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This paper has two primary objectives. The first one is to demonstrate that the solutions of master equation (∂ t - Δ) s u (x , t) = f (u (x , t)) , (x , t) ∈ B 1 (0) × R , subject to the vanishing exterior condition, are radially symmetric and strictly decreasing with respect to the origin in B 1 (0) for any t ∈ R . Another one is to establish the Liouville theorem for homogeneous master equation (∂ t - Δ) s u (x , t) = 0 , in R n × R , which states that all bounded solutions must be constant. We propose a new methodology for a direct method of moving planes applicable to the fully fractional heat operator (∂ t - Δ) s , and the proof of our main results based on this direct method involves the perturbation technique, limit argument as well as Fourier transform. This study opens up a way to investigate the geometric behavior of master equations, and provides valuable insights for establishing qualitative properties of solutions and even for deriving important Liouville theorems for other types of fractional order parabolic equations. [ABSTRACT FROM AUTHOR]

In this paper, combining Nash-Moser iteration and Saloff-Coste type Sobolev inequalities, we establish fundamental estimates for solutions to a class of nonlinear elliptic equations of the form Δ u (x) + a (x) u (x) ln ⁡ u (x) + b (x) u (x) = 0 , which possesses abundant geometric backgrounds. Utilizing these estimates which retrieve more geometric information, we obtain some further properties of such solutions. Especially, we prove a local Liouville type theorem of corresponding constant coefficient equation. [ABSTRACT FROM AUTHOR]

We investigate the validity of the Liouville property for a class of elliptic equations with a potential, posed on infinite graphs. Under suitable assumptions on the graph and on the potential, we prove that the unique bounded solution is u ≡ 0 . We also show that on a special class of graphs the condition on the potential is optimal, in the sense that if it fails, then there exist infinitely many bounded solutions. [ABSTRACT FROM AUTHOR]

In this paper, we provide a comprehensive overview of the recent developments in the field of fractional parabolic equations, with a primary focus on the underlying ideas and techniques. These results have appeared in a series of previous literatures, mainly in [30] [92] [29] [32] [97]. However, in these articles, the ideas were usually submerged in detailed calculations. What we are trying to do here is to highlight these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and hence will be able to apply them to solve a variety of other problems in this area.Actually, many techniques and estimates in fractional elliptic equations can be modified and then be applied to investigate the corresponding parabolic fractional problems. We will clearly explain how to make such transitions in relevant sections.If you find it challenging to grasp the main ideas and essences of [30], [92], [29], [32], and [97], we suggest using this paper as a primer. Beginning with this work will provide you with a solid conceptual framework, making it much easier to navigate and comprehend the additional details in the aforementioned articles. [ABSTRACT FROM AUTHOR]

Some theorems of Liouville type are given for such P-harmonic maps when target manifold have conformal vector field or convex function or have non-positive sectional curvature. [ABSTRACT FROM AUTHOR]

A theorem of Liouville asserts that the simplest angle-preserving transformations on Euclidean space—translations, dilations, reflections, and inversions—generate all angle-preserving transformations when the dimension is at least 3. This note gives a proof which uses only elementary multivariable calculus and simplifies a differential-geometric argument of Flanders. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of k-Hessian equations, we can deduce an Obata-type formula and a Liouville-type theorem by integration by parts. [ABSTRACT FROM AUTHOR]

Let \mathcal {E} be an asymptotically Euclidean end in an otherwise arbitrary connected Riemannian spin manifold (M,g). We show that if \mathcal {E} has negative ADM-mass, then there exists a constant R > 0, depending only on \mathcal {E}, such that M must become incomplete or have a point of negative scalar curvature in the R-neighborhood around \mathcal {E} in M. This gives a quantitative answer to Schoen and Yau's question on the positive mass theorem with arbitrary ends for spin manifolds. Similar results have recently been obtained by Lesourd, Unger and Yau [ Positive scalar curvature on noncompact manifolds and the liouville theorem , 2020; The positive mass theorem with arbitrary ends , 2021] without the spin condition in dimensions \leq 7 assuming Schwarzschild asymptotics on the end \mathcal {E}. We also derive explicit quantitative distance estimates in case the scalar curvature is uniformly positive in some region of the chosen end \mathcal {E}. Here we obtain refined constants reminiscent of Gromov's metric inequalities with scalar curvature. [ABSTRACT FROM AUTHOR]

In this paper, we studied an integral equation of the Allen-Cahn type$ \begin{align*} u(x) = \overrightarrow{l} +C_{*}\int_{{\mathbb{R}}^n}\frac{u(y)(1-|u(y)|^2)|1-|u(y)|^2|^{p-2}} {|x-y|^{n-\alpha}}dy. \end{align*} $Here, $ u:{\mathbb{R}}^n\rightarrow {\mathbb{R}}^k $, $ k \geq 1 $, $ n \geq 2 $, $ \alpha \in (0,n) $ and $ p-1>n/(n-\alpha) $. In addition, $ \overrightarrow{l}\in {\mathbb{R}}^k $ is a constant vector and $ C_{*} \neq 0 $ is a real constant. This equation is associated with the following fractional-order equation$ (-\Delta)^{\frac{\alpha}{2}}u = u(1-|u|^2)|1-|u|^2|^{p-2}. $Both of them play important roles in the study of phase transition in fractal media and the competition model of biological communities. We were concerned about two properties of those equations: One is the Lipschitz regularity of the bounded solutions; the other is the Liouville theorem. The regularity lifting lemma introduced by Ma-Chen-Li and the Pohozaev identity in integral form come into play here. [ABSTRACT FROM AUTHOR]

We introduce a distributional Jacobian determinant det D ⁢ V β ⁢ (D ⁢ v) in dimension two for the nonlinear complex gradient V β ⁢ (D ⁢ v) = | D ⁢ v | β ⁢ (v x 1 , − v x 2 ) for any β > − 1 , whenever v ∈ W loc 1 , 2 and β ⁢ | D ⁢ v | 1 + β ∈ W loc 1 , 2 . This is new when β ≠ 0 . Given any planar ∞-harmonic function 푢, we show that such distributional Jacobian determinant det D ⁢ V β ⁢ (D ⁢ u) is a nonnegative Radon measure with some quantitative local lower and upper bounds. We also give the following two applications. Applying this result with β = 0 , we develop an approach to build up a Liouville theorem, which improves that of Savin. Precisely, if 푢 is an ∞-harmonic function in the whole R 2 with lim inf R → ∞ inf c ∈ R 1 R ⁢ ⨍ B ⁢ (0 , R) | u ⁢ (x) − c | ⁢ d x < ∞ , then u = b + a ⋅ x for some b ∈ R and a ∈ R 2 . Denoting by u p the 푝-harmonic function having the same nonconstant boundary condition as 푢, we show that det D ⁢ V β ⁢ (D ⁢ u p) → det D ⁢ V β ⁢ (D ⁢ u) as p → ∞ in the weak-⋆ sense in the space of Radon measure. Recall that V β ⁢ (D ⁢ u p) is always quasiregular mappings, but V β ⁢ (D ⁢ u) is not in general. [ABSTRACT FROM AUTHOR]

We investigate here the nonlinear elliptic Hénon-type equation Δ²u =|x|a|u|p-1u in ℝn+ u=∂u/∂xn =0 in ∂ ℝn+, where p > 1, a ≥ 0 and n ≥ 5. Based on the approach of Hu [J. Differential Equations 256 (2014), 1817–1846], we prove Liouville-type theorems for stable solutions and solutions which are stable outside a compact set possibly unbounded and sign-changing. In contrast with the results of Hu (2014), we apply a new method to provide an implicit existence of the fourth-order Joseph– Lundgren exponent. To classify finite Morse index solutions in the supercritical case, we adopt a new method of monotonicity formula together with blowing down sequence. In addition, a difficulty stems from the fact that applying the doubling lemma leads to the singularity. For this reason, we use a more delicate approach to the interval (n + 4 + 2a; pJL2 (n, 0). Our analysis uses a combination of some integral estimates, Pohozaev-type identity, and monotonicity formula of solutions [ABSTRACT FROM AUTHOR]

This paper introduces a modified version of the Variational Iteration Method, incorporating P-transformation. We propose a novel semi-analytical technique named the modified variational iteration method for addressing fractional difierential equations featuring tempered Liouville- Caputo derivatives. The modified variational iteration method emerges as a highly eficient and powerful mathematical tool, ofiering exact or approximate solutions for a diverse range of real-world problems in en- gineering and the natural sciences, specifically those expressed through difierential equations. To assess its efiectiveness and accuracy, we scru- tinize the modified variational iteration method by applying it to three problems related to the heat-like multidimensional difiusion equation with a fractional time derivative in a tempered Liouville-Caputo form [ABSTRACT FROM AUTHOR]

The classical Liouville property says that all bounded harmonic functions in Rn$\mathbb {R}^n$, that is, all bounded functions satisfying Δf=0$\Delta f = 0$, are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator m(D)$m(D)$, such that the solutions f$f$ to m(D)f=0$m(D)f=0$ are Lebesgue a.e. constant (if f$f$ is bounded) or coincide Lebesgue a.e. with a polynomial (if f$f$ is polynomially bounded). The class of Fourier multipliers includes the (in general non‐local) generators of Lévy processes. For generators of Lévy processes, we obtain necessary and sufficient conditions for a strong Liouville theorem where f$f$ is positive and grows at most exponentially fast. As an application of our results above, we prove a coupling result for space‐time Lévy processes. [ABSTRACT FROM AUTHOR]

This paper mainly deals with the monotonicity and Liouville type theorems of nonnegative solutions for a class of fully nonlinear fractional equations in half-space. Compared with serial works in [9,14,20], the restrictions on the monotonicity of nonlinear terms and the decay assumption on solutions are not necessary here. Moreover, the monotonicity result of nonnegative bounded solutions for corresponding cooperative systems is also derived. Specifically, we establish some new types of maximum principles for fully nonlinear fractional equations and systems in unbounded domains, which will play crucial roles in carrying on the method of moving planes. [ABSTRACT FROM AUTHOR]

In this paper, we prove new Liouville-type properties of fully nonlinear subelliptic equations modelled on the Heisenberg group. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a Liouville theorem of F-harmonic maps between complete Riemannian manifolds with moderate divergent F-energy. We assume that F is a concave function and satisfies a differential inequality. We employ Ara's F-stress-energy tensor and the Hessian comparison theorem to prove the main result. [ABSTRACT FROM AUTHOR]

We study anti-symmetric solutions about the hyperplane $\{x_n=0\}$ for the following fractional Hardy–Hénon system: \[ \left\{\begin{array}{@{}ll} (-\Delta)^{s_1}u(x)=|x|^\alpha v^p(x), & x\in\mathbb{R}_+^n, \\ (-\Delta)^{s_2}v(x)=|x|^\beta u^q(x), & x\in\mathbb{R}_+^n, \\ u(x)\geq 0, & v(x)\geq 0,\ x\in\mathbb{R}_+^n, \end{array}\right. \] where $0 , $n>2\max \{s_1,s_2\}$. Nonexistence of anti-symmetric solutions are obtained in some appropriate domains of $(p,q)$ under some corresponding assumptions of $\alpha,\beta$ via the methods of moving spheres and moving planes. Particularly, for the case $s_1=s_2$ , one of our results shows that one domain of $(p,q)$ , where nonexistence of anti-symmetric solutions with appropriate decay conditions at infinity hold true, locates at above the fractional Sobolev's hyperbola under appropriate condition of $\alpha, \beta$. [ABSTRACT FROM AUTHOR]

We investigate the conditions for the existence and uniqueness of solutions in a nonlinear system of sequential fractional differential equations using the Liouville–Caputo type with varying orders. This system is enriched by nonlocal coupled integral boundary conditions. The desired outcomes are attained by employing traditional fixed-point theorems. It is essential to emphasize that the fixed-point approach proves to be an effective method for establishing the existence of solutions in boundary value problems. Furthermore, we provide constructed examples to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

This article studies the pullback dynamics and statistical solutions for the dissipative non-autonomous Zakharov equations in a bounded interval. The main obstacle in our investigating comes from the geometric constrain of global regularity of solutions for the Zakharov equations. This geometric constrain is caused essentially by its particular structure which is a special coupling of a hyperbolic equation with a dispersive nonlinear Schrödinger equation. The nonlinear term Δ | E | 2 in the hyperbolic equation is the core ingredient leading to this geometric constrain. Firstly we show the global existence and uniqueness of solutions, and prove the continuous dependence of solutions on the initial data by exploiting a certain coercivity of the operator corresponding to the linear principle part of the Zakharov equations. Then we establish that the generated process of solution mappings possesses a bounded pullback absorbing set and pullback asymptotic compactness by using a delicate decomposition of the original problem, and obtain the existence of a pullback attractor. Next we formulate an appropriate definition of τ -continuity for the process and employ the calculus of variations to prove this τ -continuity. Afterwards, we construct a family of invariant Borel probability measures for the process via the pullback attractor and the notion of generalized Banach limit. We also propose an appropriate definition of statistical solutions for the addressed Zakharov equations, and establish that the constructed family of invariant Borel probability measures is indeed a statistical solution which satisfies the Liouville theorem of Statistical Mechanics. Our definitions and approach present here can overcome effectively the difficulty caused by the geometric constrain of the Zakharov equations. Finally, we propose some open problems related to the existence and singular limiting behavior of the pullback attractors and statistical solutions for the Zakharov equations with varying coefficient. [ABSTRACT FROM AUTHOR]

In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate: any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension. [ABSTRACT FROM AUTHOR]

In this paper, we first study the dual fractional parabolic equation ∂ t α u (x , t) + (− Δ) s u (x , t) = f (u (x , t)) in  B 1 (0) × ℝ , subjected to the vanishing exterior condition. We show that for each t ∈ ℝ , the positive bounded solution u (⋅ , t) must be radially symmetric and strictly decreasing about the origin in the unit ball in ℝ n . To overcome the challenges caused by the dual nonlocality of the operator ∂ t α + (− Δ) s , some novel techniques were introduced. Then we establish the Liouville theorem for the homogeneous equation in the whole space ∂ t α u (x , t) + (− Δ) s u (x , t) = 0 in  ℝ n × ℝ. We first prove a maximum principle in unbounded domains for antisymmetric functions to deduce that u (x , t) must be constant with respect to x. Then it suffices for us to establish the Liouville theorem for the Marchaud fractional equation ∂ t α u (t) = 0 in  ℝ. To circumvent the difficulties arising from the nonlocal and one-sided nature of the operator ∂ t α , we bring in some new ideas and simpler approaches. Instead of disturbing the antisymmetric function, we employ a perturbation technique directly on the solution u (t) itself. This method provides a more concise and intuitive way to establish the Liouville theorem for one-sided operators ∂ t α , including even more general Marchaud time derivatives. [ABSTRACT FROM AUTHOR]

Non-thermal advanced fuel fusion trades the requirement of a large amount of recirculating tritium in the system for that of large recirculating power. Phase space engineering technologies utilizing externally injected electromagnetic fields can be applied to meet the challenge of maintaining non-thermal particle distributions at a reasonable cost. The physical processes of the phase space engineering are studied from a theoretical and algorithmic perspective. It is emphasized that the operational space of phase space engineering is limited by the underpinning symplectic dynamics of charged particles. The phase space incompressibility according to the Liouville theorem is just one of many constraints, and Gromov's non-squeezing theorem determines the minimum footprint of the charged particles on every conjugate phase space plane. In this sense and level of sophistication, the mathematical abstraction of phase space engineering is symplectic topology. To simulate the processes of phase space engineering, such as the Maxwell demon and electromagnetic energy extraction, and to accurately calculate the minimum footprints of charged particles, recently developed structure-preserving geometric algorithms can be used. The family of algorithms conserves exactly, on discretized spacetime, symplecticity and thus incompressibility, non-squeezability, and symplectic capacities. The algorithms apply to the dynamics of charged particles under the influence of external electromagnetic fields as well as the charged particle–electromagnetic field system governed by the Vlasov–Maxwell equations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of T-conformal transformations and T-conformal maps between Riemannian manifolds. Here, T stands for a smooth (1,1)-tensor field defined on the domain of these maps. We start by defining what it means for a map to be T-conformal and also dwell on some basic properties of such type maps. We next specialize our discussion to the situation when the map T satisfies the condition △T = 0. Accordingly, we prove Liouville's theorem for T-conformal maps between space forms Rn(c) as an application under the condition △T = 0. The proof relies upon properties of T-conformal maps proved earlier. Broadly, the paper seeks to provide a general understanding of conformal mappings in the presence of a tensor field T and show how classical results such as Liouville's theorem apply. [ABSTRACT FROM AUTHOR]

We develop a Hamiltonian theory for the nonlinear resonant interactions between energetic particles and nonlinear frequency chirping waves in the weakly inhomogeneous magnetic field. A canonical transformation is constructed to separate the fast and slowly varying scales, and the Hamiltonian of the resonant particle is transformed to the local resonance reference frames. The Vlasov equation of the local distribution function moving at the local resonance velocity is obtained using Liouville's theorem. The evolution for the slowly varying wave envelope is derived from the Ampère's law with both cold plasma and energetic particle currents. The Vlasov equation coupled with the wave envelope equation self-consistently describes the dynamics of the deeply trapped resonant particles and the slowly varying coherent wave envelope. The application of the theory to the frequency chirping chorus wave in magnetospheric plasmas is also discussed. [ABSTRACT FROM AUTHOR]

We study the equation m(D)f = 0 in a large class of sub-exponentially growing functions. Under appropriate restrictions on m ∈ C(ℝn), we show that every such solution can be analytically continued to a sub-exponentially growing entire function on ℂn if, and only if, m(ξ) ≠ 0 for ξ ≠ 0. [ABSTRACT FROM AUTHOR]

In this note, we prove interior a priori first‐ and second‐order estimates for solutions of fully nonlinear degenerate elliptic inequalities structured over the vector fields of Carnot groups, under the main assumption that u$u$ is semiconvex along the fields. These estimates for supersolutions are new even for linear subelliptic inequalities in nondivergence form, whereas in the nonlinear setting they do not require neither convexity nor concavity on the second derivatives. We complement the analysis exhibiting an explicit example showing that horizontal W2,q$W^{2,q}$ regularity of Calderón–Zygmund type for fully nonlinear subelliptic equations posed on the Heisenberg group cannot be in general expected in the range q

In this paper, we study the fractional elliptic system with coupled terms {(-Δ)su = (q+1)uqvp+1 in ℝN (-Δ)su = (q+1)vqup+1 in ℝN 0 < s < 1 and N > 2s. We first prove that if p > -1, q > -1 and p+q+1 ≤ N/N-2s, then the system has no positive supersolution. In the case p,q > 0 we establish the nonexistence result of stable positive solutions. Our results generalize some results in [Li, Yayun; Lei, Yutian; Commun. Pure Appl. Anal. 17 (2018), no. 5, 1749-1764.] to the system involving the fractional Laplacian. [ABSTRACT FROM AUTHOR]

In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: (− Δ) α / 2 u (x) = f (u (x) , v (x)) , x ∈ R n , (− Δ) α / 2 v (x) = g (u (x) , v (x)) , x ∈ R n. Under nature structure conditions on f and g, we classify the positive solutions for the semi-linear elliptic system involving the fractional Laplacian by using the direct method of the moving spheres introducing by W. Chen, Y. Li, and R. Zhang ("A direct method of moving spheres on fractional order equations," J. Funct. Anal., vol. 272, pp. 4131–4157, 2017). In the half space, we establish a Liouville type theorem without any assumption of integrability by combining the direct method of moving planes and moving spheres, which improves the result proved by W. Dai, Z. Liu, and G. Lu ("Liouville type theorems for PDE and IE systems involving fractional Laplacian on a half space," Potential Anal., vol. 46, pp. 569–588, 2017). [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space R + 2 : (− Δ) α 2 u (x) = | x | a u p 1 (x) e q 1 v (x) , x ∈ R + 2 , (− Δ) v (x) = | x | b u p 2 (x) e q 2 v (x) , x ∈ R + 2 , with Dirichlet boundary conditions, where 0 < α < 2 and p1, p2, q1, q2 > 0. First, we derived the integral representation formula corresponding to the above system under the assumption p 1 ≥ − 2 a α − 1 . Then, we prove Liouville theorem for solutions to the above system via the method of scaling spheres. [ABSTRACT FROM AUTHOR]

We prove a Liouville-type theorem for semilinear parabolic systems of the form ∂ t u i − Δ u i = ∑ j = 1 m β i j u i r u j p − r , i = 1 , 2 ,... , m in the whole space R N × R , where r > 0 , p > r + 1 and (β i j) is a real (m , m) symmetric matrix with nonnegative entries, positive on the diagonal. Due to the lack of gradient structure, a nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron is introduced. We expect to see more applications of this method to other problems with no gradient structure. [ABSTRACT FROM AUTHOR]

We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold (M , g) Δ u + a u (ln u) p + b u ln u = 0 , where a ≠ 0 , b are two constants and p = k 1 2 k 2 + 1 ≥ 2 , here k 1 and k 2 are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution. [ABSTRACT FROM AUTHOR]

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group, in this case, the irreducible representation spaces of homogeneous harmonic polynomials. In this paper, we study boundary value problems involving bosonic Laplacians in the upper-half space and the unit ball. Poisson kernels in the upper-half space and the unit ball are constructed, which give us solutions to the Dirichlet problems with L p boundary data, 1 ≤ p ≤ ∞ . We also prove the uniqueness for solutions to the Dirichlet problems with continuous data for bosonic Laplacians and provide analogs of some properties of harmonic functions for null solutions of bosonic Laplacians, for instance, Cauchy's estimates, the mean-value property, Liouville's Theorem, etc. [ABSTRACT FROM AUTHOR]

In this paper, we study the following fractional Choquard equation with weight (−Δ)su=1|x|N−α∗h(x)|u|ph(x)|u|p−2uinℝN,$$ {\left(-\Delta \right)}&#x0005E;su&#x0003D;\left(\frac{1}{{\left&#x0007C;x\right&#x0007C;}&#x0005E;{N-\alpha }}\ast h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;p\right)h(x){\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u\kern0.5em \mathrm{in}\kern0.5em {\mathrm{\mathbb{R}}}&#x0005E;N, $$where 02s,p>2,α>0$$ 02s,p>2,\alpha >0 $$ and h$$ h $$ is a positive weight function satisfying h(x)≥C|x|a$$ h(x)\ge C{\left&#x0007C;x\right&#x0007C;}&#x0005E;a $$ at infinity, for some a≥0$$ a\ge 0 $$. We establish, in this paper, a Liouville type theorem saying that if maxN−4s−2a,0<α

In this paper, we are concerned with the existence of sign-changing radial solutions with any prescribed numbers of zeros to the following Schrodinger equation with the critical exponential growth: \begin{equation*} \begin {cases} -\Delta u +u=\lambda ue^{u^2} \quad \quad \text {in } \quad \mathbb {R}^2,\\ \displaystyle \lim _{|x|\to \infty }u(x)=0, \end{cases} \end{equation*} where 0<\lambda <1. Our proof relies on the shooting method, the Sturm's comparison theorem and a Liouville type theorem in exterior domain of \mathbb {R}^2. It seems to be the first existence result of sign-changing solution for Schrodinger equation with the critical exponential growth. [ABSTRACT FROM AUTHOR]

We consider a nonautonomous stochastic $ p $-Laplacian lattice equation with multiplicative noise and a nonlinearity that is not locally Lipschitz. For each $ q\geq1 $, a pullback $ (\ell^2, \ell^q) $-attractor is obtained, and the measurability of the pullback attractor in both spaces by more complicated estimates. Then, the time-dependent invariant sample Borel probability measures are constructed and carried by the pullback random attractor. Moreover, the invariant sample measures satisfy a stochastic Liouville type theorem. [ABSTRACT FROM AUTHOR]

The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic partial differential equations with measurable coefficients and the Liouville property at infinity. Initially, we address the two‐dimensional case by proving an Alt–Caffarelli–Friedman‐type monotonicity formula, enabling the proof of optimal regularity and the Liouville property for multiphase problems. In higher dimensions, we delve into the role of monotonicity formulas in characterizing optimal regularity. By employing a hole‐filling technique, we present a distinct "almost‐monotonicity" formula that implies Hölder regularity of solutions. Finally, we explore the interplay between the least growth at infinity and the exponent of regularity by combining blow‐up and G$G$‐convergence arguments. [ABSTRACT FROM AUTHOR]

The optimization of separator masses in this paper is based on the realization of the linearity of the system under study and the consequences of Liouville's theorem. The properties of several mass separators with an ion energy of 30 keV and a beam emittance of 4 mm mrad are considered. Focusing is provided by an aberration-free lens and a magnetic corrector. Phase diagrams along the beam path are in the form of parallelograms, which indicates the absence of geometric aberrations. For each of the separators, the resolution calculated in the linear approximation coincides with the simulation results. It is shown that a mass separator based on a magnet with a rotation angle of 54.7° has a resolution of about 5000, and a separator based on two magnets with a rotation angle of 45° and 90°, respectively, has a resolution of 14 000–15 000. [ABSTRACT FROM AUTHOR]

This paper studies a priori and regularity estimates of Evans-Krylov type in Hölder spaces for fully nonlinear uniformly elliptic and parabolic equations of second order when the operator fails to be concave or convex in the space of symmetric matrices. In particular, it is assumed that either the level sets are convex or the operator is concave, convex or close to a linear function near infinity. As a byproduct, these results imply polynomial Liouville theorems for entire solutions of elliptic equations and for ancient solutions to parabolic problems. [ABSTRACT FROM AUTHOR]

Orbital invariants of integrable billiards on two-dimensional book tables are studied at constant energy values. These invariants are calculated from rotation functions defined on one-parameter families of Liouville 2-tori. For two-dimensional billiard books, a complete analogue of Liouville's theorem is proved, action–angle variables are introduced, and rotation functions are defined. A general formula for the rotation functions of such systems is obtained. For a number of examples, the monotonicity of these functions is studied, and edge orbital invariants (rotation vectors) are calculated. It turned out that not all billiards have monotonic rotation functions, as was originally assumed by A. Fomenko's hypothesis. However, for some series of billiards, this hypothesis is true. [ABSTRACT FROM AUTHOR]

Liouville theorems for scaling invariant nonlinear parabolic problems in the whole space and/or the halfspace (saying that the problem does not posses positive bounded solutions defined for all times t ∈ (- ∞ , ∞) ) guarantee optimal estimates of solutions of related initial-boundary value problems in general domains. We prove an optimal Liouville theorem for the linear equation in the halfspace complemented by the nonlinear boundary condition ∂ u / ∂ ν = u q , q > 1 . [ABSTRACT FROM AUTHOR]

In this paper we prove a Liouville theorem for the Chern–Simons–Schrödinger equation. This result is consistent with the soliton resolution conjecture for initial data that does not lie in a weighted space. See [10] for the soliton resolution result in a weighted space. [ABSTRACT FROM AUTHOR]

We study the generalized higher order Choquard-Pekar equation (⁎) (− Δ) m u = (∫ R N Ψ (| x − y |) u p (y) d y) u q in R N , where N , m ≥ 1 are integers, p , q > 0. Under suitable assumption on Ψ, when N > 2 m the Liouville theorems in the ranges (i) (p ≥ 1 or p + q ≥ 2) and ∫ | y | > 1 | y | − p (N − 2 m) Ψ (| y |) d y = ∞ ; or (ii) p + q ≥ 2 and lim sup r → ∞ r 2 N − (N − 2 m) (p + q) Ψ (r) > 0 for nonnegative classical solutions of (⁎) were got in previous studies. By using "Bootstrap" iteration procedure, we extend the above results by establishing the Liouville theorems for nonnegative classical solutions of (⁎) with N > 2 m in the ranges p ≥ 1 and p + q < N N − 2 m. Moreover, we also consider the higher order Choquard-Pekar equation, i.e., the problem (⁎) with Ψ (r) = r − σ and 0 < σ < N. By using the method of scaling sphere we also establish the Liouville theorems for nonnegative classical solutions with N > 2 m in the ranges 1 ≤ p ≤ 2 N − σ N − 2 m , 0 < q ≤ N + 2 m − σ N − 2 m , and p + q < 3 N + 2 m − 2 σ N − 2 m , which also extend the recent studies on higher order Choquard-Pekar equation. Extensions to the integral representation of (⁎) with more general assumption p > 0 are also included. [ABSTRACT FROM AUTHOR]

We develop a functional Stein-Malliavin method in a non-diffusive Poissonian setting, thus obtaining a) quantitative central limit theorems for approximation of arbitrary non-degenerate Gaussian random elements taking values in a separable Hilbert space and b) fourth moment bounds for approximating sequences with finite chaos expansion. Our results rely on an infinite-dimensional version of Stein's method of exchangeable pairs combined with the so-called Gamma calculus. Two applications are included: Brownian approximation of Poisson processes in Besov-Liouville spaces and a functional limit theorem for an edge-counting statistic of a random geometric graph. [ABSTRACT FROM AUTHOR]

In this paper, we present a unified method for deriving differential Harnack inequalities for positive solutions of the semilinear parabolic equation ∂ t u = Δ V u + H (u) on complete Riemannian manifolds with Bakry-Émery curvature bounded below. This method transforms the problem of deriving differential Harnack inequalities into solving a related ODE system. As its application, on the one hand, we obtain new and improved estimates for logarithmic type equation and Yamabe type equation. On the other hand, we establish some sharp estimates for them under non-negative Bakry-Émery curvature condition. As their natural corollary, we also obtain some sharp Harnack inequalities and Liouville type theorems. [ABSTRACT FROM AUTHOR]

We survey some recent results on minimal hypersurfaces in R n + 1 with cylindrical tangent cones. We discuss the question of the uniqueness of tangent cones, the behavior of certain minimal hypersurfaces with cylindrical tangent cones, and a Liouville type theorem for entire minimal hypersurfaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors investigate the probability distribution of solutions within the phase space for the non-autonomous tropical climate model in two-dimensional bounded domains. They first prove that the associated process possesses a pullback attractor and a family of invariant Borel probability measures. Then they establish that this family of invariant Borel probability measures satisfies Liouville's theorem and is a statistical solution of the tropical climate model. Afterwards, they prove that the statistical solution possesses degenerate Lusin's type regularity provided that the associated Grashof number is small enough. [ABSTRACT FROM AUTHOR]

In this paper we study the Liouville type problem for stationary magnetohydrodynamic equations in R n . We show that under certain anisotropic integrability conditions on the components of the velocity and the magnetic field the solution is trivial. [ABSTRACT FROM AUTHOR]

We remove (α , λ) -discretely self-similar blow up for solutions to the Euler equations for α ≥ 3 2 , for which we allow sublinear growth for the profile. More precisely, we show that there are only spatial constant (α , λ) -discretely self-similar solutions v = c (t) having the sublinear growth at the infinity. For the proof, we establish a new a priori L loc 2 (R 3) estimate for the 3D Euler equations. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. Firstly, we establish optimal Liouville type theorems for nonnegative or positive supersolutions of the fractional parabolic equation u t + (- Δ) s u = u p on the whole space R N × R , where p ∈ R and 0 < s < 1 . Secondly, we study similar questions for the fractional parabolic system u t + (- Δ) s u = v p v t + (- Δ) s v = u q , where p and q are real numbers. [ABSTRACT FROM AUTHOR]

We classify all 4-dimensional toric scalar-flat Kähler instanton metrics under a simple closure condition. This is done by reducing M 4 to its 2-dimensional metric polygon (Σ 2 , g Σ) , where a synthetic zero scalar curvature condition allows a purely analytic classification, based on solutions of a pair of boundary-degenerate elliptic functions. The proof uses a Liouville theorem for these boundary-degenerate elliptic equations, and we create a Dirichlet boundary-specification theorem for these equations. [ABSTRACT FROM AUTHOR]