Searcharxiv⌕ Search

arXiv subjects

Marek Fila

Publications and source records attributed to Marek Fila.

15 recordsLinked to original sources
arXiv2023

Fast diffusion equation: uniqueness of solutions with a moving singularity

We focus on open questions regarding the uniqueness of distributional solutions of the fast diffusion equation (FDE) with a given source term. When the source is sufficiently smooth, the uniqueness follows from standard results. Assuming that the source term is a measure, the existence of different classes of solutions is known, but in many cases, their uniqueness is an open problem. In our work, we focus on the supercritical FDE and prove the uniqueness of distributional solutions with a Dirac source term that moves along a prescribed curve.

math.AP↗
arXiv2022

Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data

We study the linear heat equation on a halfspace with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such that the problem possesses a solution. It was known before that bounded initial data guarantee solvability. Here we extend that result by showing that data from a weighted Lebesgue space will also do so.

math.AP↗
arXiv2022

Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation

The aim of this paper is to study a class of positive solutions of the fast diffusion equation with specific persistent singular behavior. First, we construct new types of solutions with anisotropic singularities. Depending on parameters, either these solutions solve the original equation in the distributional sense, or they are not locally integrable in space-time. We show that the latter also holds for solutions with snaking singularities, whose existence has been proved recently by M. Fila, J.R. King, J. Takahashi, and E. Yanagida. Moreover, we establish that in the distributional sense, isotropic solutions whose existence was proved by M. Fila, J. Takahashi, and E. Yanagida in 2019, actually solve the corresponding problem with a moving Dirac source term. Last, we discuss the existence of solutions with anisotropic singularities in a critical case.

math.AP↗
arXiv2019

A Gagliardo-Nirenberg type inequality for rapidly decaying functions

We improve the Gagliardo-Nirenberg inequality \[ \|φ\|_{L^q(\mathbb{R}^n)} \le C \|\nablaφ\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nablaφ\|_{L^r(\mathbb{R}^n)}), \] $r=2$, $0<q<\frac{rn}{(n-r)_+}$, $\mathcal{L}$ generalizing $\mathcal{L}(s)=\ln^{-1}\frac 2s$ for $0<s<1$, from [M. Fila and M. Winkler: A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, Adv. Math., 357 (2019), https://doi.org/10.1016/j.aim.2019.106823] for rapidly decaying functions ($φ\in W^{1,r}(\mathbb{R}^n)\setminus\{0\}$ with finite $K=\int_{\mathbb{R}^n} \mathcal{L}(φ)$) by specifying the dependence of $C$ on $K$ and by allowing arbitrary $r\ge1$.

math.AP↗
arXiv2019

Lack of smoothing for bounded solutions of a semilinear parabolic equation

We study a semilinear parabolic equation that possesses global bounded weak solutions whose gradient has a singularity in the interior of the domain for all $t>0$. The singularity of these solutions is of the same type as the singularity of a stationary solution to which they converge as $t\to\infty$.

math.AP↗
arXiv2019

Continuation beyond interior gradient blow-up in a semilinear parabolic equation

It is known that there is a class of semilinear parabolic equations for which interior gradient blow-up (in finite time) occurs for some solutions. We construct a continuation of such solutions after gradient blow-up. This continuation is global in time and we give an example when it never becomes a classical solution again.

math.AP↗
arXiv2017

A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation

We establish a Gagliardo-Nirenberg-type inequality in $\mathbb{R}^n$ for functions which decay fast as $|x|\to\infty$. We use this inequality to derive upper bounds for the decay rates of solutions of a degenerate parabolic equation. Moreover, we show that these upper bounds, hence also the Gagliardo-Nirenberg-type inequality, are sharp in an appropriate sense.

math.AP↗
arXiv2014

Rate of Convergence to Separable Solutions of the Fast Diffusion Equation

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class of functions which belong to their domain of attraction. For solutions in this class we establish optimal rates of convergence to separable solutions.

math.AP↗
arXiv2013

Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems.

math.AP↗
arXiv2010

A Continuum of Extinction Rates for the Fast Diffusion Equation

We find a continuum of extinction rates for solutions $u(y,τ)\ge 0$ of the fast diffusion equation $u_τ=Δu^m$ in a subrange of exponents $m\in (0,1)$. The equation is posed in $\ren$ for times up to the extinction time $T>0$. The rates take the form $\|u(\cdot,τ)\|_\infty\sim (T-τ)^θ$ \ for a whole interval of $θ>0$. These extinction rates depend explicitly on the spatial decay rates of initial data.

math.AP↗