Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber Ω. This acoustic wave equation is coupled on a boundary interface Γ₀ to a two dimensional system of thermoelasticity: this thermoelastic PDE is composed in part of a structural beam or plate equation, which governs the vibrations of flexible wall portion Γ₀ of the chamber Ω. Moreover, this elastic dynamics is coupled to a heat equation which also evolves on Γ₀, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface Γ₀, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of any additive feedback dissipation on the hard walls Γ₁ of the boundary . Under a certain geometric assumption on Γ₁, an assumption which has appeared in the literature in connection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.     

On a nonlinear wave equation with boundary damping

This paper is concerned with the existence and decay of solutions of the mixed problem for the nonlinear wave equation with boundary conditions Here, Ω is an open bounded set of with boundary Γ of class C²; Γ is constituted of two disjoint closed parts Γ₀ and Γ₁ both with positive measure; the functions μ(t), f(s), g(s) satisfy the conditions μ(t) ≥ μ₀ > 0, f(s) ≥ 0, g(s) ≥ 0 for t ≥ 0, s ≥ 0 and h(x,s) is a real function where x ∈ Γ₁, ν(x) is the unit outward normal vector at x ∈ Γ₁ and α, β are non‐negative real constants. Assuming that h(x,s) is strongly monotone in s for each x ∈ Γ₁, it is proved the global existence of solutions for the previous mixed problem. For that, it is used in the Galerkin method with a special basis, the compactness approach, the Strauss approximation for real functions and the trace theorem for nonsmooth functions. The exponential decay of the energy is derived by two methods: by using a Lyapunov functional and by Nakao's method. Copyright © 2013 John Wiley & Sons, Ltd.     

Optimal decay rate of the bipolar Euler–Poisson system with damping in dimension three

By rewriting a bipolar Euler–Poisson equations with damping into a Euler equation with damping coupled with a Euler–Poisson equation with damping and using a new spectral analysis, we obtain the optimal decay results of the solutions in L² norm. More precisely, the velocities u₁ and u₂ decay at the L²?rate , which is faster than the normal L²‐rate for the heat equation and the Navier–Stokes equations. In addition, the decay rates of the disparities of two densities ρ₁?ρ₂ and the disparity of two velocities u₁?u₂ could reach to and in L² norm, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

A conformal mapping algorithm for the Bernoulli free boundary value problem

We propose a new numerical method for the solution of the Bernoulli free boundary value problem for harmonic functions in a doubly connected domain D in where an unknown free boundary Γ₀ is determined by prescribed Cauchy data on Γ₀ in addition to a Dirichlet condition on the known boundary Γ₁. Our main idea is to involve the conformal mapping method as proposed and analyzed by Akduman, Haddar, and Kress for the solution of a related inverse boundary value problem. For this, we interpret the free boundary Γ₀ as the unknown boundary in the inverse problem to construct Γ₀ from the Dirichlet condition on Γ₀ and Cauchy data on the known boundary Γ₁. Our method for the Bernoulli problem iterates on the missing normal derivative on Γ₁ by alternating between the application of the conformal mapping method for the inverse problem and solving a mixed Dirichlet–Neumann boundary value problem in D. We present the mathematical foundations of our algorithm and prove a convergence result. Some numerical examples will serve as proof of concept of our approach. Copyright © 2015 John Wiley & Sons, Ltd.     

On the solutions of a higher‐order difference equation in terms of generalized Fibonacci sequences

This paper deals with the solutions, stability character, and asymptotic behavior of the difference equation where and the initial values x_?k,x_{?k + 1},…,x₀ are nonzero real numbers, such that their solutions are associated to Horadam numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonlinear diffusion equations as asymptotic limits of Cahn‐Hilliard systems on unbounded domains via Cauchy's criterion

This paper develops an abstract theory for subdifferential operators to give existence and uniqueness of solutions to the initial‐boundary problem P for the nonlinear diffusion equation in an unbounded domain ( ), written as which represents the porous media, the fast diffusion equations, etc, where β is a single‐valued maximal monotone function on , and T>0. In Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) existence and uniqueness of solutions for P were directly proved under a growth condition for β even though the Stefan problem was excluded from examples of P . This paper completely removes the growth condition for β by confirming Cauchy's criterion for solutions of the following approximate problem _ε with approximate parameter ε>0: which is called the Cahn‐Hilliard system, even if ( ) is an unbounded domain. Moreover, it can be seen that the Stefan problem excluded from Kurima and Yokota (J Differential Equations 2017; 263:2024‐2050 and Adv Math Sci Appl 2017; 26:221‐242) is covered in the framework of this paper.     

The form of solutions and periodicity for some systems of third‐order rational difference equations

In this paper, we deal with the system that has solutions and the periodicity character of the following systems of rational difference equations with order three with initial conditions x_?2,x_?1,x₀,y_?2,y_?1, and y₀ that are arbitrary nonzero real numbers. Some numerical examples will be given to illustrate our results. Copyright © 2015 John Wiley & Sons, Ltd.     

Dynamics of a three‐dimensional systems of rational difference equations

We investigate in this paper the solutions and the periodicity of the following rational systems of difference equations of three‐dimensional with initial conditions x_?2,x_?1,x₀,y_?2,y_?1,y₀,z_?2,z_?1andz₀ are nonzero real numbers. Copyright © 2015 John Wiley & Sons, Ltd.     

Max‐type system of difference equations with positive two‐periodic sequences

This paper deals with the form and the periodicity of the solutions of the max‐type system of difference equations where , and are positive two‐periodic sequences and initial values x₀, x _{? 1}, y₀, y _{? 1} ∈ (0, + ∞ ). Copyright © 2014 John Wiley & Sons, Ltd.     

Solution form of a higher‐order system of difference equations and dynamical behavior of its special case

The solution form of the system of nonlinear difference equations where the coefficients a ,b ,α ,β and the initial values x  _? i ,y  _? i ,i ∈{0,1,…,k } are non‐zero real numbers, is obtained. Furthermore, the behavior of solutions of the aforementioned system when p  = 1 is examined. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiplicity of positive solutions to boundary blow‐up problem with variable exponent and sign‐changing weights

In this paper, we consider the elliptic boundary blow‐up problem where Ω is a bounded smooth domain of are positive continuous functions supported in disjoint subdomains Ω₊,Ω_? of Ω, respectively, a ₊ vanishes on the boundary of satisfies p (x )≥1 in Ω,p (x ) > 1 on ? Ω and , and ε is a parameter. We show that there exists ε ^?>0 such that no positive solutions exist when ε > ε ^?, while a minimal positive solution u _ε exists for every ε ∈(0,ε ^?). Under the additional hypotheses that is a smooth N ? 1‐dimensional manifold and that a ₊,a _? have a convenient decay near Γ, we show that a second positive solution v _ε exists for every ε ∈(0,ε ^?) if , where N ^?=(N + 2)/(N ? 2) if N > 2 and if N = 2. Our results extend that of Jorge Garcá‐Melián in 2011, where the case that p > 1 is a constant and a ₊>0 on ? Ω is considered. Copyright © 2016 John Wiley & Sons, Ltd.     

The normalizer of

In this paper, the normalizers of , for k = 2 or 3, in the groups Γ^k of modular group Γ are given by using the concept of Γ^k axes of the set of left cosets of , on which Γ^k acts transitively, in Γ^k.     

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: where with b² > a² > 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p₁,p₂,p₃ in to get results for the global (in time) existence of small data solutions.     

On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

In this paper, we prove the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ? < π ∕ 2 and γ₀ > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ₀ > 0 for given 0 < ? < π ∕ 2. We also prove the maximal L_p ? L_q regularity theorem of the nonstationary Stokes problem as an application of the ‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable . Copyright © 2014 John Wiley & Sons, Ltd.     

Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion‐type source term. Based on a low‐frequency and high‐frequency decomposition, Green's function method and the classical energy method, we not only obtain L² time‐decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u₀(x) satisfies the smallness condition on , but not on . Furthermore, by taking a time‐frequency decomposition, we obtain the optimal decay estimates of solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Global well‐posedness of classical solutions to 3D compressible magnetohydrodynamic equations with large potential force

We consider the Cauchy problem of isentropic compressible magnetohydrodynamic equations with large potential force in . When the initial data (ρ₀,u₀,H₀) is of small energy, we investigate the global well‐posedness of classical solutions where the flow density is allowed to contain vacuum states.     

Qualitative behavior of two systems of higher‐order difference equations

In this paper, we study the qualitative behavior of following two systems of higher‐order difference equations: and where the parameters α,β,γ,α₁,β₁,γ₁,a,b,c,a₁,b₁,andc₁ and the initial conditions x₀, x_?1, ?, x_?k, y₀, y_?1 ,?, y_?k are positive real numbers. More precisely, we study the equilibrium points, local asymptotic stability, instability, global asymptotic stability of equilibrium points, and rate of convergence of positive solutions that converges to the equilibrium point P₀=(0,0) of these systems. Some numerical examples are given to verify our theoretical results. These examples are experimental verification of our theoretical discussions. Copyright © 2015 John Wiley & Sons, Ltd.     

Analyticity for a class of nonlocal Kuramoto‐Sivashinsky equations arising in interfacial electrohydrodynamics

In this article, we study the analyticity properties of solutions of the nonlocal Kuramoto‐Sivashinsky equations, defined on 2π‐periodic intervals, where ν is a positive constant; μ is a nonnegative constant; p is an arbitrary but fixed real number in the interval [3,4); and is an operator defined by its symbol in Fourier space, with be the Hilbert transform. We establish spatial analyticity in a strip around the real axis for the solutions of such equations, which possess universal attractors. Also, a lower bound for the width of the strip of analyticity is obtained.     

Solutions for perturbed biharmonic equations with critical nonlinearity

In this paper, we study the perturbed biharmonic equations where Δ² is the biharmonic operator, is the Sobolev critical exponent, p ∈ (2,2 ^{* *} ), P(x), and Q(x) are bounded positive functions. Under some given conditions on V, we prove that the problem has at least one nontrivial solution provided that and that for any , it has at least n ^* pairs solutions if , where and are sufficiently small positive numbers. Moreover, these solutions u_ε → 0 in as ε → 0. Copyright © 2013 The authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.     

On the Difference equation

The main objective of this paper was to study the global stability of the positive solutions and the periodic character of the difference equation where the parameters a , b , c , d , and e are positive real numbers and the initial conditions x _?t ,x _{?t + 1},...,x _?1, x ₀ are positive real numbers where t = m a x {l ,k ,s }. Some numerical examples will be given to explicate our results. Copyright © 2016 John Wiley & Sons, Ltd.