Discrete & Continuous Dynamical Systems
September 2021 , Volume 41 , Issue 9
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We deal with the bistable reaction-diffusion equation in an infinite star graph, which consists of several half-lines with a common end point. The aim of our study is to show the existence of front-like entire solutions together with the asymptotic behaviors as
In this article, we consider a non-local variant of the Kuramoto-Sivashinsky equation in three dimensions (2D interface). Besides showing the global wellposedness of this equation we also obtain some qualitative properties of the solutions. In particular, we prove that the solutions become analytic in the spatial variable for positive time, the existence of a compact global attractor and an upper bound on the number of spatial oscillations of the solutions. We observe that such a bound is particularly interesting due to the chaotic behavior of the solutions.
This paper considers a chemotaxis-convection model of capillary-sprout growth during tumor angiogenesis
under Neumann initial-boundary conditions in a smooth bounded domain. In the two-dimensional setting, introducing a generalized solution concept according to (Winkler, 2015 [
In this case we write
which counts the run length of the longest same block among the first
In this paper, we are concerned with the asymptotic behaviour of
In this paper, we study multiplicity of semi-classical solutions to nonlinear Dirac equations of space-dimension
In this paper we study the local and global existence of solutions for a class of heat equation in whole
This work investigates the dynamics of competitive Kolmogorov systems formulated in a semi-Markov regime-switching framework. The conditional holding time of each environmental regime is allowed to follow arbitrary probability distribution on the nonnegative half-line in the sense of approximations. Sharp sufficient conditions of the coexistence and competitive exclusion of species are established, and in the case of species coexistence, the convergence rate of the transition probability to the unique stationary measure is estimated. In weaker conditions, these results extend the existing results to the semi-Markov regime-switching environment. Particularly, the method of proving the exponential convergence of the transition probability to the invariant measure for the population models formulated as random differential equations driven by a semi-Markov process is proposed.
We study the quasilinear elliptic equation
where the operator
Considering the Cauchy problem of the derivative NLS
we will show its local well-posedness in modulation spaces
In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers
In this paper we study a class of singular systems with double-phase energy. The main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. In such a way, we continue the analysis introduced in [
In this work we obtain positive bounded solutions of various perturbations of
A restricted permutation of a locally finite directed graph
This paper showcases the use of PDE-based graph methods for modern machine learning applications. We consider a case study of body-worn video classification because of the large volume of data and the lack of training data due to sensitivity of the information. Many modern artificial intelligence methods are turning to deep learning which typically requires a lot of training data to be effective. They can also suffer from issues of trust because the heavy use of training data can inadvertently provide information about details of the training images and could compromise privacy. Our alternate approach is a physics-based machine learning that uses classical ideas like optical flow for video analysis paired with linear mixture models such as non-negative matrix factorization along with PDE-based graph classification methods that parallel geometric equations from PDE such as motion by mean curvature. The upshot is a methodology that can work well on video with modest amounts of training data and that can also be used to compress the information about the video scene so that no personal information is contained in the compressed data, making it possible to provide a larger group of people access to these compressed data without compromising privacy. The compressed data retains information about the wearer of the camera while discarding information about people, objects, and places in the scene.
In this paper we study the weak two-sided limit shadowing for flows on a compact metric space which is different with the usual shadowing, two-sided limit shadowing and L-shadowing, and characterize the weak two-sided limit shadowing flows from the pointwise and measurable viewpoints. Moreover, we prove that if a flow
We study the partial regularity problem for a three dimensional simplified Ericksen–Leslie system, which consists of the Navier–Stokes equations for the fluid velocity coupled with a convective Ginzburg-Landau type equations for the molecule orientation, modelling the incompressible nematic liquid crystal flows. Base on the recent studies on the Navier–Stokes equations, we first prove some new local energy bounds and an
The first aim of this paper is to show well-posedness of Dirichlet and transmission problems in bounded and exterior Lipschitz domains in
A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations
In this note we describe centralizers of volume preserving partially hyperbolic diffeomorphisms which are homotopic to identity on Seifert fibered and hyperbolic 3-manifolds. Our proof follows the strategy of Damjanovic, Wilkinson and Xu [
We consider a wide family of non-uniformly expanding maps and hyperbolic Hölder continuous potentials. We prove that the unique equilibrium state associated to each element of this family is given by the eigenfunction of the transfer operator and the eigenmeasure of the dual operator (both having the spectral radius as eigenvalue). We show that the transfer operator has the spectral gap property in some space of Hölder continuous observables and from this we obtain an exponential decay of correlations and a central limit theorem for the equilibrium state. Moreover, we establish the analyticity with respect to the potential of the equilibrium state as well as that of other thermodynamic quantities. Furthermore, we derive similar results for the equilibrium state associated to a family of non-uniformly hyperbolic skew products and hyperbolic Hölder continuous potentials.
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