Finite-Dimensionality of Tempered Random Uniform Attractors

Finite-dimensional attractors play an important role in finite-dimensional reduction of PDEs in mathematical modelization and numerical simulations. For non-autonomous random dynamical systems, Cui and Langa (J Differ Equ, 263:1225–1268, 2017) developed a random uniform attractor as a minimal compact random set which provides a certain description of the forward dynamics of the underlying system by forward attraction in probability. In this paper, we study the conditions that ensure a random uniform attractor to have finite fractal dimension. Two main criteria are given, one by a smoothing property and the other by a squeezing property of the system, and neither of the two implies the other. The upper bound of the fractal dimension consists of two parts: the fractal dimension of the symbol space plus a number arising from the smoothing/squeezing property. As an illustrative application, the random uniform attractor of a stochastic reaction–diffusion equation with scalar additive noise is studied, for which the finite-dimensionality in $$L^2$$ L 2 is established by the squeezing approach and that in $$H_0^1$$ H 0 1 by the smoothing framework. In addition, a random absorbing set that absorbs itself after a deterministic period of time is also constructed.

Many closed K-magnetic geodesics on $${\mathbb {S}}^2$$

In this paper we adopt an alternative, analytical approach to Arnol’d problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere $${\mathbb {S}}^2$$ S 2 , where $$K: {\mathbb {S}}^2 \rightarrow {\mathbb {R}}$$ K : S 2 → R is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo’s theorem [21] and Bottkoll results [7].

Large number of bubble solutions for a fractional elliptic equation with almost critical exponents

This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents: \[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \] where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$ .

Construction of solutions for a critical problem with competing potentials via local Pohozaev identities

In this paper, we consider the following critical equation: [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are two nonnegative and bounded functions. Using a finite-dimensional reduction argument and local Pohozaev type of identities, we show that if [Formula: see text], [Formula: see text] has a stable critical point [Formula: see text] with [Formula: see text] and [Formula: see text], then the above equation has infinitely many positive solutions, where [Formula: see text] is the unique positive solution of [Formula: see text] with [Formula: see text]. Combining the results of [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations; S. Peng, C. Wang and S. Yan, Construction of solutions via local Pohozaev identities, J. Funct. Anal. 274 (2018) 2606–2633], it implies that the role of stable critical points of [Formula: see text] in constructing bump solutions is more important than that of [Formula: see text] and that [Formula: see text] can influence the sign of [Formula: see text], i.e. [Formula: see text] can be nonnegative, different from that in [S. Peng, C. Wang and S. Wei, Constructing solutions for the prescribed scalar curvature problem via local Pohozaev identities, to appear in J. Differential Equations]. The concentration points of the solutions locate near the stable critical points of [Formula: see text] which include the case of a saddle point.

Bi-Lipschitz Mané projectors and finite-dimensional reduction for complex Ginzburg–Landau equation

We present a new method of establishing the finite-dimensionality of limit dynamics (in terms of bi-Lipschitz Mané projectors) for semilinear parabolic systems with cross diffusion terms and illustrate it on the model example of three-dimensional complex Ginzburg-Landau equation with periodic boundary conditions. The method combines the so-called spatial-averaging principle invented by Sell and Mallet–Paret with temporal averaging of rapid oscillations which come from cross-diffusion terms.

Multi-peak Positive Solutions of a Nonlinear Schrödinger–Newton Type System

In this paper, we study the following nonlinear Schrödinger–Newton type system:\left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right.where {\epsilon>0} and {Q(x)} is a positive bounded continuous potential on {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point {x_{0}} of {Q(x)} in {\mathbb{R}^{3}}, provided that {\epsilon>0} is sufficiently small.

Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities

We consider the following fractional Schrödinger equation involving critical exponent:\left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u+V(y)u=u^{2^{*}_{s}-1}&&% \displaystyle\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,&&\displaystyle y\in\mathbb{R}^{N},\end{aligned}\right.where {N\geq 3} and {2^{*}_{s}=\frac{2N}{N-2s}} is the critical Sobolev exponent. Under some suitable assumptions of the potential function {V(y)}, by using a finite-dimensional reduction method, combined with various local Pohazaev identities, we prove the existence of infinitely many solutions. Due to the nonlocality of the fractional Laplacian operator, we need to study the corresponding harmonic extension problem.