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pro vyhledávání: '"Sierżęga, Mikołaj"'
In this paper, we give sufficient conditions for global-in-time existence of classical solutions for the fully parabolic chemorepulsion system posed on a convex, bounded three-dimensional domain. Our main result establishes global-in-time existence o
Externí odkaz:
http://arxiv.org/abs/2303.09620
Autor:
Dembny, Mateusz, Sierżęga, Mikołaj
A sharp double-sided Harnack bound is derived for positive solutions of a fractional order heat equation.
Externí odkaz:
http://arxiv.org/abs/2303.08186
Autor:
Fabisiak, Michał, Sierżęga, Mikołaj
The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ \Delta w-\frac{1}{2} \langle x,\nabla w\rangle-\frac{\lambda}{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical or critica
Externí odkaz:
http://arxiv.org/abs/2207.07207
Autor:
Hisa, Kotaro, Sierżęga, Mikołaj
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-\Delta)^\frac{\theta}{2} u=|x|^{-\gamma} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)
Externí odkaz:
http://arxiv.org/abs/2102.04079
Autor:
Laister, Robert, Sierzega, Mikolaj
Publikováno v:
Mathematische Annalen 381 (2021) pp. 75-90
We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive
Externí odkaz:
http://arxiv.org/abs/1912.01537
Autor:
Laister, Robert, Sierzega, Mikolaj
Publikováno v:
Annales de l'Institut Henri Poincar\'e C, Analyse non lin\'eaire 37 (3), 2020, 709-725
The problem of obtaining necessary and sufficient conditions for local existence of non-negative solutions in Lebesgue spaces for semilinear heat equations having monotonically increasing source term $f$ has only recently been resolved (Laister et al
Externí odkaz:
http://arxiv.org/abs/1911.10530
In this paper, by using scalar nonlinear parabolic equations, we construct supersolutions for a class of nonlinear parabolic systems including $$ \left\{\begin{array}{ll} \partial_t u=\Delta u+v^p,\qquad & x\in\Omega,\,\,\,t>0,\\ \partial_t v=\Delta
Externí odkaz:
http://arxiv.org/abs/1510.07838
Akademický článek
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Publikováno v:
Annales de l'Institut Henri Poincar\'e - Analyse Non-Lin\'eaire, 33 (6). pp. 1519-1538 (2016)
We consider the scalar semilinear heat equation $u_t-\Delta u=f(u)$, where $f\colon[0,\infty)\to[0,\infty)$ is continuous and non-decreasing but need not be convex. We completely characterise those functions $f$ for which the equation has a local sol
Externí odkaz:
http://arxiv.org/abs/1407.2444
We give a simple proof of a lower bound for the Dirichlet heat kernel in terms of the Gaussian heat kernel. Using this we establish a non-existence result for semilinear heat equations with zero Dirichlet boundary conditions and initial data in $L^q(
Externí odkaz:
http://arxiv.org/abs/1307.6688