Piotr Biler

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 par... more 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 the 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 by S. Montgomery-Smith.

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Grzegorz Karch, laureat Nagrody Instytutu Matematycznego Polskiej Akademii Nauk za rok 2013

Wiadomości Matematyczne, 2015

An Exponential Sum

Partial Differential Equations

We study a generalization of the porous medium equation involving nonlocal terms. More precisely,... more We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit selfsimilar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem. Résumé Solutions auto-similaires pour uneéquation des milieux poreux non locale Cette note est consacréeà l'étude d'une généralisation non locale de l'équation des milieux poreux. Plus précisément, on obtient des formules explicites de solutions auto-similairesà support compact qui ressemblent fortement aux solutions de type Barenblatt. On donne aussi un argument formel qui permet d'obtenir des estimations L p des solutions faibles du problème de Cauchy. Version française abrégée Nous considérons le problème de Cauchy pour l'équation non locale suivante ∂ t u − ∇ • u∇ α−1 |u| m−1 = 0, (1) avec m > 1, x ∈ R d , α ∈ (0, 2), t > 0,à laquelle on ajoute une condition initiale u(0, x) = u 0 (x). Ici ∇ β est un opérateur intégral singulier généralisant le gradient usuel (β = 1) et lié au laplacien fractionnaire. Le résultat principal de cette note sont des formules explicites de solutions auto-similaires qui se propagent a une vitesse finie. Théorème 1 (Solutions auto-similaires) La fonction u(t, x) = Ct − d d(m−1)+α R 2 − |x| 2 t − 2 d(m−1)+α α 2 + 1 m−1 (ainsi que ses translations en x) est une solution auto-similaire de l'équation (1) pour R > 0 quelconque et une constante C > 0 convenable.

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Grzegorz Karch

Wiadomości Matematyczne, 2011

Franciszek Prus-Wisniowski, Szeregi rzeczywiste

Wiadomości Matematyczne, 2011

$Research paper thumbnail of Warsztaty zr owna nr o_zniczkowych czastkowych {Toru n, 12 {22.11. 2002$

Warsztaty zr owna nr o_zniczkowych czastkowych {Toru n, 12 {22.11. 2002

Regular and singular solutions for a model of self-gravitating particles

Warsztaty z równan rózniczkowych cza

DE L’UNIVERSITÉ DE CLERMONT-FERRAND 2 Série Probabilités et applications

Abstract. The aim of this paper is to study the problem of filtering of a Feller-Markov process f... more

Global regular and singular solutions for a model of gravitating particles

Mathematische Annalen, Jul 13, 2004

Large global solutions of the parabolic-parabolic Keller–Segel system in higher dimensions

Journal of Differential Equations, 2023

We study the global existence of the parabolic-parabolic Keller-Segel system in R d , d ≥ 2. We p... more We study the global existence of the parabolic-parabolic Keller-Segel system in R d , d ≥ 2. We prove that initial data of arbitrary size give rise to global solutions provided the diffusion parameter τ is large enough in the equation for the chemoattractant. This fact was observed before in the two-dimensional case by Biler, Guerra & Karch (2015) and Corrias, Escobedo & Matos (2014). Our analysis improves earlier results and extends them to any dimension d ≥ 3. Our size conditions on the initial data for the global existence of solutions seem to be optimal, up to a logarithmic factor in τ , when τ 1: we illustrate this fact by introducing two toy models, both consisting of systems of two parabolic equations, obtained after a slight modification of the nonlinearity of the usual Keller-Segel system. For these toy models, we establish in a companion paper [4] finite time blowup for a class of large solutions.

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The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions

Archive for Rational Mechanics and Analysis, Aug 28, 2014

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porou... more A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported selfsimilar solutions which generalize Barenblatt profiles-the well-known solutions of the classical porous medium equation.

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Around a singular solution of a nonlocal nonlinear heat equation

arXiv (Cornell University), Jul 10, 2018

We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal di... more We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey space). Then, asymptotics of subcritical solutions is determined. These results are compared with conditions on the initial data leading to a finite time blowup.

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Partition of energy in strongly damped generalized wave equations

Mathematical Methods in The Applied Sciences, Feb 1, 1990

Dissipative perturbations of hyperbolic equations such as utt + But + A2u = 0 with positive opera... more

A singular perturbation problem for nonlinear damped hyperbolic equations

Proceedings, 1989

We consider damped nonlinear hyperbolic equations u n + Au, + aAu + f3A 2 u + G(u) = 0, where A i... more We consider damped nonlinear hyperbolic equations u n + Au, + aAu + f3A 2 u + G(u) = 0, where A is a positive operator and G is the Gateaux derivative of a convex functional. We examine the asymptotic behaviour of solutions and the convergence of strong solutions to these equations when the parameter /S tends to zero.

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Barenblatt profiles for a nonlocal porous medium equation

Comptes Rendus Mathematique, Jun 1, 2011

We study a generalization of the porous medium equation involving nonlocal terms. More precisely,... more We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit selfsimilar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem. Résumé Solutions auto-similaires pour uneéquation des milieux poreux non locale Cette note est consacréeà l'étude d'une généralisation non locale de l'équation des milieux poreux. Plus précisément, on obtient des formules explicites de solutions auto-similairesà support compact qui ressemblent fortement aux solutions de type Barenblatt. On donne aussi un argument formel qui permet d'obtenir des estimations L p des solutions faibles du problème de Cauchy. Version française abrégée Nous considérons le problème de Cauchy pour l'équation non locale suivante ∂ t u − ∇ • u∇ α−1 |u| m−1 = 0, (1) avec m > 1, x ∈ R d , α ∈ (0, 2), t > 0,à laquelle on ajoute une condition initiale u(0, x) = u 0 (x). Ici ∇ β est un opérateur intégral singulier généralisant le gradient usuel (β = 1) et lié au laplacien fractionnaire. Le résultat principal de cette note sont des formules explicites de solutions auto-similaires qui se propagent a une vitesse finie. Théorème 1 (Solutions auto-similaires) La fonction u(t, x) = Ct − d d(m−1)+α R 2 − |x| 2 t − 2 d(m−1)+α α 2 + 1 m−1 (ainsi que ses translations en x) est une solution auto-similaire de l'équation (1) pour R > 0 quelconque et une constante C > 0 convenable.

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Long time behavior of solutions of Nernst-Planck and Debye-Hückel drift-diffusion systems

Zenodo (CERN European Organization for Nuclear Research), 2000

We study the convergence rates of solutions to drift-diffusion systems (arising from plasma, semi... more

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Nonlinear Singular Parabolic Equations

CRC Press eBooks, Oct 7, 2020

Self-Similar Solutions of Nonlinear PDE

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