arXiv subjects
Publications and source records attributed to Dominika Pilarczyk.
We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. As applications, we give the Yaglom limit and entrance law for the corresponding killed isotropic stable Lévy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.
We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup.
We develop mathematical methods which allow us to study asymptotic properties of solutions to the three dimensional Navier-Stokes system for incompressible fluid in the whole three dimensional space. We deal either with the Cauchy problem or with the stationary problem where solutions may be singular due to singular external forces which are either singular finite measures or more general tempered distributions with bounded Fourier transforms. We present results on asymptotic properties of such solutions either for large values of the space variables (so called the far-field asymptotics) or for large values of time.
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.
We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the existence of radial global-in-time solutions in terms of suitable Morrey norms are derived.
We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on $\mathbb{R}^d$, including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.
We consider global-in-time small mild solutions of the initial value problem to the incompressible Navier-Stokes equations in $R^3$. For such solutions, an asymptotic stability is established under arbitrarily large initial $L^2$-perturbations.
It is known that the three dimensional Navier-Stokes system for an incompressible fluid in the whole space has a one parameter family of explicit stationary solutions, which are axisymmetric and homogeneous of degree -1. We show that these solutions are asymptotically stable under any $L^2$-perturbation.
In this paper, we discuss the asymptotic stability of singular steady states of the nonlinear heat equation in the weighted Lebesgue norms.