Acta Applicandae Mathematicae最新文献

Exact solutions describing trapped modes in a plane quantum waveguide with a small rigid obstacle are constructed in the form of convergent series in powers of the small parameter characterizing the smallness of the obstacle. The terms of this series are expressed through the solution of the exterior Neumann problem for the Laplace equation describing the flow of unbounded fluid past the inflated obstacle. The exact solutions obtained describe discrete eigenvalues of the problem under certain geometric conditions, and, when the obstacle is symmetric, these solutions describe embedded eigenvalues. For obstacles symmetric with respect to the centerline of the waveguide, the existence of embedded trapped modes is known (due to the decomposition trick of the domain of the corresponding differential operator) even without the smallness assumption. We construct these solutions in an explicit form for small obstacles. For obstacles symmetric with respect to the vertical axis, we find embedded trapped modes for a specific vertical displacement of the obstacle.

In this paper we obtain the local regularity estimates in Besov spaces of weak solutions for a class of elliptic obstacle problems with variable exponents (p(x)). We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality in the following form

under some proper assumptions on the function (p(x)), (A), (varphi ) and (F). Moreover, we would like to point out that our results improve the known results for such problems.

In this paper, we consider the following Chern-Simons-Schrödinger system

where (u in H^{1}(mathbb{R}^{2})), (p > 4), (A_{alpha }: mathbb{R}^{2} rightarrow mathbb{R}) are the components of the gauge potential, (N: mathbb{R}^{2} rightarrow mathbb{R}) is a neutral scalar field, (V(x)) is a periodic potential function, the parameters (kappa , q>0) represent the Chern-Simons coupling constant and the Maxwell coupling constant, respectively, and (e>0) is the coupling constant. We prove that system ((P)) has a nontrivial solution by using a new infinite-dimensional linking theorem.

In this paper, we study higher order hyperbolic pseudo-differential equations with variable multiplicities. We work in arbitrary space dimension and we assume that the principal part is time-dependent only. We identify sufficient conditions on the roots and the lower order terms (Levi conditions) under which the corresponding Cauchy problem is (C^{infty }) well-posed. This is achieved via transformation into a first order system, reduction into upper-triangular form and application of suitable Fourier integral operator methods previously developed for hyperbolic non-diagonalisable systems. We also discuss how our result compares with the literature on second and third order hyperbolic equations.

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.