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In this paper, we provide both a preservation and breaking of symmetry theorem for $2pi$-periodic problems of the form begin{align*} begin{cases} -u(t) + g(u(t)) = f(t)cr u(0) - u(2pi) = u(0) - u(2pi) = 0 end{cases} end{align*} where $g: mathbb{R} to mathbb{R}$ is a given $C^1$ function and $f: [0,2pi] to mathbb{R}$ is continuous. We provide a preservation of symmetry result that is analogous to one given by Willem (Willem, 1989) and a generalization of the theorem given by Costa-Fang (Costa and Fang, 2019). Both of these theorems use group actions that are not normally considered in the literature.
In this paper we use a unified way studying the decay estimate for a class of dispersive semigroup given by $e^{itphi(sqrt{-Delta})}$, where $phi: mathbb{R}^+to mathbb{R}$ is smooth away from the origin. Especially, the decay estimates for the soluti
The main purpose of this paper is to establish the existence, nonexistence and symmetry of nontrivial solutions to the higher order Brezis-Nirenberg problems associated with the GJMS operators $P_k$ on bounded domains in the hyperbolic space $mathbb{
A class of covariant gauges allowing one to interpolate between the Landau, the maximal Abelian, the linear covariant and the Curci-Ferrari gauges is discussed. Multiplicative renormalizability is proven to all orders by means of algebraic renormaliz
We consider a small SO(2)-equivariant perturbation of a reaction-diffusion system on the sphere, which is equivariant with respect to the group SO(3) of all rigid rotations. We consider a normally hyperbolic SO(3)-group orbit of a rotating wave on th
In these notes we will present (a part of) the parabolic tent spaces theory and then apply it in solving some PDEs originated from the fluid mechanics. In more details, to our most interest are the incompressible homogeneous Navier-Stokes equations.