TABLE OF CONTENTS This chapter is devoted to the study of the elliptic-parabolic system of partial differential equations, arising in several areas in mathematical biology and mathematical physics. The first section is the description of the background and the motivation of mathematical study. Then, we shall establish the local wellposedness in the second section.
System of parabolic partial differential equations is proposed to describe several phenomena in mathematical biology. A typical example is
where ? ? R n is a bounded domain with smooth boundary ??, a > 0 a constant, and ? the outer unit vector on ??. It is proposed by T. Nagai in 1995 as a simplified form of the one given by E.F. Keller and L.A. Segel in 1970. Here, u = u(x, t) and v = v(x, t), respectively, stand for the density of cellular slime molds and the concentration of chemical substances secreted by themselves at the position x ? ? and the time t > 0.
The first equation describes the conservation of mass, where flux of u is given by 
, as
holds for any subdomain ? ? ? with 
. The first term ?? u of 
is the vector field with the direction where u decreases mostly, and with the rate equal to its derivative to that direction. The second term u ?v, on the other hand, indicates that u
