Marek Fila

Comenius University Bratislava · Department of Applied Mathematics and Statistics

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, these solutions either solve the original equation in the distributional sense, or they are not locally integra...

We construct solutions of the fast diffusion equation, which exist for all $t\in\mathbb{R}$ and are singular on the set $\Gamma(t):= \{ \xi(s) ; -\infty <s \leq ct \}$, $c>0$, where $\xi\in C^3(\mathbb{R};\mathbb{R}^n)$, $n\geq 2$. We also give a precise description of the behavior of the solutions near $\Gamma(t)$.

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.

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 to solutions of the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

We construct solutions with prescribed moving singularities for equations of porous medium type in two space dimensions. This complements a previous study of the problem where only dimensions higher than two were considered.

We improve the Gagliardo–Nirenberg inequality $$\begin{aligned} \Vert \varphi \Vert _{L^q({\mathbb {R}}^n)} \le C \Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)} {\mathcal {L}}^{-(\frac{1}{q} - \frac{n-2}{2n})} (\Vert \nabla \varphi \Vert _{L^r({\mathbb {R}}^n)}), \end{aligned}$$\(r=2\), \(0<q<\frac{2n}{(n-2)_+}\), \({\mathcal {L}}\) generalizin...

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 → ∞.

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.

We improve the Gagliardo-Nirenberg inequality \[ \|\varphi\|_{L^q(\mathbb{R}^n)} \le C \|\nabla\varphi\|_{L^r(\mathbb{R}^n)} \mathcal{L}^{-(\frac 1q - \frac{n-r}{rn})} (\|\nabla\varphi\|_{L^r(\mathbb{R}^n)}), \] $r=2$, $0<q<\frac{rn}{(n-r)_+}$, $\mathcal{L}$ generalizing $\mathcal{L}(s)=\ln^{-1}\frac 2s$ for $0<s<1$, from [M. Fila and M. Winkler: A...

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 $t\to\infty$.

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.

We construct positive solutions of equations of porous medium type with a singularity which moves in time along a prescribed curve and keeps the spatial profile of singular stationary solutions. It turns out that there appears a critical exponent for the existence of such solutions.

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 the Laplace equation with the same dynamical boundary condition as the diffusion coefficient tends to infinity.

We study the heat equation on a half-space with a linear dynamical boundary condition. Our main aim is to show that, if the diffusion coefficient tends to infinity, then the solutions converge (in a suitable sense) to solutions of the Laplace equation with the same dynamical boundary condition.

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 appro...

Several results on existence, nonexistence and large-time behavior of small positive solutions \(u=u(x,t)\) were obtained before for the equation \(-{\varDelta }u=u^p\), \(x\in {\mathbb R}^N_+\), \(t>0\), with a linear dynamical boundary condition. Here \({\varDelta }\) is the N-dimensional Laplacian (in x). We study the effects of the change of th...

We study properties of positive solutions of a semilinear elliptic equation with a linear dynamical boundary condition. We establish the semigroup property for minimal solutions, show that every local-in-time solution can be extended globally, and reveal a relationship between minimal solutions of the time-dependent problem and minimal solutions of...

We study positive solutions of the super-fast diffusion equation in the whole space with initial data which are unbounded as $|x|\to\infty$. We find an explicit dependence of the slow temporal growth rate of solutions on the initial spatial growth rate. A new class of self-similar solutions plays a significant role in our analysis.

We study the asymptotic behaviour of solutions of the fast diffusion equation near extinction. For a class of initial data, the asymptotic behaviour is described by a singular Barenblatt profile. We complete previous results on rates of convergence to the singular Barenblatt profile by describing a new phenomenon concerning the difference between t...

We study the asymptotic behaviour near extinction of positive solutions of the fast diffusion equation with critical and subcritical exponents. By a suitable rescaling, the equation is transformed to a nonlinear Fokker–Planck equation. We show that the rate of convergence to regular and singular steady states of the transformed equation can be arbi...

We study properties of positive solutions of a semilinear elliptic equation with a linear dynamical boundary condition. We establish the semigroup property for minimal solutions, show that every local-in-time solution can be extended globally, and reveal a relationship between minimal solutions of the time-dependent problem and minimal solutions of...

The purpose of this paper is to construct positive solutions of the semilinear elliptic equation −Δu = up in ℝ+N with a singular Dirichlet boundary condition. We show that for p > (N+1)=(N−1) there exists a positive singular solution which behaves like |x|−2/(p−1) as |x| → 0 and like the Poisson kernel as |x| → ∞.

We find a continuum of extinction rates of solutions of the Cauchy problem for the fast diffusion equation ur = δ · (u m-1 δu) with m = m* := (n-4)=(n-2), here n > 2 is the space-dimension. The extinction rates depend explicitly on the spatial decay rates of initial data and contain a logarithmic term.

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 that yields a nonlinear Fokker–Planck equation, we find a continuum of algebraic rates of convergence to a self-similar profile. These rates depend explicitly on the spati...

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 rat...

We show the existence of at least three different continuations beyond blow-up for a backward self-similar solution of a supercritical Fujita equation. One of these extended solutions cannot be approximated by classical entire solutions in a specific way given by the scaling invariance of the equation, while the minimal continuation is known to be...

We consider the following initial value problem for a two-dimensional semilinear elliptic equation with a dynamical boundary condition: [GRAPHICS] where u = u(x, t), partial derivative(t) :=partial derivative/partial derivative t,partial derivative(v) := -partial derivative/partial derivative x(2), R-2 + := {( x(1), x(2)): x(1.) R, x(2) > 0} and p...

We present conjectures on asymptotic behaviour of threshold solutions of the Cauchy problem for a semilinear heat equation with Sobolev critical nonlinearity. The conjectures say that, depending on the decay rate of initial data and the space dimension, the threshold solutions may grow up, stabilize, or decay to zero as t→∞. The rates of grow up or...

We study the large time behavior of positive solutions for the Laplace equation on the half-space with a nonlinear dynamical boundary condition. We show the convergence to the Poisson kernel in a suitable sense provided initial data are sufficiently small.

We study the existence of connecting orbits for the Fujita equation with a critical or supercritical exponent. For certain ranges of the exponent we prove the existence of heteroclinic connections from positive steady states to zero and a homoclinic orbit with respect to zero.

Consider the problem where β > 0 and Ω = B R (0) ≔ x ∈ IR N ; |x| < R. It is known ([AW]) that there is a positive number R o = R o (N,β) such that u exists globally if R < R o while for R> R o the solution u reaches zero in a finite time T (it quenches). The only point x o for which u(x o , t) → 0 as t 2192 T is x o = 0 (see [AK]).

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 to a self-similar profile. These rates depend explicitly on the spatial decay rates of initial data.

We consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the...

We find a continuum of extinction rates for solutions $u(y,\tau)\ge 0$ of the fast diffusion equation $u_\tau=\Delta u^m$ 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 $\|u(\cdot,\tau)\|_\infty\sim (T-\tau)^\theta$ \ for a whole interval of $\theta>0$. The...

It is known that diffusion together with Dirichlet boundary conditions can inhibit the occurrence of blow-up. We examine the question how strong is this stabilizing effect for reaction-diffusion equations in one space-dimension. We show that if all positive solutions of an ODE blow up in finite time then for the corresponding parabolic PDE (obtaine...

We consider the exponential reaction–diffusion equation in space-dimension n∈(2,10). We show that for any integer k≥2 there is a backward selfsimilar solution which crosses the singular steady state k-times. The same holds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense.

We analyze a mathematical model introduced by Anguige, Ward and King (J. Math. Biol. 51 (2005), 557-594) to describe a quorum sensing mechanism in spatially-structured populations of the bacteria Pseudomonas aeruginosa. In the biologically relevant limit case when the spatial distribution of bacteria is constant in time, this model reduces to a sin...

. We study global positive solutions of a supercritical parabolic equation which converge to a steady state that is singular at x = 0. We determine the rate of convergence to the singular steady state in $$L^\infty({\mathbb{R}}^{N} \ B_{\nu}(0))$$ where B ν(0) is a ball in $${\mathbb{R}}^{N}$$ with the center at the origin and radius ν.

We study the blow-up profile of radial solutions of a semilinear heat equation with an exponential source term. Our main aim is to show that solutions which can be continued beyond blow-up possess a nonconstant selfsimilar blow-up profile. For some particular solutions we determine this profile precisely.

We study the behavior of solutions of the Cauchy problem for a parabolic equation with power nonlinearity. Our concern is the rate of convergence of solutions to forward self-similar solutions. We determine the exact rate of convergence which turns out to depend on the spatial decay rate of initial data.

We give sufficient conditions which guarantee that solutions of a superlinear heat equation decay to zero at the same rate as the solutions of the linear heat equation with the same initial data. This improves a previous result of Lee and Ni by showing that this behaviour holds for a significantly larger, and rather tightly defined class of solutio...

We study the behaviour of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t → ∞. We show that any algebraic decay rate slower than the self-similar one occurs for some initial data.

We study radially symmetric classical solutions of the Dirichlet problem for a heat equation with a supercritical nonlinear source. We give a sufficient condition under which blow-up in infinite time cannot occur. This condition involves only the growth rate of the source term at infinity. We do not need the homogeneity property which played a key...

We find the rate of convergence of solutions of a semilinear parabolic equation to selfsimilar solutions of the linear heat equation. In particular, we show that the rate is not affected by the nonlinearity for some range of parameters, while in a complementary range the rate depends explicitly on the nonlinearity.

We consider a reaction-diffusion-convection equation on the halfline $(0,\infty)$ with the zero Dirichlet boundary condition at $x=0$. We find a positive selfsimilar solution $u$ which blows up in a finite time $T$ at $x=0$ while $u(x,T)$ remains bounded for $x>0$.

We construct solutions of a supercritical parabolic equation which blow up in finite time but possess multiple global continuations (as weak solutions).

We study solutions of a parabolic equation which are bounded but whose spatial derivatives blow up in finite time. We establish results on the behavior on the lateral boundary where the singularity occurs and on the rate of convergence to a singular steady state.

We consider the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the Joseph-Lundgren sense. We find the grow-up rate of solutions that approach a singular steady state from below as $t\to\infty$. The grow-up rate in the critical case contains a logarithmic term which does not appear in the Joseph-Lundgren...

For some nonlinear parabolic equations, solutions may not exist globally for t ≥ 0, but may become unbounded in finite time. This phenomenon is called “blow-up” and it has been intensively studied in connection with various fields of science, such as plasma physics, the combustion theory, and population dynamics. The chapter discusses blow-up of th...

We study the behavior of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which approach a singular steady state from below as t→∞. It is known that the grow-up rate of such solutions depends on the spatial decay rate of initial data. We give an optimal lower bound on the grow-up rate by using a comparison technique...

We study the asymptotic behavior of positive solutions of a semilinear parabolic equation with a nonlinear boundary condition. This problem admits a unique stationary solution which is not bounded and attracts all positive solutions. We find their growth rate at the singular point on the boundary.

This paper examines the following question: Suppose that we have a reaction-diffusion equation or system such that some solutions which are ho-mogeneous in space blow up in finite time. Is it possible to inhibit the occur-rence of blow-up as a consequence of imposing Dirichlet boundary conditions, or of other effects where diffusion plays a role? W...

A survey is given of results on the relation of the dynamics of a system of ordinary differential equations to the dynamics of the corresponding reaction-diffusion system when diffusion is added. The main interest here is in the influence of diffusion on the global existence of solutions. Examples are presented of systems where diffusion induces or...

We study the behavior of solutions of the Cauchy problem for a diffusion equation with supercritical nonlinearity. It is shown that if two solutions are initially close enough near the spatial infinity, then these solutions approach each other. Under some conditions, we determine the exact convergence rate, which turns out to depend on initial data...

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow-up time and if it blows up again, it can only do so finitely many times.

We study solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to a singular steady state from below as t→∞. We show that the grow-up rate of such solutions depends on the spatial decay rate of initial data.

We study the existence of positive solutions of a linear elliptic equation with critical Sobolev exponent in a nonlinear Neumann boundary condition. We prove a result which is similar to a classical result of Brezis and Nirenberg who considered a corresponding problem with nonlinearity in the equation. Our proof of the fact that the dimension three...

We study the blow-up behavior (in time and space) of positive solutions of a semilinear parabolic equation with a gradient term u t =Δu-|∇u| q +u p inΩ×(0,T),u(x,t)=0if(x,t)∈∂Ω×(0,T),u(x,0)=u 0 (x)ifx∈Ω, where p,q>1, Ω is a bounded domain in ℝ n , with C 2 boundary, and u 0 ≥0. Our main result is a sharp estimate for the spatial blow-up profile of...

We give a necessary and sufficient condition for the existence of L 1-connections between equilibria of a semilinear parabolic equation. By an L 1-connection from an equilibrium φ − to an equilibrium φ + we mean a function u(⋅, t) which is a classical solution on the interval (−∞, T) for some T ∈ ∝ and blows up at t = T but continues to exist in th...

An abstract is not available.

We derive results on blow-up rates for parabolic equations and systems from Fujita-type theorems. We complement a previous study by allowing (possibly unbounded) domains with boundary.

We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces , where is the distance to the boundary. In particular, we prove an optimal estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial value...

We consider a one-phase Stefan problem for the heat equation with a superlinear reaction term. It is known from a previous work (Ghidouche, Souplet, & Tarzia [5]) that all global solutions are bounded and decay uniformly to 0. Moreover, it was shown in Ghidouche, Souplet, & Tarzia [5] that either: (i) the free boundary converges to a finite limit a...

In this paper positive solutions of the heat equation with a nonlinear Neumann boundary conditions in an upper halfspace are studied. The optimal result on blow-up rate, valid for all solutions which blow up in finite time, is established under the assumption that the exponent of a nonlinear boundary condition is subcritical in the Sobolev sense. T...

this article we will survey briey some of the recent developments in this eld. In Section 1 we will show how the center manifold theory sheds light on the rich structure of blowup proles. We will also review some other dynamical systems approaches to blow-up phenomena. In Section 2 we will discuss whether or not solutions have a continuation beyond...

We study heteroclinic connections in a nonlinear heat equation that involves blow-up. More precisely we discuss the existence of L 1 connections among equilibrium solutions. By an L 1 -connection from an equilibrium ϕ -1 to an equilibrium ϕ + we mean a function u(·,t) which is a classical solution on the interval (-∞,T) for some T∈ℝ and blows up at...

M. CHLEB´IKCHLEB´CHLEB´IK – M. FILA Riassunto: Si espongono alcuni risultati sulla velocità di blow-up per tre sistemi parabolici e per l'equazione di Chipot-Weissler. Si ottengono questi risultati da noti teoremi del tipo di Fujita. Abstract: In this paper, we derive new results on blow-up rates for three parabolic systems and for the Chipot-Weiss...

Radial solutions of the Gelfand equation on an $N$--dimensional ball are studied for $3\le N\le 9$. It is shown that global classical solutions are uniformly bounded while unbounded global $L^1$--solutions are constructed for some parameter range.

We consider the problem (1.1) (1.2) (1.3) where Ω is a bounded domain in ℝN, v is the outer normal on δΩ, p, q > 1, λ, μ ∈ {-1,0,1}, max{λ, μ} = 1 and λp+ μq > 0

In this paper we consider the question of the long time behavior of solutions of the initial value problem for evolution equations of the formdu/dt + Au(t) = F(u(t)) in a Banach space whereAis the infinitesimal generator of an analytic semigroup andFis a nonlinear function such that the initial value problem possesses nonglobal solutions for some i...

We study the boundedness and a priori bounds of global solutions of the problem Δu=0 in Ω×(0, T), (∂u/∂t) + (∂u/∂ν) = h(u) on ∂Ω×(0, T), where Ω is a bounded domain in ℝN, ν is the outer normal on ∂Ω and h is a superlinear function. As an application of our results we show the existence of sign-changing stationary solutions. © 1997 B. G. Teubner St...

In this paper, we consider the system \arraycolsep0.14em\begin{array}{rcl {\hskip2em}rcl {\hskip2em}c}u_t&=&\Delta u+v^p,&v_t&=&\Delta v&x\in{\Bbb R}_{+}^N,t>0,\\ \displaystyle-{\partial u\over\partial x_t}&=&0,&\displaystyle-{\partial v\over\partial x_t}&=&u^q&x_1=0,t>0,\\ u(x,0)&=&u_0(x),&v(x,0)&=&v_0(x)&x\in{\Bbb R}_{+}^N, whereRN+={(x1,x′)|x′∈R...

We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions w...

We describe all nontrivial nonnegative solutions to the problem $-\Delta u=au^{(n+2)/(n-2)}$ in $H,\ \partial u/\partial\nu=bu^{n/(n-2)}$ on $\partial H$, where $H$ is a half-space of ${\bf R}^n\ (n\geq3)$.

this paper.

. In this paper, we consider the system u t = Deltau; v t = Deltav x 2 R N + ; t ? 0; Gamma @u @x 1 = v p ; Gamma @v @x 1 = u q x 1 = 0; t ? 0; u(x; 0) = u 0 (x); v(x; 0) = v 0 (x) x 2 R N + ; where R N + = f(x 1 ; x 0 ) j x 0 2 R N Gamma1 ; x 1 ? 0g, p; q ? 0, and u 0 , v 0 nonnegative. We prove that if pq 1 every nonnegative solution is global. W...

In this paper we condiser non-negative solutions of the initial value problem in ℝN for the system where 0 ⩽ δ ⩽ 1 and pq > 0. We prove the following conditions.Suppose min(p,q)≥1 but pq1.(a)If δ = 0 then u=v=0 is the only non-negative global solution of the system.(b)If δ>0, non-negative non-globle solutions always exist for suitable initial value...

L ∞ -blow-up of solutions of semilinear parabolic equations has received considerable interest. Several major problems like sufficient conditions for blow-up, the form of the blow-up set, the profile of the solution near a blow-up point or the existence after the blow-up time have been studied. The aim of this paper is to deal with similar question...

In this paper we prove that if 0 < β < 1, D ⊂ RN is bounded, and β > 0, then every element of the w-limit set of weak solutions of is a weak stationary solution of this problem. A consequence of this is that if D is a ball, β is sufficiently small, and uq is a radial, then the set ((x, t)Ɩu = 0) is a bounded subset of D × [0, ∞).

In this paper we prove that if $0 0$, then every element of the ω-limit set of weak solutions of $u_t - \Delta u + \lambda u^{-\beta} \chi_{u > 0} = 0 \quad \text{in} D \times \lbrack 0, \infty),$ \begin{equation*} u = \begin{cases} 1 \quad \text{on} \partial D \times (0, \infty), \\ u_0 > 0 \quad \text{on} \overline{D} \times \{0\}\end{cases} \end...