Acta Applicandae Mathematicae最新文献

In this paper, we consider the existence of solutions for the following nonlinear Schrödinger equation with (L^{2})-norm constraint

where (sin (0,1)), (mu ,a>0), (Nge 3), (2< q< p<2+frac{4s}{N}), ((-Delta )^{s}) is the fractional Laplacian operator, (Omega subseteq mathbb{R}^{N}) is an exterior domain, that is, (Omega ) is an unbounded domain in (mathbb{R}^{N}) with (mathbb{R}^{N}backslash Omega ) non-empty and bounded and (lambda in mathbb{R}) is Lagrange multiplier, which appears due to the mass constraint (||u||_{L^{2}(Omega )}= a). In this paper, we use Brouwer degree, barycentric functions and minimax method to prove that for any (a > 0), there exists a positive solution (uin H^{s}_{0} (Omega )) for some (lambda <0) if (mathbb{R}^{N}backslash Omega ) is contained in a small ball.

We show that the Oldroyd B fluid model is the Eulerian form of a Lagrangian model with an internal variable that satisfies the second law of thermodynamics under some conditions on the initial value of the internal variable. We similarly derive several new nonlinear versions of the Oldroyd B model as well as Lagrangian formulations of the Zaremba-Jaumann and Oldroyd A fluid models. We discuss whether or not these other models satisfy the second law.

In this paper, we consider a version of the mathematical model introduced in (Wang et al. in Commun. Nonlinear Sci. Numer. Simul. 42:571–584, 2017) to describe the interaction between vegetation and soil water in arid environments. The model corresponds to a nonlinear parabolic coupled system of partial differential equations, with non-flux boundary conditions, which incorporates, in addition to the natural diffusion of water and plants, a cross-diffusion term given by the hydraulic diffusivity due to the suction of water by the roots. The model also considers a monotonously decreasing vegetation death rate capturing the infiltration feedback between plants and ground water. We first prove the existence and uniqueness of global solutions in a large class of initial data, allowing non-regular ones. These solutions are in a mild setting and under additional regularity assumptions on the initial data and the domain, they are classical. Second, we propose a fully discrete numerical scheme, based on a semi-implicit Euler discretization in time and finite element discretization (with “mass-lumping”) in space, for approximating the solutions of the continuous model. We prove the well-posedness of the numerical scheme and some qualitative properties of the discrete solutions including, positivity, uniform weak and strong estimates, convergence towards strong solutions and optimal error estimates. Finally, we present some numerical experiments in order to showcase the good behavior of the numerical scheme including the formation of Turing patterns, as well as to validate the convergence order in the error estimates obtained in the theoretical analysis.

This work concerns with a class of chemotaxis models in which external sources, comprising nonlocal and gradient-dependent damping reactions, influence the motion of a cell density attracted by a chemical signal. The mechanism of the two densities is studied in bounded and impenetrable regions. In particular, it is seen that no gathering effect for the cells can appear in time provided that the damping impacts are sufficiently strong. Mathematically, we study this problem

for

and where (Omega ) is a bounded and smooth domain of (mathbb{R}^{n}) ((n in mathbb{N})), ({t>0}subseteq (0,infty )) an open interval, (tau in {0,1}), (m_{1},m_{2}in mathbb{R}), (chi ,a,b>0), (cgeq 0), and (alpha , beta ,delta geq 1). Herein for ((x,t)in Omega times {t>0}), (u=u(x,t)) stands for the population density, (v=v(x,t)) for the chemical signal and (f) for a regular function describing the production law. The population density and the chemical signal are initially distributed accordingly to nonnegative and sufficiently regular functions (u_{0}(x)) and (tau v_{0}(x)), respectively. For each of the expressions of (B), sufficient conditions on parameters of the models ensuring that any nonnegative classical solution ((u,v)) to system (◊) is such that ({t>0} equiv (0,infty )) and uniformly bounded in time, are established. In the literature, most of the results concerning chemotaxis models with external sources deal with classical logistics, for which (B=a u^{alpha }-b u^{beta }). Thereafter, the introduction of dissipative effects as those expressed in (B) is the main novelty of this investigation. On the other hand, this paper extends the analyses in (Chiyo et al. in Appl. Math. Optim. 89(9):1–21, 2024; Bian et al. in Nonlinear Anal. 176:178–191, 2018; Latos in Nonlocal reaction preventing blow-up in the supercritical case of chemotaxis, 2020, arXiv:2011.10764).

This article focuses on inverse problem for Hallaire-Luikov moisture transfer equation involving Hilfer fractional derivative in time. Hallaire-Luikov equation is used to study heat and mass transfer in capillary-porous bodies. Spectral expansion method is used to find the solution of the inverse problem. By imposing certain conditions on the functions involved and utilizing certain properties of multinomial Mittag-Leffler function, it is shown that the solution to the equation, known as the inverse problem, is regular and unique. Moreover, the inverse problem exhibits ill-posedness in the sense of Hadamard. The article ends with an example to demonstrate these theoretical findings.

In this paper, we consider a Lotka-Volterra competition-diffusion system with resource-dependent dispersal. We study the linear v.s. global asymptotic stability of steady states. Furthermore, how the diffusion coefficients and the dispersal strategies of two competing species affect the stability of steady states are given. This paper is a further study of (Tang and Wang in J. Math. Biol. 86:23, 2023).

For a super Korteweg-de Vries (KdV) equation introduced by Geng and Wu, nonlocal infinitesimal symmetries depending on eigenfunctions of its (adjoint) linear spectral problem are constructed from gradient of the spectral parameter, and one of such symmetries is shown to be related to a nonlocal infinitesimal symmetry of Kupershmidt’s super modified KdV equation via a Miura-type transformation. On this basis, a finite symmetry transformation is established for an enlarged system, and leads to a non-trivial exact solution and a Bäcklund transformation of Geng-Wu’s super KdV equation. A procedure is explained to generate infinitely many conservation laws. Moreover, these results could be reduced to classical situations, and their bosonic limits are briefly summarized.

In this note, we study numerically the regularity issue of the ideal MHD equations on the two-dimensional torus. By pseudo-spectral method, we provide evidence that a certain numerical solution is initially regular and eventually very singular in finite time.

In this paper, we study a Kirchhoff-type problem driven by a multi-phase operator with three variable exponents. Such problem has a right-hand side consisting of a Carathéodory perturbation which is defined only locally as well as the Kirchhoff term. Using a generalized version of the symmetric mountain pass theorem along with recent a priori upper bounds for multi-phase problems, we get whole a sequence of nontrivial solutions for our problem converging to zero in the appropriate Musielak-Orlicz Sobolev space and in (L^{infty }(Omega )).

Difference equations play a crucial role in a wide array of mathematical and physical tasks. In this article, we focus on the analysis of a second order linear homogeneous difference equation with smooth coefficients via WKB method. It is well-known that such equations exhibit two WKB solutions in a segment devoid of turning and singular points. We establish a theorem demonstrating the existence of these solutions in an unbounded domain under certain conditions regarding the smoothness and growth behavior of the coefficients at infinity. Furthermore, using this theorem, we derive the asymptotic expansion of Laguerre polynomials for large orders and values, yielding estimates that align with existing results.