Searcharxiv⌕ Search

arXiv subjects

Pavol Quittner

Publications and source records attributed to Pavol Quittner.

16 recordsLinked to original sources
arXiv2026

Oscillatory blow-up and gradient estimates for semilinear heat equations

For reaction-diffusion with blow-up nonlinearities, we consider the question whether the sup norm of any positive blow-up solution must be eventually monotone nondecreasing in time. While some sufficient conditions are known, especially for radial solutions, this natural and basic question for the blow-up theory does not seem to have been addressed so far in full generality. We construct surprising (nonradial) counter-examples of blow-up solutions with oscillatory $L^\infty$ norm, for any Sobolev supercritical power nonlinearity, which show that this property may fail. In addition, this provides examples of type II blow-up for any supercritical power, which considerably increases the known range of powers for which type II blow-up may occur. Moreover, whereas all the type II blow-up rates known so far were at most polynomial, the blow-up in our counter-examples can be arbitrarily singular. As a related question, we clarify the gradient estimates obtained and used in previous works. In particular we show that these estimates hold only at times when the $L^\infty$ norm is maximal with respect to the past.

math.AP↗
arXiv2025

Threshold, subthreshold and global unbounded solutions of superlinear heat equations

We consider the semilinear heat equation with a superlinear nonlinearity and we study the properties of threshold or subthreshold solutions, lying on or below the boundary between blow-up and global existence, respectively. For the Cauchy-Dirichlet problem, we prove the boundedness and decay to zero of any subthreshold solution. This implies, in particular, that all global unbounded solutions -- if they exist -- are threshold solutions. For the Cauchy problem, these properties fail in general but we show that they become true for a suitably modified notion of threshold. Our results strongly improve known results even in the model case of power nonlinearities, especially in the Sobolev critical and supercritical cases.

math.AP↗
arXiv2024

A priori estimates of solutions to local and nonlocal superlinear parabolic problems

We consider a priori estimates of possibly sign-changing solutions to superlinear parabolic problems and their applications (blow-up rates, energy blow-up, continuity of blow-up time, existence of nontrivial steady states etc). Our estimates are based mainly on energy, interpolation and bootstrap arguments, but we also use the Pohozaev identity, for example. We first discuss some known results on local problems and then consider problems with nonlocal nonlinearities or nonlocal differential operators. In particular, we deal with the fractional Laplacian and nonlinearities of Choquard type.

math.AP↗
arXiv2024

Liouville theorems and universal estimates for superlinear parabolic problems without scale invariance

We establish Liouville type theorems in the whole space and in a half-space for parabolic problems without scale invariance. To this end, we employ two methods, respectively based on the corresponding elliptic Liouville type theorems and energy estimates for suitably rescaled problems, and on reduction to a scalar equation by proportionality of components. We then give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant parabolic equations and systems involving superlinear nonlinearities with regular variation. To this end, we adapt methods from our preprint arXiv:2407.04154 to parabolic problems.

math.AP↗
arXiv2024

Liouville theorems and universal estimates for superlinear elliptic problems without scale invariance

We give applications of known and new Liouville type theorems to universal singularity and decay estimates for non scale invariant elliptic problems, including Lane-Emden and Schr\"odinger type systems. This applies to various classes of nonlinearities with regular variation and possibly different behaviors at $0$ and $\infty$. To this end, we adapt the method from [72] to elliptic systems, which relies on a generalized rescaling technique and on doubling arguments from [55]. This is in particular facilitated by new Liouville type theorems in the whole space and in a half-space, for elliptic problems without scale invariance, that we obtain. Our results apply to some non-cooperative systems, for which maximum principle based techniques such as moving planes do not apply. To prove these Liouville type theorems, we employ two methods, respectively based on Pohozaev-type identities combined with functional inequalities on the unit sphere, and on reduction to a scalar equation by proportionality of components. In turn we will survey the existing methods for proving Liouville-type theorems for superlinear elliptic equations and systems, and list some of the typical existing results for (Sobolev subcritical) systems. In the case of scalar equations, we also revisit the classical Gidas-Spruck integral Bernstein method, providing some improvements which turn out to be efficient for certain nonlinearities, and we next compare the performances of various methods on a benchmark example.

math.AP↗
arXiv2022

Necessary and sufficient conditions for one-dimensional variational problems with applications to elasticity

This paper deals with necessary and sufficient conditions for weak and strong minimizers of functionals $\Phi(u)=\int_a^b f(x,u(x),u'(x))\,dx$, where $u\in C^1([a,b],{\mathbb R}^N)$. We first derive conditions which are simpler than the known ones, and then apply them to several particular problems, including stability problems in the elasticity theory. In particular, we solve some open problems in [A. Majumdar, A. Raisch: Stability of twisted rods, helices and buckling solutions in three dimensions, Nonlinearity 27 (2014), 2841--2867] by finding optimal conditions for the stability of a naturally straight Kirchhoff rod under various types of endpoint constraints.

math.CA↗
arXiv2021

Liouville theorems for parabolic systems with homogeneous nonlinearities and gradient structure

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess nontrivial entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. Assume that $p>1$ is subcritical in the Sobolev sense. In the case of nonnegative solutions and the system $$U_t-\Delta U=F(U)\quad\hbox{in}\quad {\mathbb R}^n\times{\mathbb R},$$ where $U=(u_1,\dots,u_N)$, $F=\nabla G$ is $p$-homogeneous and satisfies the positivity assumptions $G(U)>0$ for $U\neq0$ and $\xi\cdot F(U)>0$ for some $\xi\in{\mathbb R}^N$ and all $U\geq0$, $U\ne 0$, it has recently been shown in [P. Quittner, Duke Math. J. 170 (2021), 1113-1136] that the parabolic Liouville theorem is true whenever the corresponding elliptic Liouville theorem for the system $-\Delta U=F(U)$ is true. By modifying the arguments in that proof we show that the same result remains true without the positivity assumptions on $G$ and $F$, and that the class of solutions can also be enlarged to contain (some or all) sign-changing solutions. In particular, in the scalar case $N=1$ and $F(u)=|u|^{p-1}u$, our results cover the main result in [T. Bartsch, P. Polacik and P. Quittner, J. European Math. Soc. 13 (2011), 219-247]. We also prove a parabolic Liouville theorem for solutions in ${\mathbb R}^n_+\times{\mathbb R}$ satisfying homogeneous Dirichlet boundary conditions on $\partial{\mathbb R}^n_+\times{\mathbb R}$ since such theorem is also needed if one wants to prove universal estimates of solutions of related systems in $\Omega\times(0,T)$, where $\Omega\subset{\mathbb R}^n$ is a smooth domain. Finally, we use our Liouville theorems to prove universal estimates for particular parabolic systems.

math.AP↗
arXiv2021

Liouville theorem and a priori estimates of radial solutions for a non-cooperative elliptic system

Liouville theorems for scaling invariant nonlinear elliptic systems (saying that the system does not possess nontrivial entire solutions) guarantee a priori estimates of solutions of related, more general systems. Assume that $p=2q+3>1$ is Sobolev subritical, $n\le3$ and $β\in{\mathbb R}$. We first prove a Liouville theorem for the system $$\left.\begin{aligned} -Δu &=|u|^{2q+2}u+β|v|^{q+2}|u|^q u, \\ -Δv &=|v|^{2q+2}v+β|u|^{q+2}|v|^q v, \end{aligned}\ \right\} \quad\hbox{in}\quad {\mathbb R}^n,$$ in the class of radial functions $(u,v)$ such that the number of nodal domains of $u,v,u-v,u+v$ is finite. Then we use this theorem to obtain a priori estimates of solutions to related elliptic systems. In the cubic case $q=0$, those solutions correspond to the solitary waves of a system of Schrödinger equations, and their existence and multiplicity have been intensively studied by various methods. One of those methods is based on a priori estimates of suitable global solutions of corresponding parabolic systems. Unlike the previous studies, our Liouville theorem yields those estimates for all $q\geq0$ which are Sobolev subcritical.

math.AP↗
arXiv2020

Optimal Liouville theorems for superlinear parabolic problems

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation $$u_t-Δu=u^p\quad\hbox{in}\quad {R}^n\times{R}, \qquad p>1, $$ the nonexistence of positive classical solutions in the subcritical range $p(n-2)<n+2$ has been conjectured for a long time, but all known results require either a more restrictive assumption on $p$ or deal with a special class of solutions (time-independent or radially symmetric or satisfying suitable decay conditions). We solve this open problem and -- by using the same arguments -- we also prove optimal Liouville theorems for a class of superlinear parabolic systems. In the case of the nonlinear heat equation, straightforward applications of our Liouville theorem solve several related long-standing problems. For example, they guarantee an optimal Liouville theorem for ancient solutions, optimal decay estimates for global solutions of the coresponding Cauchy problem, optimal blow-up rate estimate for solutions in non-convex domains, optimal universal estimates for solutions of the corresponding initial-boundary value problems. The proof of our main result is based on refined energy estimates for suitably rescaled solutions.

math.AP↗
arXiv2020

An optimal Liouville theorem for the linear heat equation with a nonlinear boundary condition

Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times $t\in(-\infty,\infty)$) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition $\partial u/\partialν=u^q$, $q>1$.

math.AP↗
arXiv2019

Entire and ancient solutions of a supercritical semilinear heat equation

We consider the semilinear heat equation $u_t=Δu+u^p$ on ${\mathbb R}^N$. Assuming that $N\ge 3$ and $p$ is greater than the Sobolev critical exponent $(N+2)/(N-2)$, we examine entire solutions (classical solutions defined for all $t\in {\mathbb R}$) and ancient solutions (classical solutions defined on $(-\infty,T)$ for some $T<\infty$). We prove a new Liouville-type theorem saying that if $p$ is greater than the Lepin exponent $p_L:=1+6/(N-10)$ ($p_L=\infty$ if $N\le 10$), then all positive bounded radial entire solutions are steady states. The theorem is not valid without the assumption of radial symmetry; in other ranges of supercritical $p$ it is known not to be valid even in the class of radial solutions. Our other results include classification theorems for nonstationary entire solutions (when they exist) and ancient solutions, as well as some applications in the theory of blowup of solutions.

math.AP↗
arXiv2019

On the multiplicity of self-similar solutions of the semilinear heat equation

In studies of superlinear parabolic equations \begin{equation*} u_t=Δu+u^p,\quad x\in {\mathbb R}^N,\ t>0, \end{equation*} where $p>1$, backward self-similar solutions play an important role. These are solutions of the form $ u(x,t) = (T-t)^{-1/(p-1)}w(y)$, where $y:=x/\sqrt{T-t}$, $T$ is a constant, and $w$ is a solution of the equation $Δw-y\cdot\nabla w/2 -w/(p-1)+w^p=0$. We consider (classical) positive radial solutions $w$ of this equation. Denoting by $p_S$, $p_{JL}$, $p_L$ the Sobolev, Joseph-Lundgren, and Lepin exponents, respectively, we show that for $p\in (p_S,p_{JL})$ there are only countably many solutions, and for $p\in (p_{JL},p_L)$ there are only finitely many solutions. This result answers two basic open questions regarding the multiplicity of the solutions.

math.AP↗
arXiv2016

Threshold and strong threshold solutions of a semilinear parabolic equation

If $p>1+2/n$ then the equation $u_t-Δu = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions lying on the borderline between global existence and blow-up.

math.AP↗
arXiv2015

Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka-Volterra systems

We consider positive solutions of cooperative parabolic Lotka-Volterra systems with equal diffusion coefficients, in bounded and unbounded domains. The systems are complemented by the Dirichlet or Neumann boundary conditions. Under suitable assumptions on the coefficients of the reaction terms, these problems possess both global solutions and solutions which blow up in finite time. We show that any solution $(u,v)$ defined on the time interval $(0,T)$ satisfies a universal estimate of the form $$u(x,t)+v(x,t)\leq C(1+t^{-1}+(T-t)^{-1}),$$ where $C$ does not depend on $x,t,u,v,T$. In particular, this bound guarantees global existence and boundedness for threshold solutions lying on the borderline between blow-up and global existence. Moreover, this bound yields optimal blow-up rate estimates for solutions which blow up in finite time. Our estimates are based on new Liouville-type theorems for the corresponding scaling invariant parabolic system and require an optimal restriction on the space dimension $n$: $n\leq5$. As an application we also prove the existence of time-periodic positive solutions if the coefficients are time-periodic. Our approach can also be used for more general parabolic systems.

math.AP↗
arXiv2014

Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure

We provide a simple method for obtaining new Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. To illustrate the method we prove Liouville theorems (guaranteeing nonexistence of positive classical solutions) for the following model problems: the scalar nonlinear heat equation $$ u_t-Δu=u^p \qquad\hbox{in}\ {\mathbb R}^n\times{\mathbb R}, $$ its vector-valued generalization with a $p$-homogeneous nonlinearity and the linear heat equation in ${\mathbb R}^n_+\times{\mathbb R}$ complemented by nonlinear boundary conditions of the form $\partial u/\partialν=u^q$. Here $ν$ denotes the outer unit normal on the boundary of the halfspace ${\mathbb R}^n_+$ and the exponents $p,q>1$ satisfy $p 2$ (or $p<(n+2)/(n-2)$ and $q 2$ and some symmetry of the solutions is assumed). As a typical application of our nonexistence results we provide optimal universal estimates for positive solutions of related problems in bounded and unbounded domains.

math.AP↗