走化性方程式 : 非線形拡散と移流項の相互作用について(<特集>移流項をもつ反応拡散系)

抄録

We consider the following reaction-diffusion equation (KS)_m. Throughout this article, we assume that N≥1, and that m, q, and γ are the constants satisfying m>1, q≥2, γ≥0, respectively. This equation is often called the Keller-Segel model describing the motion of the chemotaxis molds. Here u(x,t) and v(x,t) denote the density of amoebae and the concentration of the chemo-attractant, respectively. In the semi-linear case i.e., m=1, up to the present there have been many results for q=2. As for the quasi-linear case m>1, the balance between diffusion strength m and non-linearity effect q plays an important role for existence of global solutions to (KS)_m. In this article, we introduce recent results on global existence and blow-up in a finite time of solution of (KS)_m with the critical exponet q=m+2/N. Furthermore, we introduce a significant feature of degenerate one such as the property of finite speed of propagation for solutions, which is different phenomena from that of non-degenerate one. In addition, we characterize the interface curve as the solution of a certain ordinary differential equation associated with (KS)_m. We also state the so-called ε-regularity theorem of (KS)_m. Finally, we give several open problems.