Blow-up dynamics for the aggregation equation with degenerate diffusion
نویسندگان
چکیده
We study radially symmetric finite time blow-up dynamics for the aggregation equation with degenerate diffusion ut = ∆u m − ∇ · (u ∗ ∇(K ∗ u)) in R, where the kernel K(x) is of power-law form |x|−γ . Depending on m, d, γ and the initial data, the solution exhibits three kinds of blow-up behavior: self-similar with no mass concentrated at the core, imploding shock solution and near-self-similar blow-up with a fixed amount of mass concentrated at the core. Computation are performed for a variety of m, d and γ using an arbitrary Lagrangian Eulerian method with adaptive mesh refinement.
منابع مشابه
A note on critical point and blow-up rates for singular and degenerate parabolic equations
In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...
متن کاملBlow-up for Degenerate Parabolic Equations with Nonlocal Source
This paper deals with the blow-up properties of the solution to the degenerate nonlinear reaction diffusion equation with nonlocal source xut − (xux)x = ∫ a 0 u pdx in (0, a) × (0, T ) subject to the homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally or blows up in fini...
متن کاملAn Aggregation Equation with Degenerate Diffusion: Qualitative Property of Solutions
We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to m > 1 of the McKean–Vlasov equation where here the “diffusive” portion of the dynamics are governed by Porous medium self–interactions. We focus primarily on m ∈ (1, 2] with particular emphasis on m = 2. In general, we establish regularity properties and, ...
متن کاملLocal and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
Recently, there has been a wide interest in the study of aggregation equations and Patlak-Keller-Segel (PKS) models for chemotaxis with degenerate diffusion. The focus of this paper is the unification and generalization of the well-posedness theory of these models. We prove local well-posedness on bounded domains for dimensions d ≥ 2 and in all of space for d ≥ 3, the uniqueness being a result ...
متن کاملGlobal Solutions and Blow-up Profiles for a Nonlinear Degenerate Parabolic Equation with Nonlocal Source
This paper deals with a degenerate parabolic equation vt = ∆v + av1 ∥v∥1 α1 subject to homogeneous Dirichlet condition. The local existence of a nonnegative weak solution is given. The blow-up and global existence conditions of nonnegative solutions are obtained. Moreover, we establish the precise blow-up rate estimates for all the blow-up solutions.
متن کامل