Asymptotic behaviour of strongly damped nonlinear hyperbolic equations
Blowup in nonlinear parabolic equations
The Cauchy problem and self-similar solutions for a nonlinear parabolic equation
The existence of solutions to the Cauchy problem for a nonlinear parabolic equation describing the gravitational interaction of particles is studied under minimal regularity assumptions on the initial conditions. Self-similar solutions are constructed for some homogeneous initial data.
Existence and nonexistence of solutions for a model of gravitational interaction of particles, III
On the stationary solutions of Burgers' equation
Growth and accretion of mass in an astrophysical model
We study asymptotic behavior of radial solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles. In particular, we consider stationary solutions in balls and in the whole space, self-similar solutions defined globally in time, blowing up self-similar solutions, and singularities of solutions that blow up in a finite time.
A note on the paper of Y. Naito
This note contains some remarks on the paper of Y. Naito concerning the parabolic system of chemotaxis and published in this volume.
Radially symmetric solutions of a chemotaxis model in the plane-the supercritical case
The existence, uniqueness and large time behaviour of radially symmetric solutions to a chemotaxis system in the plane ℝ² are studied for the (supercritical) value of mass greater than 8π.
Convergence of solutions of generalized Korteweg-de Vries-Burgers equations to those of first order equations
Asymptotic behaviour in time of solutions to some equations generalizing the Korteweg-de Vries-Burgers equation
A Neumann problem for a convection-diffusion equation on the half-line
We study solutions to a nonlinear parabolic convection-diffusion equation on the half-line with the Neumann condition at x=0. The analysis is based on the properties of self-similar solutions to that problem.
A class of nonlocal parabolic problems occurring in statistical mechanics
We consider parabolic equations with nonlocal coefficients obtained from the Vlasov-Fokker-Planck equations with potentials. This class of equations includes the classical Debye system from electrochemistry as well as an evolution model of self-attracting clusters under friction and fluctuations. The local in time existence of solutions to these equations (with no-flux boundary conditions) and properties of stationary solutions are studied.
Existence and nonexistence of solutions for a model of gravitational interaction of particles, I
We study the existence of stationary and evolution solutions to a parabolic-elliptic system with natural (no-flux) boundary conditions describing the gravitational interaction of particles.
Nonlocal quadratic evolution problems
Nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered. The inertial type nonlinearity is quadratic and nonlocal while the diffusive term, also nonlocal, is anomalous and fractal, i.e., represented by a fractional power of the Laplacian. Conditions for global in time existence versus finite time blow-up are studied. Self-similar solutions are constructed for certain homogeneous initial data. Monte Carlo approximation schemes by interacting...
Growth and accretion of mass in an astrophysical model, II
Radially symmetric solutions of a nonlocal Fokker-Planck equation describing the evolution of self-attracting particles in a bounded container are studied. Conditions ensuring either global-in-time existence of solutions or their finite time blow up are given.
Global existence versus blow up for some models of interacting particles
We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and Debye-Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method due to S. Montgomery-Smith.
An elementary approach to nonexistence of solutions of linear parabolic equations
This note presents an elementary approach to the nonexistence of solutions of linear parabolic initial-boundary value problems considered in the Feller test.
On the parabolic-elliptic limit of the doubly parabolic Keller-Segel system modelling chemotaxis
We establish new results on convergence, in strong topologies, of solutions of the parabolic-parabolic Keller-Segel system in the plane to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.
Generalized Fokker-Planck equations and convergence to their equilibria
We consider extensions of the classical Fokker-Planck equation uₜ + ℒu = ∇·(u∇V(x)) on with ℒ = -Δ and V(x) = 1/2|x|², where ℒ is a general operator describing the diffusion and V is a suitable potential.
Existence of Solutions for the Keller-Segel Model of Chemotaxis with Measures as Initial Data
A simple proof of the existence of solutions for the two-dimensional Keller-Segel model with measures with all the atoms less than 8π as the initial data is given. This result was obtained by Senba and Suzuki (2002) and Bedrossian and Masmoudi (2014) using different arguments. Moreover, we show a uniform bound for the existence time of solutions as well as an optimal hypercontractivity estimate.
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