Summary

Abstract

Our study is concerned with the system of elliptic-parabolic partial differential equations arising in mathematical biology and statistical mechanics. A typical example is

$$ \begin{gathered} \left. {\begin{array}{*{20}c} {u_t = \nabla \cdot \left( {\nabla u - u\nabla v} \right)} \\ {0 = \Delta v - av + u} \\ \end{array} {\mathbf{ }}} \right\}{\mathbf{ }}{\text{in}}{\mathbf{ }}\Omega {\mathbf{ }} \times \left( {0,T} \right), \hfill \\ \frac{{\partial u}} {{\partial v}} = \frac{{\partial u}} {{\partial v}} = 0{\mathbf{ }}{\text{on}}{\mathbf{ }}\partial \Omega \times \left( {0,T} \right), \hfill \\ u|_{t = {\text{0}}} = u_0 \left( x \right){\mathbf{ }}{\text{in}}{\mathbf{ }}\Omega \hfill \\ \end{gathered} $$

(1.1)

where ΩX2283;Rⁿ ? Rn is a bounded domain with smooth boundary ∂Ω, a > 0 is a constant, and ν is the outer unit vector on ∂Ω. This system was proposed by Nagai [106] in the context of chemotaxis in mathematical biology. Here, u = u(x, t) and v = v(x, t) stand for the density of cellular slime molds and the concentration of chemical substances secreted by themselves, respectively, at the position x ∂Ω and the time t >0.