Communications on Pure & Applied Analysis
February 2021 , Volume 20 , Issue 2
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This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators
This article deals with some generalizations of the Cahn–Hilliard equation with mass source endowed with Neumann boundary conditions. This equation has many applications in real life e.g. in biology and image inpainting. The first part of this article, discusses the stationary problem of the Cahn–Hilliard equation with mass source. We prove the existence of a unique solution of the associated stationary problem. Then, in the latter part of this article, we consider the evolution problem of the Cahn–Hilliard equation with mass source. We construct a numerical scheme of the model based on a finite element discretization in space and backward Euler scheme in time. Furthermore, after obtaining some error estimates on the numerical solution, we prove that the semi discrete scheme converges to the continuous problem. In addition, we prove the stability of our scheme which allows us to obtain the convergence of the fully discrete problem to the semi discrete one. Finally, we perform the numerical simulations that confirm the theoretical results and demonstrate the performance of our scheme for cancerous tumor growth and image inpainting.
A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the positive half–line while the nonlinearity has some boundedness and Lipschitz conditions. In addition, we provide new characterizations for the resulting homeomorphisms. When the linear system is asymptotically stable and the nonlinear system has a unique equilibrium, we deduce sharper results for the smoothness of the topological equivalence. Finally, we study the asymptotic stability and its preservation by topological equivalence.
We study a Dirichlet problem in periodic porous medium depending on two small parameters, the hydraulic permeability of the porous inclusions
This article is concerned with a strong unique continuation property of a forward differential inequality abstracted from parabolic equations proposed on a convex domain
We study the classification and evolution of bifurcation curves for the porous-medium combustion problem
This paper considers two problems: the initial boundary value problem of nonlinear Caputo time-fractional pseudo-parabolic equations with fractional Laplacian, and the Cauchy problem (initial value problem) of Caputo time-fractional pseudo-parabolic equations. For the first problem with the source term satisfying the globally Lipschitz condition, we establish the local well-posedness theory including existence, uniqueness and regularity of the local solution, and the further local existence theory related to the finite time blow-up are also obtained for the problem with logarithmic nonlinearity. For the second problem with the source term satisfying the globally Lipschitz condition, we prove the global existence theorem.
The asymptotic behavior of dissipative evolution problems, determined by complex networks of reaction-diffusion systems, is investigated with an original approach. We establish a novel estimation of the fractal dimension of exponential attractors for a wide class of continuous dynamical systems, clarifying the effect of the topology of the network on the large time dynamics of the generated semi-flow. We explore various remarkable topologies (chains, cycles, star and complete graphs) and discover that the size of the network does not necessarily enlarge the dimension of attractors. Additionally, we prove a synchronization theorem in the case of symmetric topologies. We apply our method to a complex network of competing species systems modeling an heterogeneous biological ecosystem and propose a series of numerical simulations which underpin our theoretical statements.
We consider the cubic nonlinear fourth-order Schrödinger equation
In this paper, we generalize the Doob's maximal inequality for mixed-norm
We investigate regular elliptic boundary-value problems in boun\-ded domains and show the Fredholm property for the related operators in an extended scale formed by inner product Sobolev spaces (of arbitrary real orders) and corresponding interpolation Hilbert spaces. In particular, we can deal with boundary data with arbitrary low regularity. In addition, we show interpolation properties for the extended scale, embedding results, and global and local a priori estimates for solutions to the problems under investigation. The results are applied to elliptic problems with homogeneous right-hand side and to elliptic problems with rough boundary data in Nikolskii spaces, which allows us to treat some cases of white noise on the boundary.
We prove an almost global existence result for the Klein-Gordon equation with the Kirchhoff-type nonlinearity on
We present a periodic nonlinear scalar delay differential equation model for a population with short reproduction period. By transforming the equation to a discrete dynamical system, we reduce the infinite dimensional problem to one dimension. We determine the basic reproduction number not merely as the spectral radius of an operator, but as an explicit formula and show that is serves as a threshold parameter for the stability of the trivial equilibrium and for permanence.
The aim of this paper is to establish new regularity results for a non-isothermal Cahn-Hilliard system in the two dimensional setting. The main achievement is a crucial
The minimizers of the anisotropic fractional isoperimetric inequality with respect to a convex body
In this paper, we establish the boundedness on
In this paper, we consider the existence of a ground state nodal solution and a ground state solution, energy doubling property and asymptotic behavior of solutions of the following fractional critical problem
We prove a Hopf's lemma in the point-wise sense for fractional
For coupled Schrödinger equations with nonhomogeneous perturbations we give several results on the existence of multiple positive solutions. In particular in one case we consider perturbations of the permutation symmetry.
Given a quantum graph
In this paper we present a very short proof of inequalities of Hermite-Hadamard type obtained by M. Bessenyei and Zs. Páles. This proof is based on the recently developed method connected with use of stochastic orderings of random variables. In the second part of the paper we present a way to extend these known inequalities. Namely, we describe completely the possible inequalities of Hermite-Hadamard type for longer expression than it was the case in the results of Bessenyei and Páles.
In this paper, we study a priori derivative estimates (up to the second order) of solutions for the Monge-Ampère equation
In this paper, we study a class of quasilinear Schrödinger equation of the form
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