Asymptotic expansion of solutions to the drift-diffusion equation with large initial data
type: Item
instance of: scholarly article
title: Asymptotic expansion of solutions to the drift-diffusion equation with large initial data
zbMATH Open document ID: 1201.35050
author: Masakazu Yamamoto
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-07-07T00:00:00Z
review text: The Cauchy problem to the drift-diffusion equation is considered. \[ \begin{aligned} &\frac{\partial u}{\partial t}+\nabla\cdot(u\nabla \psi)-\Delta u=0, \quad -\Delta\psi=-ku,\quad x\in\mathbb R^n,\;t>0\\ &u(x,0)=u_0(x),\quad x\in\mathbb R^n, \end{aligned} \] where \(n\geq 3\), \(k=\pm 1\), \(u(x,t)\) is the unknown function. This equation with \(k=1\) is known as the model of the plasma dynamics. When \(k=-1\) the equation stands for the motion of the gravitational interaction particles in astronomy. The large time behavior of the solution to the problem is shown in the paper. This behavior is different for odd an even \(n\). If \(k=1\) the initial condition \(u_0\) can be large, but for \(k=-1\) \(u_0\) must be small in \(L_p\) norm.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 5731672
zbMATH Keywords: drift-diffusion equation
zbMATH Keywords: asymptotic behavior of the solution
zbMATH Keywords: Keller-Segel equation
zbMATH Keywords: elliptic-parabolic system
reviewed by: Il'ya Sh. Mogilevskii
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.02.049
OpenAlex ID: W2025495093
cites work: Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems
cites work: Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws
cites work: A nonlocal singular parabolic problem modelling gravitational interaction of particles
cites work: Large-Time Behavior in Incompressible Navier–Stokes Equations
cites work: Large time behavior for convection-diffusion equations in \(\mathbb{R}{} ^ N\)
cites work: Global solutions of the time-dependent drift-diffusion semiconductor equations
cites work: Asymptotic profiles of nonstationary incompressible Navier-Stokes flows in the whole space
cites work: Two-dimensional Navier-Stokes flow with measures as initial vorticity
cites work: The decay of the solutions for the heat equation with a potential
cites work: On Explosions of Solutions to a System of Partial Differential Equations Modelling Chemotaxis
cites work: QUALITATIVE BEHAVIOR OF SOLUTIONS OF A DEGENERATE NONLINEAR DRIFT-DIFFUSION MODEL FOR SEMICONDUCTORS
cites work: Sharp asymptotics for a parabolic system of chemotaxis in one space dimension.
cites work: Commutator estimates and the euler and navier-stokes equations
cites work: Decay Estimates and Large Time Behavior of Solutions to the Drift-Diffusion System
cites work: Initiation of slime mold aggregation viewed as an instability
cites work: The Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}(\mathbb{R}^n)$ and its application to self-similar solutions
cites work: The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system
cites work: Well-posedness for the drift-diffusion system in \(L^p\) arising from the semiconductor device simulation
cites work: Asymptotic behavior of solutions of transport equations for semiconductor devices
cites work: Global existence and blowup of solutions to a chemotaxis system.
cites work: Q4381941
cites work: Decay Properties and Asymptotic Profiles of Bounded Solutions to a Parabolic System of Chemotaxis in Rn
cites work: Large time behavior of bounded solutions to a parabolic system of chemotaxis in the whole space
cites work: ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO DRIFT-DIFFUSION SYSTEM WITH GENERALIZED DISSIPATION
cites work: Behavior of solutions to a system related to chemotaxis.
cites work: A decay property of the Fourier transform and its application to the Stokes problem
cites work: Q4338490
cites work: Higher-order asymptotic expansions for a parabolic system modeling chemotaxis in the whole space
cites work: Q3568185
cites work: Q3568194
cites work: Change of variables for absolutely continuous functions
DOI: 10.1016/J.JMAA.2010.02.049
description: scientific article; zbMATH DE number 5731672