Acta Applicandae Mathematicae最新文献

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.

We consider the classical Holling–Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling–Tanner system is studied by means of symmetry based methods. Lie and (Q)-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.

This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity:

in a bounded domain (Omega subset mathbb{R}^{N}) with (2le Nle 3), where (Din C^{0}([0,infty ))cap C^{2}((0,infty ))) and (Sin C^{2}(bar{Omega }times [0,infty )^{2};mathbb{R}^{Ntimes N})). The global solvability of the system in a natural weak sense is obtained under the conditions that (D(n)ge k_{D}n^{m-1}) and (|S(x,n,c)|le frac{S_{0}(c)}{c^{alpha }}) for all ((x,n,c)in Omega times (0,infty )^{2}) with some (k_{D}>0), (m>frac{3N-2}{2N}), (alpha in [0,1)) and some nondecreasing (S_{0}:(0,infty )rightarrow (0,infty )). Moreover, in the case that (m=frac{3N-2}{2N}) and (alpha in [0,1)), we also get the global weak solutions under smallness assumptions on the initial data (|n_{0}|_{L^{1}(Omega )}) and (|c_{0}|_{L^{infty }(Omega )}).

In this paper, we investigate the vanishing micro-rotation and angular viscosities limit of solutions to the 2D incompressible micropolar equations in a bounded domain with Navier-type boundary conditions satisfied by the velocity field. In a general bounded smooth domain (Omega ), we establish the uniform (H^{2}(Omega )) estimates (independent of the micro-rotation and angular viscosities) of global strong solutions and prove the rate of convergence of viscosity solutions to the inviscid solutions in (C(0,T;H^{1}(Omega ))) for any (T>0).

We study the homogenization limit of the renewal equation with heterogeneous external constraints by means of the two-scale convergence theory. We prove that the homogenized limit satisfies an equation involving non-local terms, which are the consequence of the oscillations in the birth and death terms. We have moreover shown that the numerical approximation of the homogenized equation via the two-scale limit gives an alternative way for the numerical study of the solution of the limiting problem.

We study the long-time behaviour of a non conservative piecewise deterministic measure-valued Markov process modelling the proliferation of an age-and-size structured population, which generalises the “adder” model of bacterial growth. Firstly, we prove the existence of eigenelements of the associated infinitesimal generator, which are used to bring ourselves back to the study of a conservative Markov process using a Doob (h)-transform. Finally, we obtain the exponential ergodicity of the process via drift-minorisation arguments. Specifically, we show the “petiteness” of the compact sets of the state space. This permits to circumvent the difficulties encountered when trying to construct mixing trajectories at a fixed uniform time on an unbounded two-dimensional space with only advection and degenerate jump terms.