Acta Applicandae Mathematicae最新文献

In this work, we study a class of degenerate Dirichlet problems, whose prototype is

where (Omega ) is a bounded open subset of (mathbb{R}^{N}). (0<gamma <1), (0<theta leq 1) and (0leq beta <1). We prove existence of bounded solutions when (f) and (c) belong to suitable Lebesgue spaces. Moreover, we investegate the existence of renormalized solutions when the function (f) belongs only to (L^{1}(Omega )).

The aim of this paper is to conduct two group reductions for matrix spectral problems simultaneously. We formulate reduced Ablowitz-Kaup-Newell-Segur matrix spectral problems under two local group reductions, and construct associated hierarchies of matrix integrable models, which keep the corresponding zero curvature equations invariant. In this way, various integrable models can be generated via zero curvature equations.

A new Bresse system with hybrid damping coming from elasticity, thermoelasticity, and viscoelasticity, is analyzed. The uniform (exponential) stabilization of semigroup solution is proved under the dynamic response of each hybrid damping effect.

The global well-posedness of the 3D inviscid MHD-Boussinesq system, with large axisymmetric initial data, in the Sobolev space (H^{m}) is given. To overcome difficulties that arise in the time-uniform (H^{1}) estimate, a reformulated system of good unknowns is discovered and an intermediate estimate is shown. Based on the reformulated system, a Beale-Kato-Majda-type criterion of the inviscid MHD-Boussinesq system is verified. Then higher-order estimates are concluded by the classical energy method and estimates of commutators. At last, we show the (H^{m}) norm of the global-in-time solution temporally grows no faster than a four times exponential function ((forall min mathbb{N})).

In this paper we study the dispersive properties related to a model of peridynamic evolution, governed by a non local initial value problem, in the cases of two and three spatial dimensions. The features of the wave propagation characterized by the nontrivial interactions between nonlocality and the regimes of low and high frequencies are studied and suitable numerical investigations are exposed.

We study a one-dimensional cross diffusion system with a free boundary modeling the Physical Vapor Deposition. Using the flatness approach, we show several results of boundary controllability for this system in spaces of Gevrey class functions. One of the main difficulties consists in the physical constraints on the state and on the control. More precisely, the state corresponds to volume fractions of the (n+1) chemical species and to the thickness of the film produced in the process, whereas the controls are the fluxes of the chemical species. We obtain the local controllability in the case where we apply (n+1) nonnegative controls and a controllability result for large time in the case where we apply (n) controls without any sign constraints. We also show in this last case that the controllability may fail for small times. We illustrate these results with some numerical simulations.

In this paper, we prove the existence of the fractal interpolation function corresponding to a general data set using the Rakotch contraction theory and iterated function system. We also prove the existence of the fractal measure supported on the graph of the fractal interpolation function. We emphasize the fact that our theory covers the fractal interpolation theory for finite cases, countably infinite cases, and many more. We establish dimensional results for the graph of the fractal interpolation function for the general data sets.

We revisit the following Moser-Trudinger problem

where (Omega subset mathbb{R}^{2}) is a smooth bounded domain and (lambda >0) is sufficiently small. Qualitative analysis of peaked solutions for Moser-Trudinger type equation in (mathbb{R}^{2}) has been widely studied in recent decades. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions for the corresponding linearized Moser-Trudinger problem by using a variety of local Pohozaev identities combined with some elliptic theory in dimension two. Here we give some fine estimates for the first eigenvalue and eigenfunction of the linearized Moser-Trudinger problem. Since this problem is a critical exponent for dimension two and will lose compactness, we have to obtain some new and technical estimates.

This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ. 319 (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in Arch. Ration. Mech. Anal. 217 (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.

To uncover that the magnetic field mechanism can stabilize electrically conducting turbulent fluids, we investigate the stability of a special two dimensional anisotropic MHD system with vertical dissipation in the horizontal velocity component and partial magnetic damping near a background magnetic field. Since the MHD system has only vertical dissipation in the horizontal velocity and vertical magnetic damping, the stability issue and large time behavior problem of the linearized magneto-hydrodynamic system is not trivial. By performing refined energy estimates on the linear system coupled with a careful analysis of the nonlinearities, the stability of a MHD-type system near a background magnetic field is justified for the initial data belonging to (H^{3}(mathbf{R}^{2})) space. The authors also build the explicit decay rates of the linearized system.