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Piotr Biler

Publications and source records attributed to Piotr Biler.

At least 19 recordsLinked to original sources
arXiv2022

Sharp well-posedness and blowup results for parabolic systems of the Keller-Segel type

We study two toy models obtained after a slight modification of the nonlinearity of the usual doubly parabolic Keller-Segel system. For these toy models, both consisting of a system of two parabolic equations, we establish that for data which are, in a suitable sense, smaller than the diffusion parameter $τ$ in the equation for the chemoattractant, we obtain global solutions, and for some data larger than $τ$ , a finite time blowup. In this way, we check that our size condition for the global existence is sharp for large $τ$ , up to a logarithmic factor.

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arXiv2022

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

We study the global existence of the parabolic-parabolic Keller-Segel system in $\R^d , d \ge 2$. 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 $d \ge 3$. 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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arXiv2020

Sharp Sobolev estimates for concentration of solutions to an aggregation-diffusion equation

We consider the drift-diffusion equation $u_t-εΔu + \nabla \cdot(u\nabla K^*u)=0$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselves to the model case $K(x)=-|x|$. We quantify the mass concentration phenomenon, a genuinely nonlinear effect, for radially symmetric solutions of this equation for small diffusivity $ε$ studied in our previous paper [3], obtaining optimal sharp upper and lower bounds for Sobolev norms.

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arXiv2020

Concentration phenomena in a diffusive aggregation model

We consider the drift-diffusion equation $$ u_t-\varepsilon Δu+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied.

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arXiv2018

Around a singular solution of a nonlocal nonlinear heat equation

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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arXiv2017

Blowup versus global in time existence of solutions for nonlinear heat equations

This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.

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arXiv2017

Large global-in-time solutions to a nonlocal model of chemotaxis

We consider the parabolic-elliptic model for the chemotaxis with fractional (anomalous) diffusion. Global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.

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arXiv2016

Local criteria for blowup in two-dimensional chemotaxis models

We consider two-dimensional versions of the Keller--Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Moreover, the impact of the consumption term on the global-in-time existence of solutions is analyzed for the classical Keller--Segel system.

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arXiv2015

Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems

A simple proof of concentration of mass equal to $8π$ for blowing up $N$-symmetric solutions of the Keller--Segel model of chemotaxis in two dimensions with large $N$ is given. Moreover, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is derived for radial solutions of chemotaxis in several dimensions. This condition is, in a sense, complementary to the one guaranteeing the global-in-time existence of solutions.

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arXiv2015

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 has been obtained by Senba--Suzuki and Bedrossian--Masmoudi 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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arXiv2014

Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane

As it is well known, the parabolic-elliptic Keller-Segel system of chemotaxis on the plane has global-in-time regular nonnegative solutions with total mass below the critical value $8π$. Solutions with mass above $8π$ blow up in a finite time. We show that the case of the parabolic-parabolic Keller-Segel is different: each mass may lead to a global-in-time-solution, even if the initial data is a finite signed measure. These solutions need not be unique, even if we limit ourselves to nonnegative solutions.

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arXiv2013

Nonlocal porous medium equation: Barenblatt profiles and other weak solutions

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 self-similar solutions which generalize Barenblatt profiles --- the well-known solutions of the classical porous medium equation.

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arXiv2011

Barenblatt profiles for a nonlocal porous media equation

We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar 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.

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arXiv2009

Large mass self-similar solutions of the parabolic-parabolic Keller--Segel model of chemotaxis

In two space dimensions, the parabolic-parabolic Keller--Segel system shares many properties with the parabolic-elliptic Keller--Segel system. In particular, solutions globally exist in both cases as long as their mass is less than 8?. However, this threshold is not as clear in the parabolic-parabolic case as it is in the parabolic-elliptic case, in which solutions with mass above 8? always blow up. Here we study forward self-similar solutions of the parabolic-parabolic Keller--Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above 8?, which is forbidden in the parabolic-elliptic case.

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