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Our study is concerned with the system of elliptic-parabolic partial differential equations arising in mathematical biology and statistical mechanics. A typical example is where ΩX2283;R ? 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.

This chapter is a short description of mathematical modelling of the problem. First, we describe the physical motivation. In fact, parabolic-elliptic systems with drift terms are found in several areas of science involved with the transport theory; statistical mechanics, quantum mechanics, physical chemistry, and so forth. Here, we mention two of them, the and .

We study the system of chemotaxis, or the adiabatic limit of the Fokker-Planck equation, and thus, Ω R is a bounded domain with smooth boundary ∂Ω, and V log W stands for the potential of the outer force, where W = W(x)> 0 is a smooth function of x ε Ω.

This chapter studies the existence of the solution to (3.1) globally in time: where Ω ⊂ R is a bounded domain with smooth boundary ΩΩ and W = W(x) Ω 0 is a smooth positive function defined on Ω. The initial value u = u(x) is a nonnegative function not identically equal to 0, and in the case of t Ω 0, the additional initial condition v|t=0 = v(x) is imposed. Then, we shall show that Ω = u1 Ω 4π implies T = +#x8734; in the case of n = 2.

The criterion is sharp for T = +8∞ the simplified system of the (N) field, and in this chapter we prove the following theorem [146].

In this chapter, we begin the study of the stationary problem to (3.1): where Ω ⊂ Rn is a bounded domain with smooth boundary ∂Ω, W = W(x) > 0 is a smooth function of x ? Ω, and A > 0 is a self-adjoint operator in L2(Ω) with compact resolvent. This study provides several heuristic supports Theorems 1.1 and 1.2, although it does not bring any rigorous proof or tool to them.

In this chapter we show that the mass quantization of collapse occurs if the solution [148]. It is uncertain whether such a solution exists or not. Actually, it is suspected that the blowup in infinite time occurs only when the solution converges to a singular stationary solution, and therefore, in that case, the total mass λ = ||O|| must be quantized as λ ∈ 4N. This question is open, but an important tool is exploited, which we call the (This is different from the lemma given by [23].)

I deal first with legislation as a source of law for a simple reason: Legislation appears intuitively as the paradigm source of law. As we shall see by the end of this chapter, this intuition needs careful handling if it is not to mislead the legal theorist. But it is an excellent starting-point. When we think of law in an ordinary, or pre-philosophical, context, we think of a set of rules controlling behaviour, which are stipulated in one place (the legislature) and applied in another (the courts). Laws presuppose law-makers. Laws tell us what to do, and so there must be some one person or body of persons who does the telling. In H. L. A. Hart’s famous tale of the transition from a pre-legal to a legal world (Hart 1994, 91ff.), legislation plays a prominent part. For there to be law, we need a way of identifying which norms are legal norms; we need an agency to introduce new laws, or amend the ones that exist. Only when all this is in place, is there point to turning to the remaining issue of an agency for the settlement of disputes under these laws. The legislature is also deeply implicated in the central ways in which law differs from morality. Laws typically come into existence, are changed, or cease to exist at specific points in time, and as a result of the following of specific procedures. These are not features of moral norms, and as features of laws all typically occur as a result of legislative activity.

This chapter returns to the general () and describes the fact that the varia-tional structure F defined on and that of on V = dom() stated in Chapter 6 are equivalent up to Morse indices. This fact was known concerning the stability in the case that A is equal to - Δ with the Dirichlet boundary condition, but actually general theory holds true. This structure is not restricted to the Keller-Segel system; it is valid for several mean field theories.

In this chapter, we conclude the study of stationary solutions and describe several suggestions obtained by this for the dynamics of (3.1), where Ω x2282; 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.