Asymptotic Analysis of Spectrum and Stability for One Class of Singularly Perturbed Neutral-Type Time-Delay Systems

In this study, a singularly perturbed linear time-delay system of neutral type is considered. It is assumed that the delay is small of order of a small positive parameter multiplying a part of the derivatives in the system. This system is decomposed asymptotically into two much simpler parameter-free subsystems, the slow and fast ones. Using this decomposition, an asymptotic analysis of the spectrum of the considered system is carried out. Based on this spectrum analysis, parameter-free conditions guaranteeing the exponential stability of the original system for all sufficiently small values of the parameter are derived. Illustrative examples are presented.

Spectra of Elliptic Operators on Quantum Graphs with Small Edges

We consider a general second order self-adjoint elliptic operator on an arbitrary metric graph, to which a small graph is glued. This small graph is obtained via rescaling a given fixed graph γ by a small positive parameter ε. The coefficients in the differential expression are varying, and they, as well as the matrices in the boundary conditions, can also depend on ε and we assume that this dependence is analytic. We introduce a special operator on a certain extension of the graph γ and assume that this operator has no embedded eigenvalues at the threshold of its essential spectrum. It is known that under such assumption the perturbed operator converges to a certain limiting operator. Our main results establish the convergence of the spectrum of the perturbed operator to that of the limiting operator. The convergence of the spectral projectors is proved as well. We show that the eigenvalues of the perturbed operator converging to limiting discrete eigenvalues are analytic in ε and the same is true for the associated perturbed eigenfunctions. We provide an effective recurrent algorithm for determining all coefficients in the Taylor series for the perturbed eigenvalues and eigenfunctions.

Dynamics of solitary waves and periodic waves for a generalized KP-MEW-Burgers equation with damping

<p style='text-indent:20px;'>This paper discusses the existence of solitary waves and periodic waves for a generalized (2+1)-dimensional Kadomtsev-Petviashvili modified equal width-Burgers (KP-MEW-Burgers) equation with small damping and a weak local delay convolution kernel by using the dynamical systems approach, specifically based on geometric singular perturbation theory and invariant manifold theory. Moreover, the monotonicity of the wave speed is proved by analyzing the ratio of Abelian integrals. The upper and lower bounds of the limit wave speed are given. In addition, the upper and lower bounds and monotonicity of the period <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> of traveling wave when the small positive parameter <inline-formula><tex-math id="M2">\begin{document}$ \tau\rightarrow 0 $\end{document}</tex-math></inline-formula> are also obtained. Perhaps this paper is the first discussion on the solitary waves and periodic waves for the delayed KP-MEW-Burgers equations and the Abelian integral theory may be the first application to the study of the (2+1)-dimensional equation.</p>

Concentration behavior of semiclassical solutions for Hamiltonian elliptic system

In this paper, we study the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} -\epsilon^{2}{\it\Delta} \psi +\epsilon \vec{b}\cdot \nabla \psi +\psi+V(x)\varphi=f(|\eta|)\varphi~~\hbox{in}~\mathbb{R}^{N},\\ -\epsilon^{2}{\it\Delta} \varphi -\epsilon \vec{b}\cdot \nabla \varphi +\varphi+V(x)\psi=f(|\eta|)\psi~~\hbox{in}~\mathbb{R}^{N},\\ \end{array} \right. \end{array}$$ where η = (ψ, φ) : ℝN → ℝ2, ϵ is a small positive parameter and b⃗ is a constant vector. We require that the potential V only satisfies certain local condition. Combining this with other suitable assumptions on f, we construct a family of semiclassical solutions. Moreover, the concentration phenomena around local minimum of V, convergence and exponential decay of semiclassical solutions are also explored. In the proofs we apply penalization method, linking argument and some analytical techniques since the local property of the potential and the strongly indefinite character of the energy functional.

Concentration of positive solutions for a class of fractional p-Kirchhoff type equations

We study the existence and concentration of positive solutions for the following class of fractional p-Kirchhoff type problems: $$ \left\{\begin{array}{@{}ll} \left(\varepsilon^{sp}a+\varepsilon^{2sp-3}b \,[u]_{s, p}^{p}\right)(-\Delta)_{p}^{s}u+V(x)u^{p-1}=f(u) & \text{in}\ \mathbb{R}^{3},\\ \noalign{ u\in W^{s, p}(\mathbb{R}^{3}), \quad u>0 & \text{in}\ \mathbb{R}^{3}, \end{array}\right.$$ where ɛ is a small positive parameter, a and b are positive constants, s ∈ (0, 1) and p ∈ (1, ∞) are such that $sp \in (\frac {3}{2}, 3)$ , $(-\Delta )^{s}_{p}$ is the fractional p-Laplacian operator, f: ℝ → ℝ is a superlinear continuous function with subcritical growth and V: ℝ3 → ℝ is a continuous potential having a local minimum. We also prove a multiplicity result and relate the number of positive solutions with the topology of the set where the potential V attains its minimum values. Finally, we obtain an existence result when f(u) = uq−1 + γur−1, where γ > 0 is sufficiently small, and the powers q and r satisfy 2p < q < p* s ⩽ r. The main results are obtained by using some appropriate variational arguments.

Positive Solutions for a Singular Superlinear Fourth-Order Equation with Nonlinear Boundary Conditions

We show the existence of positive solutions for a singular superlinear fourth-order equation with nonlinear boundary conditions. u⁗x=λhxfux, x∈0,1,u0=u′0=0,u″1=0, u⁗1+cu1u1=0, where λ > 0 is a small positive parameter, f:0,∞⟶ℝ is continuous, superlinear at ∞, and is allowed to be singular at 0, and h: [0, 1] ⟶ [0, ∞) is continuous. Our approach is based on the fixed-point result of Krasnoselskii type in a Banach space.

Sign-changing tower of bubbles to an elliptic subcritical equation

This paper is concerned with the following nonlinear elliptic problem involving nearly critical exponent [Formula: see text]: [Formula: see text] in [Formula: see text], [Formula: see text] on [Formula: see text], where [Formula: see text] is a bounded smooth domain in [Formula: see text], [Formula: see text], [Formula: see text] is a small positive parameter. As [Formula: see text] goes to zero, we construct a solution with the shape of a tower of sign changing bubbles.

Numerical Solution of Heat Equation with Singular Robin Boundary Condition

In this work we study the numerical solution of one-dimensional heatdiffusion equation with a small positive parameter subject to Robin boundary conditions. The simulations examples lead us to conclude that the numerical solutionsof the differential equation with Robin boundary condition are very close of theanalytic solution of the problem with homogeneous Dirichlet boundary conditionswhen tends to zero

Singularly Perturbed Fractional Schrödinger Equations with Critical Growth

We are concerned with the following singularly perturbed fractional Schrödinger equation:\left\{\begin{aligned} &\displaystyle{\varepsilon^{2s}}{(-\Delta)^{s}}u+V(x)u=% f(u)&&\displaystyle{\text{in }}{\mathbb{R}^{N}},\\ &\displaystyle u\in{H^{s}}({\mathbb{R}^{N}}),&&\displaystyle u>0{\text{ on }}{% \mathbb{R}^{N}},\end{aligned}\right.where ε is a small positive parameter,{N>2s}, and{{(-\Delta)^{s}}}, with{s\in(0,1)}, is the fractional Laplacian. Using variational technique, we construct a family of positive solutions{{u_{\varepsilon}}\in{H^{s}}({\mathbb{R}^{N}})}which concentrates around the local minima ofVas{\varepsilon\to 0}under general conditions onfwhich we believe to be almost optimal.

A multiplicity result for a fractional Kirchhoff equation in ℝN with a general nonlinearity

In this paper, we deal with the following fractional Kirchhoff equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is a small positive parameter and [Formula: see text] is an odd function satisfying Berestycki–Lions type assumptions. By using minimax arguments, we establish a multiplicity result for the above equation, provided that [Formula: see text] is sufficiently small.