### A geometric approach to invariant sets for dynamical systems.

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Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized...

We consider the semilinear Lane–Emden problem $$ where $p>1$ and $\Omega $ is a smooth bounded domain of ${\mathbb{R}}^{2}$. The aim of the paper is to analyze the asymptotic behavior of sign changing solutions of ${(}_{p})$, as $p\to +\infty $. Among other results we show, under some symmetry assumptions on $\Omega $, that the positive and negative parts of a family of symmetric solutions concentrate at the same point, as $p\to +\infty $, and the limit profile looks like a tower of two bubbles given by a superposition of a regular and a singular solution of the Liouville...

A linear, uniformly stratified ocean model is used to investigate propagation of baroclinic Kelvin waves in a cylindrical basin. It is found that smaller wave amplitudes are inherent to higher mode individual terms of the obtained solutions that are also evanescent away of a costal line toward the center of the circular basin. It is also shown that the individual terms if the obtained solutions can be visualized as spinning patterns in rotating stratified fluid confined in a circular basin. Moreover,...

By a sub-super solution argument, we study the existence of positive solutions for the system ⎧$-{\Delta}_{p}u=a\u2081\left(x\right)F\u2081(x,u,v)$ in Ω, ⎪$-{\Delta}_{q}v=a\u2082\left(x\right)F\u2082(x,u,v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in ${\mathbb{R}}^{N}$ with smooth boundary or $\Omega ={\mathbb{R}}^{N}$. A nonexistence result is obtained for radially symmetric solutions.

We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations $\Delta u-u+f\left(u\right)=0$ in ${\mathbb{R}}^{N}$, $u\in {H}^{1}\left({\mathbb{R}}^{N}\right)$, where $N\ge 2$. Under natural conditions on the nonlinearity $f$, we prove the existence of $\mathrm{\mathit{i}\mathit{n}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{e}\mathit{l}\mathit{y}\mathit{m}\mathit{a}\mathit{n}\mathit{y}\mathit{n}\mathit{o}\mathit{n}\mathit{r}\mathit{a}\mathit{d}\mathit{i}\mathit{a}\mathit{l}\mathit{s}\mathit{o}\mathit{l}\mathit{u}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{s}}$ in any dimension $N\ge 2$. Our result complements earlier works of Bartsch and Willem $(N=4\mathrm{\U0001d698\U0001d69b}N\ge 6)$ and Lorca-Ubilla $(N=5)$ where solutions invariant under the action of $O\left(2\right)\times O(N-2)$ are constructed. In contrast, the solutions we construct are invariant under the action of ${D}_{k}\times O(N-2)$ where ${D}_{k}\subset O\left(2\right)$ denotes the dihedral group...

We study rolling maps of the Euclidean ellipsoid, rolling upon its affine tangent space at a point. Driven by the geometry of rolling maps, we find a simple formula for the angular velocity of the rolling ellipsoid along any piecewise smooth curve in terms of the Gauss map. This result is then generalised to rolling any smooth hyper-surface. On the way, we derive a formula for the Gaussian curvature of an ellipsoid which has an elementary proof and has been previously known only for dimension two....

In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

New Q-conditional symmetries for a class of reaction-diffusion-convection equations with exponential diffusivities are derived. It is shown that the known results for reaction-diffusion equations with exponential diffusivities follow as particular cases from those obtained here but not vice versa. The symmetries obtained are applied to construct exact solutions of the relevant nonlinear equations. An application of exact solutions to solving a boundary-value problem with constant Dirichlet conditions...

In the paper we present an identity of the Picone type for a class of nonlinear differential operators of the second order involving an arbitrary norm $H$ in ${\mathbb{R}}^{n}$ which is continuously differentiable for $x\ne 0$ and such that ${H}^{p}$ is strictly convex for some $p>1$. Two important special cases are the $p$-Laplacian and the so-called pseudo $p$-Laplacian. The identity is then used to establish a variety of comparison results concerning nonlinear degenerate elliptic equations which involve such operators. We also get criteria...

We study the existence of positive solutions for the $p$-Laplace Emden-Fowler equation. Let $H$ and $G$ be closed subgroups of the orthogonal group $O\left(N\right)$ such that $HG\subset O\left(N\right)$. We denote the orbit of $G$ through $x\in {\mathbb{R}}^{N}$ by $G\left(x\right)$, i.e., $G\left(x\right):=\{gx:g\in G\}$. We prove that if $H\left(x\right)G\left(x\right)$ for all $x\in \overline{\Omega}$ and the first eigenvalue of the $p$-Laplacian is large enough, then no $H$ invariant least energy solution is $G$ invariant. Here an $H$ invariant least energy solution means a solution which achieves the minimum of the Rayleigh quotient among all $H$ invariant functions. Therefore...

This article recalls the results given by A. Dutrifoy, A. Majda and S. Schochet in [1] in which they prove an uniform estimate of the system as well as the convergence to a global solution of the long wave equations as the Froud number tends to zero. Then, we will prove the convergence with weaker hypothesis and show that the life span of the solutions tends to infinity as the Froud number tends to zero.

We classify nonconstant entire local minimizers of the standard Ginzburg–Landau functional for maps in ${H}_{\text{loc}}^{1}({\mathbb{R}}^{3};{\mathbb{R}}^{3})$ satisfying a natural energy bound. Up to translations and rotations,such solutions of the Ginzburg–Landau system are given by an explicit solution equivariant under the action of the orthogonal group.

We consider the functional $${\mathcal{I}}_{\Omega}\left(v\right)={\int}_{\Omega}\left[f\right(\left|Dv\right|)-v]dx,$$ where $\Omega $ is a bounded domain and $f$ is a convex function. Under general assumptions on $f$, Crasta [Cr1] has shown that if ${\mathcal{I}}_{\Omega}$ admits a minimizer in ${W}_{0}^{1,1}\left(\Omega \right)$ depending only on the distance from the boundary of $\Omega $, then $\Omega $ must be a ball. With some restrictions on $f$, we prove that spherical symmetry can be obtained only by assuming that the minimizer has one level surface parallel to the boundary (i.e. it has only a level surface in common with the distance). We then discuss how these...