The notion of relative maximal pattern entropy is investigated in this paper. It turns out that in both topological and measure-theoretical settings, the relative maximal pattern entropy coincides with the supremum of relative sequence entropies over all infinite increasing sequences. Furthermore, leveraging the development of the local theory for relative maximal pattern entropy, we demonstrate that for any topological dynamical system or any ergodic measure-preserving system, the relative maximal pattern entropy is precisely $ \log k $ for some $ k\in\mathbb{N}\cup \{\infty\} $ with $ k $ the superior length of essential relative maximal pattern entropy tuples.

The purpose of this paper is to establish the existence of ground states without restricting ourselves to the space of radial functions for a class of fourth order nonlinear Schrödinger systems of the form:$ \left\{\begin{array}{ll} \Delta^2 u_1 - \alpha_1 \Delta u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{2}u_1 + \beta |u_2|^{2}u_1 \\ \Delta^2 u_2 - \alpha_2 \Delta u_2 + \lambda_2 u_2 = \mu_2 |u_2|^{2}u_2 + \beta |u_1|^{2}u_2 \end{array} \right. \quad \hbox{in} \ \mathbb{R}^N. $For our particular problem, neither the maximum principle nor the Schwarz symmetric rearrangement can be applied, as is usually performed for second order problems. Consequently, in order to arrive at the results obtained here, we remove the radial symmetry by applying the Fourier rearrangement, following a totally different approach from the usual one used for these types of problems. A classification of whether the ground states are semi-trivial or fully-nontrivial is also presented, obtaining a precise analysis depending on the parameters involved in the system. Our results complement those given by Álvarez-Caudevilla, Colorado and Galaktionov in [1] for the case $ \alpha_j = 0 $ $ (j = 1, 2) $.

We prove up to the boundary regularity estimates in Morrey-Lorentz spaces for weak solutions of the linear system of differential forms with regular anisotropic coefficients$ \begin{equation*} d^{\ast} \left( A d\omega \right) + B^{\intercal}d d^{\ast} \left( B\omega \right) = \lambda B\omega + f \text{ in } \Omega, \end{equation*} $with either $ \nu\wedge \omega $ and $ \nu\wedge d^{\ast} \left( B\omega \right) $ or $ \nu\lrcorner B\omega $ and $ \nu\lrcorner \left( A d\omega \right) $ prescribed on $ \partial\Omega. $ We derive these estimates from the $ L^{p} $ estimates obtained in [32] in the spirit of Campanato's method. Unlike Lorentz spaces, Morrey spaces are neither interpolation spaces nor rearrangement invariant. So, Morrey estimates can not be obtained directly from the $ L^{p} $ estimates using interpolation. We instead adapt an idea of Lieberman [19] to our setting to derive the estimates. Applications to Hodge decomposition in Morrey-Lorentz spaces, Gaffney type inequalities, and estimates for related systems such as Hodge-Maxwell systems and 'div-curl' systems are discussed.

In the stationary case, atomistic interaction energies can be proved to $ \Gamma $-converge to classical elasticity models in the simultaneous atomistic-to-continuum and linearization limit [19,41]. The aim of this note is that of extending the convergence analysis to the dynamic setting. Moving within the framework of [41], we prove that solutions of the equation of motion driven by atomistic deformation energies converge to the solutions of the momentum equation for the corresponding continuum energy of linearized elasticity. By recasting the evolution problems in their equivalent energy-dissipation-inertia-principle form, we directly argue at the variational level of evolutionary $ \Gamma $-convergence [33,37]. This in particular ensures the pointwise in time convergence of the energies.

A reaction-diffusion system in a domain with randomly located obstacles was considered. When studying the problem, we sat the homogeneous Dirichlet condition on the outer boundary of the domain and the Neumann condition on the boundary of the cavities. Under such assumptions, it was proven that random trajectory attractors of this system with random coefficients converge in some weak topology to the deterministic trajectory attractor of a homogenized reaction-diffusion system with deterministic coefficients in a homogeneous domain without obstacles. In the case of uniqueness, we obtained weak convergence of random global attractors to a deterministic global attractor.Bibliography: 19 titles.

For any $ x\in (0,1] $, let $ [d_1(x), d_2(x), \ldots] $ be its Lüroth expansion. We prove the exact Hausdorff dimension of the set defined as $ \begin{align*} \left\{x\in(0,1]:c_iA_i^n\leq d_{n+i}(x)<2c_iA_i^n, 0\leq i\leq m-1, {\rm{for\;infinitely\;many\;}}n\in\mathbb{N}\right\}, \end{align*} $ where each partial quotient grows exponentially and the base is given by the parameters $ A_i>1 $, and $ c_i>0 $ are fixed real numbers. This set generalises several known sets and by fixing specific choices of parameters we obtain lower bounds for the Hausdorff dimension for various limsup sets. In particular, we obtain lower bounds for the Hausdorff dimension of the following sets: $ E_m (B) : = \{ x\in(0,1] : \prod\limits_{i = 0}^{m-1}d_{n+i}(x) \ge B^n {\rm{\;infinitely\;many\;}} n\in{\mathbb N} \}, $ and$ \begin{equation*} \mathcal F_{B_1,B_2} = \left\{x\in(0,1]: \begin{split} &d_n(x)d_{n+1}(x) \geq B_1^n {\rm{\;for\;infinitely\;many\;}} n\in{\mathbb N} \\ &d_{n+1}(x) < B_2^n {\rm{\;for\;all\;sufficiently\;large\;}} n\in{\mathbb N} \end{split} \right\}, \end{equation*} $where $ m\in{\mathbb N} $, $ 1\leq B,B_1,B_2<\infty $.

We study strange non-chaotic attractors in a class of quasiperiodically forced monotone interval maps known as pinched skew products. We prove that the probability of positive time-N Lyapunov exponents-with respect to the unique physical measure on the attractor-decays exponentially as N → ∞. The motivation for this work comes from the study of finite-time Lyapunov exponents as possible early-warning signals of critical transitions in the context of forced dynamics. 2020 MSC numbers: 37C60, 37G35 * We say v ∈ T D is totally irrational if there is no non-zero n ∈ Z d with v, n ∈ Z.

This paper belongs to a series of papers devoted to the study of the structure of the non-wandering set of an A-diffeomorphism. We study such set N W (f ) for an Ω-stable diffeomorphism f , given on a closed connected 3-manifold M 3 . Namely, we prove that if all basic sets in N W (f ) are trivial except attractors, then every non-trivial attractor is either one-dimensional non-orientable or two-dimensional expanding.

In this paper, we propose a Hamiltonian regularization of scalar conservation laws, which is parametrized by ℓ > 0 and conserves an H 1 energy. We prove the existence of global weak solutions for this regularization. Furthermore, we demonstrate that as ℓ approaches zero, the unique entropy solution of the original scalar conservation law is recovered, providing justification for the regularization. This regularization belongs to a family of non-diffusive, non-dispersive regularizations that were initially developed for the shallow-water system and extended later to the Euler system. This paper represents a validation of this family of regularizations in the scalar case.

The paper deals with a parametrized $ p $-Kirchhoff type problem involving a criticalnonlinearity on a closed Riemannian manifold $ (M,{{{g}}}) $ of dimension $ n\geq 3 $. Existence and multiplicity of solutions are obtained under suitable conditions by variation methods. Our results extend and enrich the previous work of E. Hebey [Comm. Partial Differential Equations, 41 (2016), 913-924] from the specific case of $ p = 2 $ to the general case of $ 1<p<n $.