Studies on linear and nonlinear differential equations with time delay

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The differential-integral equation with time lag, Chen's equation, was studied by constructing the fundamental solutions using Puiseux and Laplace expansions. Also, long-time behavior of the equation was determined in detail by the asymptotic series characterized by the Mittag-Leffler function. Chen's equation originally describes the motion of a spherical particle in fluid have a close connection with the equations of dynamics of 2-species fluids mixture.Professor Radkevich in Moscow University had been staying at our department for four and half months through March 1998. The Muskat problem, nonlinear elliptic partial differential equation describing the interfacial motion between oil and water, has been studied jointly with him. The joint work is still going on. Our main idea is to characterize the weak solution to the Muskat problem as the weak-limit of the smooth solution of the nonlinear parabolic partial differential equation which has the small parameter corresponding the width of the interface. This idea is actually true for the case of the symmetric domain as a ball. These results will appear in the journal, Applicable Analysis.