22 citations · 35 across the 13 of their papers we have counts for
15 papers
Fast diffusion equation: uniqueness of solutions with a moving singularity
Marek Fila, Petra Macková
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 smoot…
Solvability of the heat equation on a half-space with a dynamical boundary condition and unbounded initial data
Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami
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 s…
Anisotropic and isotropic persistent singularities of solutions of the fast diffusion equation
Marek Fila, Petra Macková, Jin Takahashi +1
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 solution…
Solutions with snaking singularities for the fast diffusion equation
M. Fila, J. R. King, J. Takahashi +1
We construct solutions of the fast diffusion equation, which exist for all and are singular on the set , , where $ξ\in…
The large diffusion limit for the heat equation in the exterior of the unit ball with a dynamical boundary condition
Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami +1
We study the heat equation in the exterior of the unit ball with a linear dynamical boundary condition. Our main aim is to find upper and lower bounds for the rate of convergence t…
A Gagliardo-Nirenberg type inequality for rapidly decaying functions
Marek Fila, Johannes Lankeit
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(\mathb…
Lack of smoothing for bounded solutions of a semilinear parabolic equation
Marek Fila, Johannes Lankeit
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 . The singularity…
Continuation beyond interior gradient blow-up in a semilinear parabolic equation
Marek Fila, Johannes Lankeit
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 s…
Rate of convergence in the large diffusion limit for the heat equation with a dynamical boundary condition
Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami +1
We study the heat equation on a half-space or on an exterior domain with a linear dynamical boundary condition. Our main aim is to establish the rate of convergence to solutions of…
A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation
Marek Fila, Michael Winkler
We establish a Gagliardo-Nirenberg-type inequality in for functions which decay fast as . We use this inequality to derive upper bounds for the decay r…
Slow growth of solutions of super-fast diffusion equations with unbounded initial data
Marek Fila, Michael Winkler
We study positive solutions of the super-fast diffusion equation in the whole space with initial data which are unbounded as . We find an explicit dependence of the s…
Rate of Convergence to Separable Solutions of the Fast Diffusion Equation
Marek Fila, Michael Winkler
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 so…
Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent
Marek Fila, John R. King, Michael Winkler
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 w…
Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation
Marek Fila, Juan Luis Vazquez, Michael Winkler +1
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence…
A Continuum of Extinction Rates for the Fast Diffusion Equation
Marek Fila, Juan Luis Vazquez, Michael Winkler
We find a continuum of extinction rates for solutions of the fast diffusion equation in a subrange of exponents . The equation is posed in $\re…