arXiv subjects
Publications and source records attributed to Mikołaj Sierżęga.
We establish a connection between a sharp double-sided Harnack bound for positive solutions of a fractional heat equation and the circular geometry in higher dimensions. The present work extends and generalizes the results obtained in the preceding paper.
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 of regular nonnegative solutions provided that $\nabla\sqrt{u} \in L^4(0, T; L^2(Ω))$. Our method is related to the Bakry--Émery calculation and appears to be new in this context.
A sharp double-sided Harnack bound is derived for positive solutions of a fractional order heat equation.
The question of triviality of solutions of the semilinear Ornstein-Uhlenbeck equation, \[ Δw-\frac{1}{2} \langle x,\nabla w\rangle-\fracλ{p-1}w+|w|^{p-1}w=0, \] is considered. It is shown, that if $p>1$ is Sobolev subcritical or critical and $λ\leq 1$, then all bounded entire solutions are constant. Moreover, in the critical case, the same conclusion holds in the subclass of radial solutions provided that $n\geq 4$ and $λ\in \left[\frac{3 n}{2(n-1)},2\right]$.
In this paper, we obtain necessary conditions and sufficient conditions on the initial data for the local-in-time solvability of the Cauchy problem \[ \partial_t u +(-Δ)^\fracθ{2} u=|x|^{-γ} u^p ,\quad x\in{\bf R}^N, t>0, \qquad u(0)=μ\quad \mbox{in} \quad {\bf R}^N, \] where $N\ge 1$, $0<θ\le2$, $p>1$, $γ>0$ and $μ$ is a nonnegative Radon measure on ${\bf R}^N$. Using these conditions, we attempt to identify the optimal strength of the singularity of $μ$ for the existence of solutions to this problem.
We discuss conditions for well-posedness of the scalar reaction-diffusion equation $u_{t}=Δu+f(u)$ equipped with Dirichlet boundary conditions where the initial data is unbounded. Standard growth conditions are juxtaposed with the no-blow-up condition $\int_{1}^{\infty}1/f(s) \d s=\infty$ that guarantees global solutions for the related ODE $\dot u=f(u)$. We investigate well-posedness of the toy PDE $u_{t}=f(u)$ in $L^{p}$ under this no-blow-up condition. An example is given of a source term $f$ and an initial condition $ψ\in L^{2}(0,1)$ such that $\int_{1}^{\infty}1/f(s)\d s=\infty$ and the toy PDE blows-up instantaneously while the reaction-diffusion equation is globally well-posed in $L^{2}(0,1)$.