Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0709.2705

Classification of connecting solutions of semilinear parabolic equations

Michael Robinson
View original source ↗Read source document ↗Export metadata ↓

Abstract

For a given semilinear parabolic equation with polynomial nonlinearity, many solutions blow up in finite time. For a certain large class of these equations, we show that some of the solutions which do not blow up actually tend to equilibria. The characterizing property of such solutions is a finite energy constraint, which comes about from the fact that this class of equations can be written as the $L^2$ gradient of a certain functional.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael Robinson. 2007-09-17. Classification of connecting solutions of semilinear parabolic equations. https://arxiv.org/abs/0709.2705

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On the dynamics of leapfrogging vortex rings

The evolution of highly concentrated vorticity around rings in the three-dimensional axisymmetric Euler equations is studied in a regime for which the leapfrogging dynamics predicted by Helmholtz is expected to occur. We provide in this paper the first result deriving this phenomenon for a general class of initial data in the suitable regime. The singular interaction of rings requires significant improvements of weak and strong localization estimates obtained in prior works. Our method is based on the combination of a new variational argument and a recently introduced double iterative procedure.

math.AP↗
arXiv2026

Lipschitz regularity of fractional $p$-Laplacian

In this article, we investigate the Hölder regularity of the fractional $p$-Laplace equation of the form $(-Δ_p)^s u=f$ where $p>1, s\in (0, 1)$ and $f\in L^\infty_{\rm loc}(Ω)$. Specifically, we prove that $u\in C^{0, γ_\circ}_{\rm loc}(Ω)$ for $γ_\circ=\min\{1, \frac{sp}{p-1}\}$, provided that $\frac{sp}{p-1}\neq 1$. In particular, it shows that $u$ is locally Lipschitz for $\frac{sp}{p-1}>1$. Moreover, we show that for $\frac{sp}{p-1}=1$, the solution is locally Lipschitz, provided that $f$ is locally Hölder continuous. Additionally, we discuss further regularity results for the fractional double-phase problems.

math.AP↗
arXiv2026

Mixed Local-Nonlocal Equations in the Hyperbolic Space and Sharp Sobolev Constants

This paper explores the existence of solutions to a class of nonlinear elliptic equations involving a mixed local-nonlocal operator of the form $-Δ_{\mathbb{B}^N} + (-Δ_{\mathbb{B}^N})^s$, with $0 < s < 1$, set in the hyperbolic space. By employing variational methods, we address both subcritical and critical nonlinearities, establishing the existence of weak solutions under appropriate conditions. We also compare the best Sobolev constants for the mixed and local operators on $\mathbb{B}^N$ with the Euclidean Sobolev constant.

math.AP↗