy on the x1 direction by the diffusion on other directions, we introduce the frequency decomposition method and obtain the low frequency estimate and the high frequency estimate of the solution by the Green's function method and energy method respectively.

In this paper we are interested in a general class of hyperbolic balance laws. Within the frame work of existence of a convex entropy function that symmetrizes the system in certain sense and the Kawashima–Shizuta condition, we study the large time behavior of the solution to the Cauchy problem in one space dimension. For a solution around a constant equilibrium state, we predetermine the asymptotic solution as a superposition of the constant state and diffusion waves along the equilibrium characteristic directions. We estimate the remainder in the pointwise sense both in space and in time. The decay rates in various directions are optimal, which further give the optimal Lp rates with 1≤p≤∞. Applications to several examples including the Kerr–Debye model are given.

In this paper, we study the time-asymptotic behavior of the solution for the Cauchy problem of the damped wave equation with a nonlinear convection term in the multi-dimensional space. When the initial data is a small perturbation around a constant state $u^*$, we obtain the point-wise decay estimates of the solution under the so-called dissipative condition $|b| < 1$, where $b$ depends on $u^*$ and the nonlinear term.

In this paper we explore the classical solutions to the conservation law with degenerate diffusion term (ut−Δx′u=divf(u),x∈Ω⊂Rn,t>0, with x=(x1,x′)). We establish the global existence and exponential decay estimates to the solutions of the initial boundary value problem in domain Ω=R×∏i=2n(0,Li). Meanwhile, to clarify the viscous effect of the degenerate diffusion term, we also investigate the classical solutions to the Cauchy problem of the modified equation ut−Δx′u=(1−χ(D))divf(u),x∈Rn,t>0, with χ(D) a Fourier multiplier operator, we use the frequency decomposition method to establish the global existence and the polynomial decay estimates.