Blowup of solutions to a diffusive aggregation model
The nonexistence of global in time solutions is studied for a class of aggregation equations involving Lévy diffusion operators and general interaction kernels.
math.AP↗
arXiv subjects
Piotr Biler
Publications and source records attributed to Piotr Biler.
On the parabolic-elliptic limit of the doubly parabolic Keller--Segel system modelling chemotaxis
We establish new convergence results, in strong topologies, for 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.
math.AP↗
Blow up of solutions to generalized Keller--Segel model
The existence and nonexistence of global in time solutions is studied for a class of equations generalizing the chemotaxis model of Keller and Segel. These equations involve Lévy diffusion operators and general potential type nonlinear terms.
math.AP↗
Nonlinear diffusion of dislocation density and self-similar solutions
We study a nonlinear pseudodifferential equation describing the dynamics of dislocations. The long time asymptotics of solutions is described by the self-similar profiles.
math.AP↗
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 the Debye-Hukel 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.
math.AP↗