TITLE

# Critical mass for a Patlakï¿½Kellerï¿½Segel model with degenerate diffusion in higher dimensions

AUTHOR(S)
Blanchet, Adrien; Carrillo, Jos�; Lauren�ot, Philippe
PUB. DATE
June 2009
SOURCE
Calculus of Variations & Partial Differential Equations;Jun2009, Vol. 35 Issue 2, p133
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlakï¿½Kellerï¿½Segel system with d = 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M c such that if $${M \in (0, M_c]}$$ solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for $${M \in (0, M_c)}$$ . While characterising the possible infinite time blowing-up profile for M = M c, we observe that the long time asymptotics are much more complicated than in the classical Patlakï¿½Kellerï¿½Segel system in dimension two.
ACCESSION #
36479340

## Related Articles

• Numerical Simulation of Shock Induced Polarization in Binary Electrolytes. Skryl, Yuri; Belak, Anna A.; Kuklja, Maija M. // AIP Conference Proceedings;2006, Vol. 845 Issue 1, p355

Simulation of the shock-induced polarization in aqueous solution of KF is performed. To describe the polarization, a complete system of diffusion equations for charge carriers has been coupled with the Poisson equation and solved simultaneously. The results obtained in this study demonstrate...

• Diffusion and convection of gaseous and fine particulate from a chimney. MACKAY, CAMERON; MCKEE, SEAN; MULHOLLAND, ANTHONY J. // IMA Journal of Applied Mathematics;Oct2006, Vol. 71 Issue 5, p670

Particle dispersion from a high chimney is considered and an expression for the subsequent concentration of the particulate deposited on the ground is derived. We consider the general case wherein the effects of both diffusion and convection on the steady state ground concentration of...

• Diffusion description of the kinetics of isotopic phase separation of 3Heâ€“4He solid solutions. Grigor’ev, V. N.; Degtyarev, I. A.; Sokolov, S. S. // Low Temperature Physics;Nov2005, Vol. 31 Issue 11, p951

The process of the separation of solid solutions of helium isotopes is examined using a diffusion approach with boundary conditions prescribed on the boundary of the inclusion of a new phase, the position of the boundary being a function of time. The diffusion processes occurring simultaneously...

• Modeling the effect of diffusion of the ambient medium on the long-term strength of a hollow cylinder under uniaxial tension. Lokoshchenko, A. M.; Nazarov, V. V. // Journal of Applied Mechanics & Technical Physics;Jul2007, Vol. 48 Issue 4, p542

The problem of the long-term strength of an extended thick-walled tube containing a corrosive medium in the internal cavity is solved. The diffusion of this medium into the tube material is analyzed. The diffusion equation is solved approximately by introducing the diffusion front, and the error...

• Internal Stabilization of a Mutualistic Reaction Diffusion System. Wang Yuan Dong // Acta Mathematica Sinica;Feb2007, Vol. 23 Issue 2, p373

In this paper, we study the internal stabilization of steady-state solutions to a 2-species mutualistic reaction diffusion system via finite-dimensional feedback controllers. Our main idea is to use different internal controllers to stabilize different steady-state solutions. The controllers are...

• Localized patterns in reaction-diffusion systems. Vanag, Vladimir K.; Epstein, Irving R. // Chaos;Sep2007, Vol. 17 Issue 3, p037110

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

• Beryllium-Diffused Blue Sapphire - An Overview. Fox, Karen // Canadian Gemmologist;Jun2007, Vol. 28 Issue 2, p63

The article presents an overview of the structure of beryllium-diffused blue sapphire. It relates that the color in natural sapphire has been enhanced through intentional diffusion of the light element beryllium (Be) into the crystal lattice. Subsequent investigation indicates that the process...

• Equation of State at the Adiabat of Unloading. Kokhanenko, I. K.; Levchenko, E. M. // Technical Physics;Jan2001, Vol. 46 Issue 1, p20

On the assumption that condensed mater at high pressures and temperatures behaves as an ideal gas, a functional dependence of the adiabatic exponent on the GrÃ¼neisen constant was obtained, as well as an equation of state that relates the velocity of the shock front and the mass velocity in...

• Characteristics of local compliance of an elastic body under a small punch indented into the plane part of its boundary. Argatov, I. I. // Journal of Applied Mechanics & Technical Physics;Jan/Feb2002, Vol. 43 Issue 1, p147

An asymptotic solution of the contact problem of an elastic body indented (without friction) by a circular punch with a flat base is obtained under the assumption of a small relative size of the contact zone. The resulting formulas involve integral characteristics of the elastic body, which...

Share