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This paper extends the results from a work of Hytönen, Petermichl, and Volberg about sharp A 2 A_2 - A ∞ A_\infty estimates with matrix weights to the non-homogeneous situation.

The driver tries to keep the car in the center of the lane. If the car is too near the left edge, this causes the driver to make a “corrective” right turn. If the car is near the right edge, a “corrective” left turn is made. Therefore, a quantity which decreases with increasing distance ΔL from the left edge may be considered as a stimulusSR producing the reactionRR of turning to the right. A similar situation holds for the distance ΔR from the right edge. When the car is in the center of the lane, ΔL = ΔR andSR=SL, the stimuli are equal. We thus have here a situation analogous to the one studied by H. D. Landahl in his theory of psychophysical discrimination. In general a reactionRR (resp.RL) will occur only ifRR−RL≥h* (resp.RL−RR≥h*) whereh* is a threshold. Applying Landahl’s theory to this situation, we find thath* determines the distance from the edge, at which a corrective turn is made. This distance is not constant, but a function of the speedv of the car. The requirement that a corrective turn should be madebeforre the car runs off the road leads to an expression for the maximum safe speed. Because of the transcendency of the equations involved, closed solutions cannot be obtained. It is, however, shown that the expression for maximum safe speed, given in a previous paper (Bull. Math. Biophysics,21, 299–308, 1959), is a rough first approximation to the expressions found now.

In this paper, we shall establish sufficient conditions for the existence of solutions for semi-linear integrodifferential inclusions in Banach spaces with nonlocal conditions. By using suitable fixed point theorems we study the case when the multi-valued map has convex as well as nonconvex values. Rocky Mountain Journal of Mathematics

For positive functions $u\in C^{2}(\Omega) $, where $\Omega$ is an open subset of $\mathbb{R}^{n}$, the Symmetric Minimal Surface Equation (SME), is $\sum_{i=1}^{n}D_{i}\bigl(\frac{D_{i}u}{\sqrt{1+|Du|^{2}}}\bigr)=\frac{m-1}{u\sqrt{1+|Du|^{2}}}$. Geometrically, the SME expresses the fact that the ``symmetric graph'' $SG(u)$, defined by $SG(u)=\bigl\{(x,\xi)\in \Omega\times\mathbb{R}^{m}:|\xi|=u(x)\bigr\}$, is a minimal (i.e.\ zero mean curvature) hypersurface in $\Omega\times\mathbb{R}^{m}$. A function $u\in C^{1}(\Omega)$ is said to be a singular solution if $u^{-1}\{0\}\neq \emptyset$, and if $u=\lim_{j\to\infty}u_{j}$, uniformly on each compact subset of $\Omega$, where each $u_{j}$ is a positive $C^{2}(\Omega)$ solution of the SME. The present paper develops are theory of singular solutions of the SME, including existence, H\"older and Lipschitz estimates for bounded solutions, and a compactness and regularity theory. We also prove that the singular set $u^{-1}{\{0\}}$ is codimension at most 2.

Surface Green’s functions for semi-infinite systems play an important role in the design of nano-scale electronic devices. In particular, functions in multilayered periodic structures are useful for practical applications. This paper describes numerical issues associated with the computation of these functions and offers an approach to addressing these issues. The proposed approach is based on the observation that numerical errors, which can occur in the existing algorithm to compute the functions, are caused by inaccurate computation of eigenpairs. We consider an alternative approach to eigenpairs to derive an algorithm to compute the functions. Numerical experiments show that the proposed algorithm can compute functions that satisfy physical requirements.

In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation. We first verify that the QCHE has global peakon and periodic peakon solutions. Then by the invariants of the equation and controlling the extrema of the solution, we prove that the shapes of the peakons and periodic peakons are stable under small perturbations in the energy space.

In this paper we study the homogenization of an eigenvalue problem for a cooperative system of weakly coupled elliptic partial differential equations, called the neutronic multigroup diffusion model, in a periodic heterogeneous domain. Such a model is used for studying the criticality of nuclear reactor cores. In a recent work in collaboration with Grégoire Allaire, it is proved that, under a symmetry assumption, the first eigenvector of the multigroup system in the periodicity cell controls the oscillatory behaviour of the solutions, whereas the global trend is asymptotically given by a homogenized diffusion eigenvalue problem. It is shown here that when this symmetry condition is not fulfilled, the asymptotic behaviour of the neutron flux, corresponding to the first eigenvector of the multigroup system, is dramatically different. This result enables to consider new types of geometrical configurations in practical nuclear reactor core computations.

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H withGD U.p;qIC/ andHD U.1IC/ U.p 1;qIC/. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroupH . We obtain the Plancherel formula by a special method which is also suitable for other problems, for example, for quantization in the spirit of Berezin. © 2000 Editions scientifiques et medicales Elsevier SAS Keywords: Plancherel formula, Spherical distributions, Complex hyperbolic spaces In this paper we obtain the Plancherel formula for the spaces of L 2 -sections of line bundles over the complex projective hyperboloids G=H with GD U.p;qIC/ and H D U.1IC/ U.p 1;qIC/, i.e. we present the decomposition of L 2 into irreducible represen- tations of the group G of class . In order to leave aside the well-known case of a hyperboloid with compact stabilizer subgroup, see (14), we assumep>1 ;q >0. The Plancherel formula is given in an explicit form by means of spherical distributions associated with a character of the subgroup H. We obtain the Plancherel formula by Molchanov's method, see (9). Namely, we follow the detailed scheme in (1), Sections 4, 7. This method deals with the spectral resolution of the radial part of the Laplace operator. The essential step is setting the boundary conditions at certain special points. Those conditions are prescribed by the behaviour of spherical distributions. Finally, it is necessary to perform various analytic continuations. This method is also suitable for other problems, for example, for quantization in the spirit of Berezin, namely, for the decomposition of the Berezin form. It is therefore why this method has to be preferred to the existing methods, described in (3). We use our results from (13). There we define -spherical distributions, study their asymptotic behaviour and express them by means of hypergeometric functions. We describe the irreducible unitary representations of the group G ,o f class associated with an isotropic cone. We give constructions for the Fourier and Poisson transform, define intertwining operators and diagonalize them. Some of those results are presented in Section 1.

We prove Lp estimates (Theorem 1.2) for the ‘‘biest’’, a trilinear multiplier operator with singular symbol. The methods used are based on the treatment of the Walsh analogue of the biest in the prequel [13] of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.

We study the obstruction space T 2 {T^2} for minimally elliptic surface singularities. We apply the main lemma of our previous paper [3] which relates T 2 {T^2} to deformations of hypersurface sections. To use this we classify general hypersurface sections of minimally elliptic singularities. As in the rational singularity case there is a simple formula for the minimal number of generators for T 2 {T^2} as a module over the local ring. This number is in many cases (e.g. for cusps of Hilbert modular surfaces) equal to the vector space dimension of T 2 {T^2} .

The first purpose of this paper is to solve completely the finite field cone restriction conjecture in four dimensions with $-1$ non-square. The second is to introduce a new approach to study incidence problems via restriction theory. More precisely, using the cone restriction estimates, we will prove sharp point-sphere incidence bounds associated with complex-valued functions for sphere sets of small size. Our incidence bounds with a specific function improve significantly a result given by Cilleruelo, Iosevich, Lund, Roche-Newton, and Rudnev., Title was changed

In this expository paper we discuss Bowen’s thermodynamicformalism for conformal repellers, and possible connections withthe theory of asymptotic Teichm¨uller spaces and uniformly asymptoticallyconformal dynamical systems.

Consider a stochastic process that lives on n-semiaxes emanating from a common origin. On each semiaxis it behaves as a Brownian motion and at the origin it chooses a semiaxis randomly. In this paper we study the first hitting time of the process. We derive the Laplace transform of the first hitting time, and provide the explicit expressions for its density and distribution functions. Numerical examples are presented to illustrate the application of our results.

This paper is devoted to the study of the qualitative properties of solutions to boundary-value problems for strongly elliptic differential-difference equations. Some results for these equations such as existence and smoothness of generalized solutions in certain subdomains of Q were obtained earlier. Nevertheless, the smoothness of generalized solutions of such problems can be violated near the boundary of these subdomains even for infinitely differentiable right-hand side. The subdomains are defined as connected components of the set that is obtained from the domain Q by throwing out all possible shifts of the boundary Q by vectors of a certain group generated by shifts occurring in the difference operators. For the one dimensional Neumann problem for differential-difference equations there were obtained conditions on the coefficients of difference operators, under which for any continuous right-hand side there is a classical solution of the problem that coincides with the generalized solution. 2 Also there was obtained the smoothness (in Sobolev spaces W k ) of generalized solutions of the second and the third boundary-value problems for strongly elliptic differential-difference equations in subdomains excluding -neighborhoods of certain points. However, the smoothness (in Ho lder spaces) of generalized solutions of the second boundary-value problem for strongly elliptic differential-difference equations on the boundary of adjacent subdomains was not considered. In this paper, we study this question in Ho lder spaces. We establish necessary and sufficient conditions for the coefficients of difference operators that guarantee smoothness of the generalized solution on the boundary of adjacent subdomains for any right-hand side from the Ho lder space.

This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly K\"ahler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface of nearly K\"ahler manifold. It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.

This paper studies the overall evolution of fronts propagating with a normal velocity that depends on position, υn = f(x), where f is rapidly oscillating and periodic. A level-set formulation is used to rewrite this problem as the periodic homogenization of a Hamilton–Jacobi equation. The paper presents a series of variational characterization (formulae) of the effective Hamiltonian or effective normal velocity. It also examines the situation when f changes sign.

Let $\Omega \subset {{\mathbb {R}}}^{n+1}$, $n\geq 2$, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, merely bounded and possibly nonsymmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partial \Omega $ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $\Omega $ is $\varepsilon $-approximable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $\Omega $ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called “$S

The purpose of this work is to show the well-posedness in L2-Sobolev spaces of the Poisson-transmission problem for the Oseen and Brinkman systems on complementary Lipschitz domains in a compact Riemannian manifold. The Oseen system appears as a perturbation of order one of the Stokes system, given in terms of the Levi-Civita connection, while the Brinkman system is a zero order perturbation of the Stokes system. The technical details of this paper rely on the layer potential theory for the Stokes system and the invertibility of some perturbed zero index Fredholm operators by a first order differential operator given in terms of the Levi-Civita connection. The compactness of this differential operator requires to restrict ourselves to low dimensional compact Riemannian manifolds.

It is known, after Jabin (J Differ Equ 260(5):4739–4757, 2016) and Alberti et al. (Ann PDE 5(1):9, 2019), that ODE flows and solutions of the transport equation associated to Sobolev vector fields do not propagate Sobolev regularity, even of fractional order. In this paper, we improve the result at Clop and Jylha (J Differ Equ 266(8):4544–4567, 2019) and show that some kind of propagation of Sobolev regularity happens as soon as the gradient of the drift is exponentially integrable. We provide sharp Sobolev estimates and new examples. As an application of our main theorem, we generalize a regularity result for the 2D Euler equation obtained by Bahouri and Chemin in Bahouri and Chemin (Arch Ration Mech Anal 127(2):159–181, 1994).

In this paper, we study a predator-prey system with general response function and nonlocal diffusion in high dimensional space and investigate the propagation properties of its solution. More precisely, we study the invasion speed of the predator into habitat of the aborigine prey and obtain the existence of the planar waves by constructing the upper and lower solutions. Finally, we present some numerical simulations to support our results.

In this paper, we prove the global existence and asymptotic behavior, as time tends to infinity, of solutions in Hi (i=1, 2) to the initial boundary value problem of the compressible Navier–Stokes equations of one-dimensional motion of a viscous heat-conducting gas in a bounded region with a non-autonomous external force and a heat source. Some new ideas and more delicate estimates are used to prove these results. Copyright © 2008 John Wiley & Sons, Ltd.

In this paper, we consider a class of nonlocal equations where the convolution kernel is given by a Bessel potential symbol of order $\alpha$ for $\alpha > 1$. Based on the properties of the convolution operator, we apply a global bifurcation technique to show the existence of a highest, even, $2\pi$-periodic traveling-wave solution. The regularity of this wave is proved to be exactly Lipschitz., Comment: 22 pages. arXiv admin note: text overlap with arXiv:1810.00248 by other authors

The well-established Timoshenko system is characterized by a particular relation between shear stress and bending moment from its constitutive equations. Accordingly, a (thermal) dissipation added on the bending moment produces exponential stability if and only if the so called “equal wave speeds” condition is satisfied. This remarkable property extends to the case of non-homogeneous coefficients. In this paper, we consider a non-homogeneous thermoelastic system with dissipation restricted to the shear stress. To this new problem, by means of a delicate control observability analysis, we prove that a local version of the equal wave speeds condition is sufficient for the exponential stability of the system. Otherwise, we study the polynomial stability of the system with decay rate depending on the regularity of initial data.

In a recent paper [10], Peter A. Loeb showed how to convert non-standard measure spaces into standard ones and gave applications to probability theory. We apply these results to Brownian Motion and Ito integration. We first develop a number of new tools about Loeb spaces. We then show that Brownian Motion can be obtained as the Loeb process corresponding to a non-standard random walk obtained from a*-finite number of coin tosses. This permits a very constructive proof of a special case of Donsker's Theorem. The Ito integral with respect to this Brownian Motion is a non-standard Stieltjes integral with respect to the random walk. As a consequence, an easy proof of Ito’s Lemma is possible. The results in this paper were announced in [1].

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.

In this paper, a complete analytical solution to the integro-differential model describing the nucleation and growth of ellipsoidal crystals in a supersaturated solution is obtained. The asymptotic solution of the model equations is constructed using the saddle-point method to evaluate the Laplace-type integral. Numerical simulations carried out for physical parameters of real solutions show that the first four terms of the asymptotic series give a convergent solution. The developed theory was compared with the experimental data on desupersaturation kinetics in proteins. It is shown that the theory and experiments are in good agreement. © 2021 John Wiley & Sons, Ltd. Ministry of Education and Science of the Russian Federation, Minobrnauka: FEUZ-2020-0057; Russian Science Foundation, RSF: 18-19-00008 This work was supported by the Russian Science Foundation (grant no. 18-19-00008). This article contains two parts: (i) a new theory of the growth of an ensemble of ellipsoidal crystals in a metastable liquid and (ii) a computational simulation of crystal growth based on the developed theory. Part (i) was supported by the Russian Science Foundation (grant no. 18-19-00008), whereas part (ii) was made possible due to the financial support from the Ministry of Science and Higher Education of the Russian Federation (project no. FEUZ-2020-0057).

This paper applies two computational techniques for constructing novel solitary wave solutions of the ill-posed Boussinesq dynamic wave (IPB) equation. Jacques Hadamard has formulated this model for studying the dynamic behavior of waves in shallow water under gravity. Extended simple equation (ESE) method and novel Riccati expansion (NRE) method have been applied to the investigated model's converted nonlinear ordinary differential equation through the wave transformation. As a result of this research, many solitary wave solutions have been obtained and represented in different figures in two-dimensional, three-dimensional, and density plots. The explanation of the methods used shows their dynamics and effectiveness in dealing with certain nonlinear evolution equations.

This research paper deals with two novel varieties of boundary value problems for nonlinear hybrid fractional differential equations involving generalized fractional derivatives known as the $ \Psi $-Caputo fractional operators. Such operators are generated by iterating a local integral of a function with respect to another increasing positive function $ \Psi $. The existence results to the proposed systems are obtained by using Dhage's fixed point theorem. Two pertinent examples are provided to confirm the feasibility of the obtained results. Our presented results generate many special cases with respect to different values of a $ \Psi $ function.

In this paper, for the first time, we apply symmetric $ q $ -calculus operator theory to define symmetric Salagean $ q $-differential operator. We introduce a new class $ \widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) $ of harmonic univalent functions $ f $ associated with newly defined symmetric Salagean $ q $-differential operator for complex harmonic functions. A sufficient coefficient condition for the functions $ f $ to be sense preserving and univalent and in the same class is obtained. It is proved that this coefficient condition is necessary for the functions in its subclass $ \overline{\widetilde{\mathcal{H}}_{q}^{m}\left(\alpha \right) } $ and obtain sharp coefficient bounds, distortion theorems and covering results. Furthermore, we also highlight some known consequence of our main results.

This paper deals with a well known open problem raised by Kannan (Bull. Calcutta Math. Soc., 60: 71–76, 1968) and B. E. Rhoades (Contemp. Math., 72: 233–245, 1988) on the existence of general contractions which have fixed points, but do not force the continuity at the fixed point. We propose some new affirmative solutions to this question using two new contractions called $ (\psi, \varphi) $-$ \mathcal{A} $-contraction and $ (\psi, \varphi) $-$ \mathcal{A^{\prime}} $-contraction inspired by the results of H. Garai et al. (Applicable Analysis and Discrete Mathematics, 14(1): 33–54, 2020) and P. D. Proinov (J. Fixed Point Theory Appl. (2020) 22: 21). Some new fixed point and common fixed point results in compact metric spaces and also in complete metric spaces are proved in which the corresponding contractive mappings are not necessarily continuous at their fixed points. Moreover, we show that new solutions to characterize the completeness of metric spaces. Several examples are provided to verify the validity of our main results.

In this paper, we mainly investigate the random convolution sampling stability for signals in multiply generated shift invariant subspace of weighted mixed Lebesgue space. Under some restricted conditions for the generators and the convolution function, we conclude that the defined multiply generated shift invariant subspace could be approximated by a finite dimensional subspace. Furthermore, with overwhelming probability, the random convolution sampling stability holds for signals in some subset of the defined multiply generated shift invariant subspace when the sampling size is large enough.

In this paper, the stability of $ (\omega, c) $-periodic solutions of non-instantaneous impulses differential equations is studied. The exponential stability of homogeneous linear non-instantaneous impulsive problems is studied by using Cauchy matrix, and some sufficient conditions for exponential stability are obtained. Further, by using Gronwall inequality, sufficient conditions for exponential stability of $ (\omega, c) $-periodic solutions of nonlinear noninstantaneous impulsive problems are established. Finally, some examples are given to illustrate the correctness of the conclusion.

In this paper, we discuss standard approaches to the Hyers-Ulam Mittag Leffler problem of fractional derivatives and nonlinear fractional integrals (simply called nonlinear fractional differential equation), namely two Caputo fractional derivatives using a fractional Fourier transform. We prove the basic properties of derivatives including the rules for their properties and the conditions for the equivalence of various definitions. Further, we give a brief basic Hyers-Ulam Mittag Leffler problem method for the solving of linear fractional differential equations using fractional Fourier transform and mention the limits of their usability. In particular, we formulate the theorem describing the structure of the Hyers-Ulam Mittag Leffler problem for linear two-term equations. In particular, we derive the two Caputo fractional derivative step response functions of those generalized systems. Finally, we consider some physical examples, in the particular fractional differential equation and the fractional Fourier transform.

In this paper, we investigate a nonlinear generalized fractional differential equation with two-point and integral boundary conditions in the frame of $ \kappa $-Hilfer fractional derivative. The existence and uniqueness results are obtained using Krasnoselskii and Banach's fixed point theorems. We analyze different types of stability results of the proposed problem by using some mathematical methodologies. At the end of the paper, we present a numerical example to demonstrate and validate our findings.

In this paper, we are interested in studying the spread of infectious disease using a fractional-order model with Caputo's fractional derivative operator. The considered model includes an infectious disease that includes two types of infected class, the first shows the presence of symptoms (symptomatic infected persons), and the second class does not show any symptoms (asymptomatic infected persons). Further, we considered a nonlinear incidence function, where it is obtained that the investigated fractional system shows some important results. In fact, different types of bifurcation are obtained, as saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation, where it is discussed in detail through the research. For the numerical part, a proper numerical scheme is used for the graphical representation of the solutions. The mathematical findings are checked numerically.

In present paper, several conditions ensuring existence of three distinct solutions of a system of over-determined Fredholm fractional integro-differential equations on time scales are derived. Variational methods are utilized in the proofs.

In the present work, the authors developed the scheme for time Fractional Partial Diffusion Differential Equation (FPDDE). The considered class of FPDDE describes the flow of fluid from the higher density region to the region of lower density, macroscopically it is associated with the gradient of concentration. FPDDE is used in different branches of science for the modeling and better description of those processes that involve flow of substances. The authors introduced the novel concept of fractional derivatives in term of both time and space independent variables in the proposed FPDDE. We provided the approximate solution for the underlying generalized non-linear time PFDDE in the sense of Caputo differential operator via Laplace transform combined with Adomian decomposition method known as Laplace Adomian Decomposition Method (LADM). Furthermore, we established the general scheme for the considered model in the form of infinite series by aforementioned techniques. The consequent results obtained by the proposed technique ensure that LADM is an effective and accurate technique to handle nonlinear partial differential equations as compared to the other available numerical techniques. At the end of this paper, the obtained numerical solution is visualized graphically by Matlab to describe the dynamics of desired solution.

Riemann-Liouville fractional differential equations with impulses are useful in modeling the dynamics of many real world problems. It is very important that there are good and consistent theoretical proofs and meaningful results for appropriate problems. In this paper we consider a boundary value problem for integro-differential equations with Riemann-Liouville fractional derivative of orders from $ (1, 2) $. We consider both interpretations in the literature on the presence of impulses in fractional differential equations: With fixed lower limit of the fractional derivative at the initial time point and with lower limits changeable at each impulsive time point. In both cases we set up in an appropriate way impulsive conditions which are dependent on the Riemann-Liouville fractional derivative. We establish integral presentations of the solutions in both cases and we note that these presentations are useful for furure studies of existence, stability and other qualitative properties of the solutions.

In this paper, we describe generalized fractional integral operator and its inverse with generalized Bessel-Maitland function (BMF-Ⅴ) as its kernel. We discuss its convergence, boundedness, its relation with other well known fractional operators (Saigo fractional integral operator, Riemann-Liouville fractional operator), and establish its integral transform. Moreover, we have given the relationship of BMF-Ⅴ with Mittag-Leffler functions.

In this paper we revisit a discrete predator-prey model with Holling Ⅳ functional response. By using the method of semidiscretization, we obtain new discrete version of this predator-prey model. Some new results, besides its stability of all fixed points and the transcritical bifurcation, mainly for codimension two 1:1 strong resonance bifurcation, are derived by using the center manifold theorem and bifurcation theory, showing that this system possesses complicate dynamical properties.

In this paper we develop a potential theory for strongly degenerate parabolic operators of the form L : = ∇ X ⋅ ( A ( X , Y , t ) ∇ X ) + X ⋅ ∇ Y − ∂ t , in unbounded domains of the form Ω = { ( X , Y , t ) = ( x , x m , y , y m , t ) ∈ R m − 1 × R × R m − 1 × R × R | x m > ψ ( x , y , y m , t ) } , where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L . Concerning A = A ( X , Y , t ) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in R m ). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

We study a fundamental model in fluid mechanics--the 3D gravity water wave equation, in which an incompressible fluid occupying half the 3D space flows under its own gravity. In this paper we show long-term regularity of solutions whose initial data is small but not localized. Our results include: almost global wellposedness for unweighted Sobolev initial data and global wellposedness for weighted Sobolev initial data with weight $|x|^\alpha$, for any $\alpha > 0$. In the periodic case, if the initial data lives on an $R$ by $R$ torus, and $\epsilon$ close to the constant solution, then the life span of the solution is at least $R/(\epsilon^2(\log R)^2)$., Comment: 88 pages

The $(k,a)$-generalised Fourier transform is the unitary operator defined using the $a$-deformed Dunkl harmonic oscillator.The main aim of this paper is to prove $L^p$-$L^q$ boundedness of $(k, a)$-generalised Fourier multipliers. To show the boundedness we first establish Paley inequality and Hausdorff-Young-Paley inequality for $(k, a)$-generalised Fourier transform. We also demonstrate applications of obtained results to study the well-posedness of nonlinear partial differential equations., 20 pages. arXiv admin note: text overlap with arXiv:2108.01146

In Leonov and Kuznetsov (2013), the authors shown numerically the existence of a limit cycle surrounding the unstable node that system (1) has in the positive quadrant for specific values of the parameters. System (1) is one of the Belousov-Zhabotinsky dynamical models. The objective of this paper is to prove that system (1), when in the positive quadrant Q has an unstable node or focus, has at least one limit cycle, and when (Formula presented.), (Formula presented.), and ϵ > 0 sufficiently small this limit cycle is unique.

Let M be a complete C ~~ Riemannian manifold of non-positive sectional curvature. We say that a geodesic 9: IR~ M bounds a fiat strip of width c > 0 (a fiat half plane) if there is a totally geodesic, isometric immersion i: [0, c) x IR~M(i: [0, oo) x IR~M) such that i(0, t) = 9(0. A 9eodesic without fiat strip (without fiat half plane) is a geodesic, which does not bound a flat strip (a flat half plane). We will prove that the existence of a closed geodesic without flat half plane has rather strong consequences for the geometry and topology of M. In fact, many of the properties of a manifold of strictly negative curvature (resp. of a visibility manifold) still remain true if one assumes only the existence of a closed geodesic without flat half plane. We will discuss the existence of free (non-Abelian) subgroups of gl(M), the existence of infinitely many closed geodesics, the density of closed geodesics, and a transitivity property of the geodesic flow. It is, therefore, interesting to give conditions which ensure the existence of a closed geodesic without flat half plane. We will prove that M has a closed geodesic without flat half plane if vol(M)< oo and if M contains a geodesic without flat half plane. Note that a geodesic is not boundary of a flat strip (and a fortiori not boundary of a flat half plane) if it passes through a point p e M such that the sectional curvature of all tangent planes at p is negative. In the proofs of our results we investigate the action of rtl(M ) as group of isometries on the universal covering space H of M. In the proofs of many of our results we do not use the fact that this action is properly discontinuous and free. We, therefore, formulate these results for arbitrary groups D of isometries of H. The paper is organized as follows: In Sect. 1 we fix some definitions and notations and quote some standard results of non-positive curvature. Section 2 is the central section of this paper. We investigate the properties of those isometries of H which correspond to closed geodesics in M. We also prove

Let S be a smooth algebraic surface satisfying the following property: H^i(\oc_S(C))=0 (i=1,2) for any irreducible and reduced curve C of S. The aim of this paper is to provide a characterization of special linear systems on S which are singular along a set of double points in general position. As an application, the dimension of such systems is evaluated in case S is an Abelian, an Enriques, a K3 or an anticanonical rational surface., Comment: 10 pages, LaTeX

This paper deals with the long-time behavior of viscosity solutions of evolutionary contact Hamilton-Jacobi equations w t + H ( x , w , w x ) = 0 , \begin{equation*} w_t+H(x,w,w_x)=0, \end{equation*} where H ( x , u , p ) H(x,u,p) is strictly decreasing in u u and satisfies Tonelli conditions in p p . We show that each viscosity solution of the ergodic contact Hamilton-Jacobi equation H ( x , u , u x ) = 0 H(x,u,u_x)=0 can be reached by many different viscosity solutions of the above evolutionary equation in a finite time.

In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficient in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof., Comment: 25 pages

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)