Multidimensional Degenerate Keller–Segel System with Critical Diffusion Exponent $2n/(n+2)$

Multidimensional Degenerate Keller-Segel System with Critical Diffusion Exponent 2n/(n+2)

Abstract. This paper deals with a degenerate diffusion Patlak-Keller-Segel system in n ≥ 3 dimension. The main difference between the current work and many other recent studies on the same model is that we study the diffusion exponent m = 2n/(n+2) which is smaller than the usually used exponentm∗ = 2−2/n in other studies. With the exponentm = 2n/(n+2), the associated free energy is conformal in...

Fast Diffusion Equation with Critical Sobolev Exponent in a Ball

Solutions of an Elliptic System with a Nearly Critical Exponent

This problem has positive solutions for ǫ > 0 (with pqǫ > 1) and no non-trivial solution for ǫ ≤ 0. We study the asymptotic behaviour of least energy solutions as ǫ → 0. These solutions are shown to blow-up at exactly one point, and the location of this point is characterized. In addition, the shape and exact rates for blowing up are given. Résumé. Considéré le problème −∆uǫ = v p ǫ vǫ > 0 en Ω...

Elliptic Equations with Critical Exponent

where As3 is the Laplace-Beltrami operator on B' . Let 0* C (0, 7r) be the radius o r B ' , i.e., the geodesic distance of the North pole to OBq The values 0 < 0* < 7r/2 correspond to a spherical cap contained in the Northern hemisphere, 0* -7r/2 corresponds to B ~ being the Northern hemisphere and the values rr/2 < 0* < ~c correspond to a spherical cap which covers the Northern hemisphere. Fin...

A Critical Exponent of Fujita Type for a Nonlinear Reaction-Diffusion System on Riemannian Manifold

In this paper, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction-diffusion system  ut −∆u = W (x)v + S(x) in M × (0,∞), vt −∆v = F (x)u + G(x) in M × (0,∞), u(x, 0) = u0(x) in M, v(x, 0) = v0(x) in M, where M (n ≥ 3) is a non-compact complete Riemannian manifold, ∆ is the Laplace-Beltrami operator, and S(x), G(x) are non-negative...