arXiv subjects

Rafał Celiński

Publications and source records attributed to Rafał Celiński.

3 recordsLinked to original sources

arXiv2021

Mathematical treatment of PDE model describing chemotactic E. coli colonies

We consider an initial-boundary value problem describing the formation of colony patterns of bacteria Escherichia coli. This model consists of reaction-diffusion equations coupled with the Keller-Segel system from the chemotaxis theory in a bounded domain, supplemented with zero-flux boundary conditions and with non-negative initial data. We answer questions on the global in time existence of solutions as well as on their large time behaviour. Moreover, we show that solutions of a related model may blow up in a finite time.

math.AP↗

arXiv2013

Stability of solutions to aggregation equation in bounded domains

We consider the aggregation equation $u_t= ÷(\nabla u-u\nabla \K(u))$ in a bounded domain $Ω\subset \R^d$ with supplemented the Neumann boundary condition and with a nonnegative, integrable initial datum. Here, $\K=\K(u)$ is an integral operator. We study the local and global existence of solutions and we derive conditions which lead us to either the stability or instability of constant solutions.

math.AP↗

arXiv2011

Asymptotics behaviour in one dimensional model of interacting particles

We consider the equation u_t=εu_{xx}+(u\ K'*u)_x for x\in\mathbb{R}, t>0 and with ε\geq 0, supplemented with a nonnegative, integrable initial datum. We present a class of interaction kernels K' such that the large time behaviour of solutions to this initial value problem is described by a compactly supported self-similar profile.

math.AP↗