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Peter Poláčik

Publications and source records attributed to Peter Poláčik.

7 recordsLinked to original sources
arXiv2022

Large-time behavior of solutions of parabolic equations on the real line with convergent initial data III:unstable limit at infinity

This is a continuation, and conclusion, of our study of bounded solutions $u$ of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line whose initial data $u_0=u(\cdot,0)$ have finite limits $θ^\pm$ as $x\to\pm\infty$. We assume that $f$ is a locally Lipschitz function on $\mathbb{R}$ satisfying minor nondegeneracy conditions. Our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In the first two parts of this series we mainly considered the cases where either $θ^-\neq θ^+$; or $θ^\pm=θ_0$ and $f(θ_0)\ne0$; or else $θ^\pm=θ_0$, $f(θ_0)=0$, and $θ_0$ is a stable equilibrium of the equation $\dot ξ=f(ξ)$. In all these cases we proved that the corresponding solution $u$ is quasiconvergent -- if bounded -- which is to say that all limit profiles of $u(\cdot,t)$ as $t\to\infty$ are steady states. The limit profiles, or accumulation points, are taken in $L^\infty_{loc}(\mathbb{R})$. In the present paper, we take on the case that $θ^\pm=θ_0$, $f(θ_0)=0$, and $θ_0$ is an unstable equilibrium of the equation $\dot ξ=f(ξ)$. Our earlier quasiconvergence theorem in this case involved some restrictive technical conditions on the solution, which we now remove. Our sole condition on $u(\cdot,t)$ is that it is nonoscillatory (has only finitely many critical points) at some $t\geq 0$. Since it is known that oscillatory bounded solutions are not always quasiconvergent, our result is nearly optimal.

math.AP↗
arXiv2020

The existence of partially localized periodic-quasiperiodic solutions and related KAM-type results for elliptic equations on the entire space

We consider the equation $Δ_x u+u_{yy}+f(u)=0,\ x=(x_1,\dots,x_N)\in\mathbb{R}^N,\ y\in \mathbb{R},$ where $N\geq 2$ and $f$ is a sufficiently smooth function satisfying $f(0)=0$, $f'(0)<0$, and some natural additional conditions. We prove that the equation possesses uncountably many positive solutions (disregarding translations) which are radially symmetric in $x'=(x_1,\dots,x_{N-1})$ and decaying as $|x'|\to\infty$, periodic in $x_N$, and quasiperiodic in $y$. Related theorems for more general equations are included in our analysis as well. Our method is based on center manifold and KAM-type results.

math.AP↗
arXiv2020

Large-time behavior of solutions of parabolic equations on the real line with convergent initial data II: equal limits at infinity

We continue our study of bounded solutions of the semilinear parabolic equation $u_t=u_{xx}+f(u)$ on the real line, where $f$ is a locally Lipschitz function on $\mathbb{R}.$ Assuming that the initial value $u_0=u(\cdot,0)$ of the solution has finite limits $θ^\pm$ as $x\to\pm\infty$, our goal is to describe the asymptotic behavior of $u(x,t)$ as $t\to\infty$. In a prior work, we showed that if the two limits are distinct, then the solution is quasiconvergent, that is, all its locally uniform limit profiles as $t\to\infty$ are steady states. It is known that this result is not valid in general if the limits are equal: $θ^\pm=θ_0$. In the present paper, we have a closer look at the equal-limits case. Under minor non-degeneracy assumptions on the nonlinearity, we show that the solution is quasiconvergent if either $f(θ_0)\ne0$, or $f(θ_0)=0$ and $θ_0$ is a stable equilibrium of the equation $\dot ξ=f(ξ)$. If $f(θ_0)=0$ and $θ_0$ is an unstable equilibrium of the equation $\dot ξ=f(ξ)$, we also prove some quasiconvergence theorem making (necessarily) additional assumptions on $u_0$. A major ingredient of our proofs of the quasiconvergence theorems---and a result of independent interest---is the classification of entire solutions of a certain type as steady states and heteroclinic connections between two disjoint sets of steady states.

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↗
arXiv2013

Equilibria with a nontrivial nodal set and the dynamics of parabolic equations on symmetric domains

We consider the Dirichlet problem u_t &= Δu + f(x, u, \nabla u)+ h(x, t),& \qquad &(x, t) \in Ω\times (0, \infty), u &= 0, & \qquad &(x, t) \in \partialΩ\times (0, \infty), on a bounded domain $Ω\subset \mathbb{R}^N$. The domain and the nonlinearity $f$ are assumed to be invariant under the reflection about the $x_1$-axis, and the function $h$ accounts for a nonsymmetric decaying perturbation: $h(\cdot, t)\to 0$ as $t\to\infty$. In one of our main theorems, we prove the asymptotic symmetry of each bounded positive solution $u$. The novelty of this result is that the asymptotic symmetry is established even for solutions that are not assumed uniformly positive. In particular, some equilibria of the limit time-autonomous problem (the problem with $h\equiv 0$) with a nontrivial nodal set may occur in the $ω$-limit set of $u$ and this prevents one from applying common techniques based on the method of moving hyperplanes. The goal of our second main theorem is to classify the positive entire solutions of the time-autonomous problem. We prove that if $U$ is a positive entire solution, then one of the following applies: (i) for each $t\in \mathbb{R}$, $U(\cdot,t)$ is even in $x_1$ and decreasing in $x_1>0$, (ii) $U$ is an equilibrium, (iii) $U$ is a connecting orbit from an equilibrium with a nontrivial nodal set to a set consisting of functions which are even in $x_1$ and decreasing in $x_1>0$, (iv) is a heteroclinic connecting orbit between two equilibria with a nontrivial nodal set.

math.AP↗