Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation

Autor: Grzegorz Karch, Alexandre Boritchev, Philippe Laurençot, Piotr Biler

Publikováno v: Journal of Dynamics and Differential Equations
Journal of Dynamics and Differential Equations, 2022, 34, pp.3131--3141. ⟨10.1007/s10884-021-09998-w⟩

We consider the drift-diffusion equation $$\begin{aligned} u_t-\varepsilon \varDelta u+\nabla \cdot (u\ \nabla K*u)=0 \end{aligned}$$ in the whole space with global-in-time solutions bounded in all Sobolev spaces; for simplicity, we restrict ourselve

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::f9ca171a6c80ee15d5dfee04cb900a2c
https://doi.org/10.1007/s10884-021-09998-w

Blowup of solutions for nonlinear nonlocal heat equations

Autor: Piotr Biler

Publikováno v: Monatshefte für Mathematik. 189:611-624

Blowup analysis for solutions of a general evolution equation with nonlocal diffusion and localized source is performed. By comparison with recent results on global-in-time solutions, a dichotomy result is obtained.
Comment: 12

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::04be99bfec75da1f34bd6a4c11bc2af9
https://doi.org/10.1007/s00605-019-01269-7

Problems and Examples in Differential Equations

Autor: Piotr Biler, Tadeusz Nadzieja

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_________::2b30c6915c31c3f4182fddced69b2d53
https://doi.org/10.1201/9781003066835

Concentration phenomena in a diffusive aggregation model

Autor: Philippe Laurençot, Grzegorz Karch, Piotr Biler, Alexandre Boritchev

Publikováno v: Journal of Differential Equations
Journal of Differential Equations, Elsevier, 2021, 271, pp.1092--1108
Journal of Differential Equations, 2021, 271, pp.1092--1108. ⟨10.1016/j.jde.2020.09.035⟩

We consider the drift-diffusion equation u t − e Δ u + ∇ ⋅ ( u ∇ K ⁎ 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 ar

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::da22cdf987f457078eebcb6bf3b3e008

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

Autor: Grzegorz Karch, Jacek Zienkiewicz, Piotr Biler

Publikováno v: Advances in Mathematics. 330:834-875

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 radi

Externí odkaz: https://explore.openaire.eu/search/publication?articleId=doi_dedup___::600fca3a905ba16f292848dc03769846
https://doi.org/10.1016/j.aim.2018.03.036