Professor Ding has attributed systematically and creatively in numerous fields of mathematical sciences. His great contributions include nearly 100 publications, among which there are over 90 academical papers, 4 monographs, and 3 pop-scientific books. He is among the first generation of mathematicians brought up by the new China. As a promising young man, he was selected for further study in the Institute of Mathematics in 1951 after college years. In the newly established institute, he had the opportunity of working under the guidance of Professor Hua Luogeng.

The authors consider the global existence and the blow-up phenomenon of classical solutions with small amplitude to the Cauchy problem for general quasilinear hyperbolic systems with characteristics with constant multiplicity and given some applications.

In this paper, spectral collocation and Nyström methods for integral equations are discussed. Some multilevel correction schemes are presented for infinitely accelerating the covergence even if the exact solution is not so smooth.

This paper regards the existence of entropy waves for a model example of nonstrictly hyperbolic system which degenerates on a sonic line. This construction is an important tool to approach the convergence of the vanishing viscosity and of other similar approximations to the weak solutions of the hyperbolic systems, via the compensated compactness techniques.

We list a hierarchy of hyperbolic-parabolic partial differential equations in terms of the regularization properties of their solution operators. This ranges from the most regularizing of the heat operator to the least, that of the hyperbolic conservation laws. We illustrate this with physical examples in gas dynamics and mechanics.

The asymptotic behavior of solutions of the damped p-system is known to be described by a nonlinear diffusion equation. The previous works on this topic concern either the case of small smooth data where estimates of high-order derivatives are available by energy methods or the case of special and small rough data. For large data, the existence of solutions is proved by using the method of compensated compactness. Thus the above mentioned energy estimates are not expected. However, the compensated compactness gives a very weak justification (in the mean in time) of the asymptotics. In the present paper we prove that the natural energy estimates, which does not involve derivatives, combined with this “convergence in the mean”, gives the strong convergence in L^p_{loc} space (p is finite) as expected.

We construct global solutions in L^∞ for the equations of motion or one-dimensional viscoelastic media, in Lagrangian coordinates, with arbitrarily large L^∞ initial data, via the vanishing viscosity method. A priori estimates for approximate solutions, with artificial viscosity, are derived through entropy inequalities. The convergence of the approximate solutions to a weak solution compatible with the entropy condition is demonstrated. This also establishes the compactness of the corresponding solution operators, which indicates that the memory effect does not affect the hyperbolic behavior.