Search

Traveling wave solutions are a class of invariant solutions which are critical for shallow water wave equations. In this paper, traveling wave solutions for two perturbed KP-MEW equations with a local delay convolution kernel are examined. The model equation is reduced to a planar near-Hamiltonian system via geometric singular perturbation theorem, and the qualitative properties of the corresponding unperturbed system are analyzed by using dynamical system approach. The persistence of the bounded traveling wave solutions for the perturbed KP-MEW equations with delay is investigated. By using a criterion for the monotonicity of ratio of two Abelian integrals and Melnikov’s method, the existence of kink (anti-kink) wave solutions and periodic wave solutions of the model equation are established. The result shows that the delayed KP-MEW equations with positive perturbation and the one with negative perturbation exhibit completely diverse dynamical properties. These new findings greatly enrich the understanding of dynamical properties of the traveling wave solutions of perturbed nonlinear wave equations with local delay convolution kernel. Numerical experiments further confirm and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we construct the solutions to the following nonlinear Schrödinger system - ϵ 2 Δ u + P (x) u = μ 1 u p + β u p - 1 2 v p + 1 2 in R N , - ϵ 2 Δ v + Q (x) v = μ 2 v p + β u p + 1 2 v p - 1 2 in R N , where 3 < p < + ∞ , N ∈ { 1 , 2 } , ϵ > 0 is a small parameter, the potentials P, Q satisfy 0 < P 0 ≤ P (x) ≤ P 1 and Q(x) satisfies 0 < Q 0 ≤ Q (x) ≤ Q 1 . We construct the solution for attractive and repulsive cases. When x 0 is a local maximum point of the potentials P and Q and P (x 0) = Q (x 0) , we construct k spikes concentrating near the local maximum point x 0 . When x 0 is a local maximum point of P and x ¯ 0 is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x 0 and m spikes of v concentrating at the local maximum point x ¯ 0 when x 0 ≠ x ¯ 0. This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N = 3 , p = 3 . [ABSTRACT FROM AUTHOR]

In this paper we consider the following focusing mass-critical Hartree equation with a defocusing perturbation and harmonic potential i ∂ t ψ = - Δ ψ + | x | 2 ψ - (| x | - 2 ∗ | ψ | 2 ) ψ + ε | ψ | p - 2 ψ , in R + × R N , where N ≥ 3 , 2 < p < 2 ∗ = 2 N / (N - 2) and ε > 0 . We mainly focus on the normalized ground state solitary waves of the form ψ (t , x) = e i μ t u ε , ρ (x) , where u ε , ρ (x) is radially symmetric-decreasing and ∫ R N | u ε , ρ | 2 d x = ρ . Firstly, we prove the existence and nonexistence of normalized ground states under the L 2 -subcritical, L 2 -critical ( p = 4 / N + 2 ) and L 2 -supercritical perturbations. Secondly, we characterize perturbation limit behaviors of ground states u ε , ρ as ε → 0 + and find that the ε -blow-up phenomenon happens for ρ ≥ ρ c = ‖ Q ‖ L 2 2 , where Q is a positive radially symmetric ground state of - Δ u + u - (| x | - 2 ∗ | u | 2) u = 0 in R N . We prove that ∫ R N | ∇ u ε , ρ (x) | 2 d x ∼ ε - 4 N (p - 2) + 4 for ρ = ρ c and 2 < p < 2 ∗ , while ∫ R N | ∇ u ε , ρ | 2 d x ∼ ε - 4 N (p - 2) - 4 for ρ > ρ c and 4 / N + 2 < p < 2 ∗ , and obtain two different blow-up profiles corresponding to two limit equations. Finally, we study the limit behaviors as ε → + ∞ , which corresponds to a Thomas–Fermi limit. The limit profile is given by the Thomas–Fermi minimizer u TF = μ TF - | x | 2 + 1 p - 2 , where μ TF is a suitable Lagrange multiplier with exact value. Moreover, we obtain a sharp vanishing rate for u ε , ρ that ‖ u ε , ρ ‖ L ∞ ∼ ε - N N (p - 2) + 4 as ε → + ∞ . [ABSTRACT FROM AUTHOR]

This paper is devoted to establishing the convergence rate of the hypersonic similarity for the inviscid steady irrotational Euler flow over a two-dimensional Lipschitz slender wedge in B V ∩ L 1 space. The rate we established is the same as the one predicted by Newtonian-Busemann law (see (3.29) in [2, p 67] for more details) as the incoming Mach number M ∞ → ∞ for a fixed hypersonic similarity parameter K. The hypersonic similarity, which is also called the Mach-number independence principle, is equivalent to the following Van Dyke's similarity theory: For a given hypersonic similarity parameter K, when the Mach number of the flow is sufficiently large, the governing equations after the scaling are approximated by a simpler equation, that is called the hypersonic small-disturbance equation. To achieve the convergence rate, we approximate the curved boundary by piecewisely straight lines and find a new Lipschitz continuous map P h such that the trajectory can be obtained by piecing together the Riemann solutions near the approximated boundary. Next, we derive the L 1 difference estimates between the approximate solutions U h , ν (τ) (x , ·) to the initial-boundary value problem for the scaled equations and the trajectories P h (x , 0) (U 0 ν) by piecing together all the Riemann solvers. Then, by the uniqueness and the compactness of P h and U h , ν (τ) , we can further establish the L 1 estimates of order τ 2 between the solutions to the initial-boundary value problem for the scaled equations and the solutions to the initial-boundary value problem for the hypersonic small-disturbance equations, if the total variations of the initial data and the tangential derivative of the boundary are sufficiently small. Based on it, we can further establish a better convergence rate by considering the hypersonic flow past a two-dimensional Lipschitz slender wing and show that for the length of the wing with the effect scale order O (τ - 1) , that is, the L 1 convergence rate between the two solutions is of order O (τ 3 2) under the assumption that the initial perturbation has compact support. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence of mild solutions as well as uniform h-stability for non-autonomous evolution equations in a Banach space, where the operators in linear part (possibly unbounded) depend on time t and generate an evolution family. Under some estimates on the perturbation terms, we discuss the global uniform h-stability, the uniform h-stability and the global practical uniform h-stability using Gronwall integral inequalities. Some examples are provided to show the validity of the obtained results. [ABSTRACT FROM AUTHOR]

The paper investigates the robustness of pullback attractors and pullback exponential attractors of the non-autonomous Kirchhoff wave models with strong nonlinear damping: u tt - (1 + ϵ ‖ ∇ u ‖ 2 ) Δ u - σ (‖ ∇ u ‖ 2) Δ u t + f (u) = g (x , t) , where ϵ ∈ [ 0 , 1 ] is an extensibility parameter. It shows that when the growth exponent p of the nonlinearity f(u) is up to the supercritical range: 1 ≤ p < p ∗ ∗ ≡ N + 4 (N - 4) + , (i) the related process has a pullback attractor A ϵ in natural energy space H = (H 0 1 ∩ L p + 1) × L 2 for each ϵ , and it is upper semicontinuous on the perturbation ϵ ; (ii) the related process has a partially strong pullback exponential attractor M ϵ for each ϵ , and it is Hölder continuous on ϵ ∈ [ 0 , 1 ] . These results deepen and extend those in recent literatures (Chueshov in J Diff Equ 252:1229–1262, 2012; Ding et al. in Appl Math Lett 76:40–45, 2018; Wang and Zhong in Discrete Contin Dyn Syst 7:3189–3209, 2013). The main novelty of this paper is that it provides a new method to investigate the upper semicontinuity of pullback attractors and the stability of pullback exponential attractors in supercritical nonlinearity case. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability problem on perturbation near a physically steady state solution of the 3D generalized incompressible magnetohydrodynamic system in Lei-Lin space. The global stability and analytic estimates for small perturbation are established by the semigroup method in the critical space χ 1 - 2 α (R 3) with 1 2 ≤ α ≤ 1 , where linear terms from perturbation incur much difficulty. By introducing a diagonalization process we successfully eliminate the linear terms. Then, by virtue of the analytic estimates for a solution, the temporal decay rate (1 + t) - (5 4 α - 1) of the global solution is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in studying the existence or non-existence of solutions for a class of elliptic problems involving the N-Laplacian operator in the whole space. The nonlinearity considered involves critical Trudinger–Moser growth. Our approach is non-variational, and in this way we can address a wide range of problems not yet contained in the literature. Even W 1 , N (R N) ↪ L ∞ (R N) failing, we establish ‖ u ‖ L ∞ (R N) ≤ C ‖ u ‖ W 1 , N (R N) Θ (for some Θ > 0 ), when u is a solution. To conclude, we explore some asymptotic properties. [ABSTRACT FROM AUTHOR]

In this paper, a classical Stefan problem is studied. It is assumed that a small, time-dependent heat influx is present at the boundary, and that the initial values are small. By using a multiple timescales perturbation approach, it is shown analytically (most likely for the first time in the literature) how the moving interface and its stability are influenced by the time-dependent heat influx at the boundary and by the initial conditions. Accurate approximations of the solution of the problem are constructed, which are valid on long timescales. The constructed approximations turn out to agree very well with solutions of problems for which similarity solutions are available (in numerical form). [ABSTRACT FROM AUTHOR]

This paper studies bistable wavefronts of a diffusive time-periodic Lotka-Volterra system. We obtain a new condition for the existence, uniqueness and stability of bistable time-periodic traveling waves. This condition is sharp and greatly improves the result established in the reference (X. Bao and Z. Wang, Journal of Differential Equations, 255(2013) 2402–2435). An example is given to demonstrate our consequence. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of time periodic solutions for the quantum version of the nonlinear Schrödinger equation. The uniqueness can also be proved by applying the Runge-Kutta-Matsaev’s theory Bai (J. Math. Anal. Appl. 444(1):721–736, 2016) and Phragmén-Lindelöf method Li and Li (Acta Math. Sci. Ser. A. 40(5):1248–1258, 2020). In particular, the steady equation and the quantum effects are taken into account which play an exceptionally important role in time decay estimates, especially in the the case when the equation is periodic with respect to the Schrödinger norm. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of response solution (i.e., quasi-periodic solution with the same frequency as the forcing) for the quasi-periodically forced generalized ill-posed Boussinesq equation: y tt (t , x) = μ y xxxx + y xx + y 3 + ε f (ω t , x) xx , x ∈ [ 0 , π ] , μ > 0 , subject to the hinged boundary conditions y (t , 0) = y (t , π) = y xx (t , 0) = y xx (t , π) = 0 , where ω = (1 , α) with α being any irrational numbers. The proof is based on a modified Kolmogorov–Arnold–Moser (KAM) iterative scheme. We will, at every step of KAM iteration, construct a symplectic transformation in a such way that the composition of these transformations reduce the original system to a new system which possesses zero as equilibrium. Note that we allow α to be any irrational numbers, and thus the frequency ω = (1 , α) is beyond Diophantine or Brjuno frequency, which we call as Liouvillean frequency. Moreover, the model under consideration is ill-posed and has complicated Hamiltonian structure. This makes homological equations appearing in KAM iteration are different from the ones in the classical infinite-dimensional KAM theory. The result obtained in this paper strengthens the existing results in the literature where the system is well-posed or the forcing frequency is assumed to be Diophantine. [ABSTRACT FROM AUTHOR]

In this paper, we study the following Choquard equations with small perturbation f − Δ u + V (x) u = (I α * | u | p) | u | p − 2 u + f (x) , x ∈ R N. where N ≥ 3 and Iα denotes the Riesz potential. As is known that the above equation has a ground state uα and a bound state vα by fibering maps (see [22] or [23]), our aim is to show that for fixed p ∈ (1 , N N − 2) , uα and vα converge to a ground state and a bound state of the limiting local problem respectively, as α → 0. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the multiplicity of solutions for a class of noncooperative Schrödinger systems in R N . The systems involves a variable exponent elliptic operators with critical nonlinearity. By applying the Limit Index Theory developed by Li (Nonlinear Anal 25, 1371–1389, 1995) and utilizing a version of the concentration-compactness principle and the principle of symmetric criticality of Krawcewicz and Marzantowicz, we obtain a sequence of solutions under appropriate assumptions. [ABSTRACT FROM AUTHOR]

In this paper we adopt an alternative, analytical approach to Arnol'd problem [4] about the existence of closed and embedded K-magnetic geodesics in the round 2-sphere S 2 , where K : S 2 → R is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo's theorem [21] and Bottkoll results [7]. [ABSTRACT FROM AUTHOR]

This paper deals with a class of nonlinear scalar field equations with an inhomogeneous perturbation. Two positive solutions were obtained using the variational methods. [ABSTRACT FROM AUTHOR]

In this paper, the Schrödinger operator with a localized potential in a multidimensional cylinder is considered. The boundary of the cylinder is split into three parts, two of which are "sleeves" going to infinity, and the third (central) part is located between them. On the sleeves and the central part, respectively, the Neumann and Dirichlet boundary conditions are posed. We examine the situation where the distance between the sleeves increases. We assume that the same Schrödinger operator in the same cylinder endowed with the Dirichlet condition on the whole boundary has an isolated double eigenvalue. We show that for a sufficiently large distance between the sleeves, this double eigenvalue splits into a pair of resonances of the original operator. For these resonances, we explicitly obtain the first terms of their asymptotic expansions and describe the behavior of the imaginary part of the resonances. [ABSTRACT FROM AUTHOR]

In this paper, the speed sign of the traveling wave to the bistable Lotka–Volterra competitive lattice system is investigated via the upper–lower solution method as well as the comparison principle. We provide an interval estimation for the bistable speed firstly. Two comparison principles are further established to obtain new conditions to the determinacy of the sign of the bistable speed. To our knowledge, this is the first investigation to the lattice system for the propagation direction. [ABSTRACT FROM AUTHOR]

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued $$L^\infty (\varOmega ,\mathbb {R}^{N\times N})$$ -controls in its coefficients and a nonlinear equation of Hammerstein type. Using the direct method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization. [ABSTRACT FROM AUTHOR]

In this paper we give the complete description of the structure of compact global (forward) attractors for non-autonomous perturbations of autonomous gradient-like dynamical systems under the assumption that the original autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed non-autonomous (in particular τ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular the perturbed systems has the same number of τ -periodic, quasi-periodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff solutions). It is shown the compact global (forward) attractor of non-autonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections. [ABSTRACT FROM AUTHOR]

We consider reaction–diffusion equations on closed surfaces in R 3 having genus 1. Stable nonconstant stationary solutions are often called patterns. The purpose of this paper is to construct closed surfaces together with patterns having as many critical points as one wants. [ABSTRACT FROM AUTHOR]

In this paper we consider the quasilinear critical problem - Δ p u = λ u q - 2 u + u p ⋆ - 2 u in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N with smooth boundary, - Δ p u : = div (∇ u p - 2 ∇ u) is the p-Laplacian, N ≥ 3 , 1 < p ≤ q < p ⋆ , p ⋆ : = N p / N - p , and λ > 0 is a parameter. We investigate the multiplicity of sign-changing solutions to the problem and find the phenomenon depending on the positive solutions. Precisely, we show that the problem admits infinitely many pairs of sign-changing solutions when a positive solution exists. These results complete those obtained in Schechter and Zou (On the Brézis–Nirenberg problem, Arch Rational Mech Anal 197:337–356, 2010) for the cases p = q = 2 and N ≥ 7 , and in Azorero and Peral (Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues, Commun Partial Differ Equ 12:1389–1430, 1987) for the case of one positive solution. Our approach is based on variational methods combining upper-lower solutions and truncation techniques, and flow invariance arguments. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global existence and stability of strong shock front for one-dimensional piston problem. We use the exothermically reacting Euler equations as a mathematical model to describe the gas motion. Under the assumptions that the total variations of initial data and the perturbation of piston velocity are sufficiently small, we establish the global existence and asymptotic behavior of entropy solutions including a strong shock front without restriction on the strength. A modified fractional wave front tracking scheme is developed and a modified Glimm-type functional is carefully designed. [ABSTRACT FROM AUTHOR]

This paper concerns ecological invasion phenomenon of species based on the diffusive Lotka–Volterra competition model. We investigate the spreading speed (or the minimal wave speed of traveling waves) selection to the model and concentrate on the conjecture raised by Roques et al. (J Math Biol 71(2):465–489, 2015). By using an abstract implicit function theorem in a weighted functional space coupled with a perturbation technique, we not only prove this conjecture, but also show that the fast decay behavior of the first species is necessary and sufficient for the nonlinear speed selection of the whole system. This may lead to further significant results on the answer to the original Hosono's conjecture, a problem that has been outstanding for more than twenty years. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of elliptic problems of p-Kirchhoff type with critical exponent in bounded domains and new results as regards the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and variational method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the boundary-value problem on the Boltzmann equation in bounded domains with diffuse-reflection boundary where the boundary temperature is time-periodic. We establish the existence of time-periodic solutions with the same period for both hard and soft potentials, provided that the time-periodic boundary temperature is sufficiently close to a stationary one which has small variations around a positive constant. The dynamical stability of time-periodic profiles is also proved under small perturbations, and this in turn yields the non-negativity of the profile. For the proof, we develop new estimates in the time-periodic setting. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to study the long time behavior, in terms of upper semicontinuous property of trajectory attractor of perturbed 3D Navier-Stokes equations by adding a high-order term of hyperdissipation ( l-Lapalcian). We show that the trajectory attractor of the perturbed system converges to the trajectory attractor of the 3D Navier-Stokes system as time tends to infinity and the artificial dissipation parameter approaches zero. [ABSTRACT FROM AUTHOR]

This paper is the first attempt to systematically study properties of the effective Hamiltonian H¯ arising in the periodic homogenization of some coercive but nonconvex Hamilton-Jacobi equations. Firstly, we introduce a new and robust decomposition method to obtain min-max formulas for a class of nonconvex H¯. Secondly, we analytically and numerically investigate other related interesting phenomena, such as “quasi-convexification” and breakdown of symmetry, of H¯ from other typical nonconvex Hamiltonians. Finally, in the appendix, we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex Hamilton-Jacobi equations. Some conjectures and problems are also proposed. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of multi-bump solutions for the following Kirchhoff equation - a + b ∫ R 3 | ∇ u | 2 d x Δ u + u = (1 - ε q (x)) | u | p - 2 u , x ∈ R 3 , where a , b > 0 , 2 < p < 6 and q (x) ∈ C (R 3 , R +) satisfies some suitable conditions. By using the Lyapunov–Schmidt reduction method, we extend the results in [14] to the Kirchhoff problem. [ABSTRACT FROM AUTHOR]

We investigate the following problem - div (v (x) | ∇ u | m - 2 ∇ u) + V (x) | u | m - 2 u = | x | - θ ∗ | u | b | x | α | u | b - 2 | x | α u + λ | x | - γ ∗ | u | c | x | β | u | c - 2 | x | β u in R N , where b , c , α , β > 0 , θ , γ ∈ (0 , N) , N ≥ 3 , 2 ≤ m < ∞ and λ ∈ R . Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following nonlinear Schrödinger system in $$\mathbb {R}^{3}$$ where $$\mu _{1}>0$$ , $$\mu _{2}>0$$ , $$2\le p,q\le 5$$ , $$p+q\le 6$$ and $$\beta \in \mathbb {R}$$ is a coupling constant. We show that the nonlinear Schrödinger system has infinitely many non-radial positive segregated solutions. [ABSTRACT FROM AUTHOR]

Under suitable conditions on a family (I(t))t≥ 0 of Lipschitz mappings on a complete metric space, we show that, up to a subsequence, the strong limit S (t) : = lim n → ∞ (I (t 2 − n)) 2 n exists for all dyadic time points t, and extends to a strongly continuous semigroup (S(t))t≥ 0. The common idea in the present approach is to find conditions on the generating family (I(t))t≥ 0, which can be transferred to the semigroup. The construction relies on the Lipschitz set, which is invariant under iterations and allows to preserve Lipschitz continuity to the limit. Moreover, we provide a verifiable condition which ensures that the infinitesimal generator of the semigroup is given by lim h ↓ 0 I (h) x − x h , whenever this limit exists. The results are illustrated with several examples of nonlinear semigroups such as robustifications and perturbations of linear semigroups. [ABSTRACT FROM AUTHOR]

We consider the stability analysis of a two-dimensional model for post-burn contraction. The model is based on morphoelasticity for permanent deformations and combined with a chemical-biological model that incorporates cellular densities, collagen density, and the concentration of chemoattractants. We formulate stability conditions depending on the decay rate of signaling molecules for both the continuous partial differential equations-based problem and the (semi-)discrete representation. We analyze the difference and convergence between the resulting spatial eigenvalues from the continuous and semi-discrete problems. [ABSTRACT FROM AUTHOR]

Four classical examples on mathematical modeling in the life-sciences are summarized. These include Turing's diffusion-reaction systems in morphogenesis, Hodkin's and Huxley's model on the initiation and propagation of action potentials in a nerve fibre, first rigorous chemotaxis models, and the mathematical analysis of molecular sequence characteristics. [ABSTRACT FROM AUTHOR]

A system with equation and dynamic boundary condition of Cahn–Hilliard type is considered. This system comes from a derivation performed in Liu–Wu (Arch. Ration. Mech. Anal., 233:167–247, 2019) via an energetic variational approach. Actually, the related problem can be seen as a transmission problem for the phase variable in the bulk and the corresponding variable on the boundary. The asymptotic behavior as the coefficient of the surface diffusion acting on the boundary phase variable goes to 0 is investigated. By this analysis we obtain a forward-backward dynamic boundary condition at the limit. We can deal with a general class of potentials having a double-well structure, including the non-smooth double-obstacle potential. We illustrate that the limit problem is well-posed by also proving a continuous dependence estimate. Moreover, in the case when the two graphs, in the bulk and on the boundary, exhibit the same growth, we show that the solution of the limit problem is more regular and we prove an error estimate for a suitable order of the diffusion parameter. [ABSTRACT FROM AUTHOR]

We study the stationary Keller-Segel chemotaxis models with logistic cellular growth over a one-dimensional region subject to the Neumann boundary condition. We show that nonconstant solutions emerge in the sense of Turing's instability as the chemotaxis rate $${\chi}$$ surpasses a threshold number. By taking the chemotaxis rate as the bifurcation parameter, we carry out bifurcation analysis on the system to obtain the explicit formulas of bifurcation values and small amplitude nonconstant positive solutions. Moreover, we show that solutions stay strictly positive in the continuum of each branch. The stabilities of these steady-state solutions are well studied when the creation and degradation rate of the chemical is assumed to be a linear function. Finally, we investigate the asymptotic behaviors of the monotone steady states. We construct solutions with interesting patterns such as a boundary spike when the chemotaxis rate is large enough and/or the cell motility is small. [ABSTRACT FROM AUTHOR]

Cholera is a water- and food-borne infectious disease caused by V. cholerae. To investigate multiple effects of human behavior change, seasonality and spatial heterogeneity on cholera spread, we propose a reaction-advection-diffusion model that incorporates human hosts and aquatic reservoir of V. cholerae. We derive the basic reproduction number R 0 for this system and then establish a threshold type result on its global dynamics in terms of R 0 . Further, we show that the bacterial loss at the downstream end of the river due to water flux can reduce the disease risk, and describe the asymptotic behavior of R 0 for small and large diffusion in a special case (where the diffusion rates of infected human and the pathogen are constant). We also study the transmission dynamics at the early stage of cholera outbreak numerically, and find that human behavior change may lower the infection level and delay the disease peak. Moreover, the relative rate of bacterial loss, together with convection rate, plays an important role in identifying the severely infected areas. Meanwhile spatial heterogeneity may dilute or amplify cholera infection, which in turn would increase the complexity of disease spread. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Schrödinger equation with steep potential well and sublinear perturbation. By virtue of variational methods, the existence criteria of infinitely many nontrivial high or small energy solutions are established. In addition, the phenomenon of the concentration of solutions is also explored. We also give some examples to demonstrate the main results. [ABSTRACT FROM AUTHOR]

We consider the elliptic system problems arising from the Maxwell-Chern-Simons model. In this system, there are two important parameters related to Chern-Simons mass scale and electric charge. Under almost optimal conditions on these two parameters, we show the existence of nontopological condensates with magnetic field concentrated at multi-bubbling points. [ABSTRACT FROM AUTHOR]

In this article, we present a highly-accurate wavelet-based approximation to study and analyze the physical and numerical aspects of two-parameter singularly perturbed problems with Robin boundary conditions. To explore the swiftly changing behavior of such problems, we have used a special type of non-uniform mesh known as Shishkin mesh. Using Shishkin mesh with the Haar wavelet scheme contains a novelty in itself. We comprehensively explain an approach to solve the Robin boundary conditions involving the proposed Haar wavelet scheme. Through rigorous analysis, the order of convergence of the present scheme is shown quadratic and linear in the spatial and temporal directions, respectively. The robustness and proficiency of the contributed scheme are conclusively demonstrated with three test examples. Irrespective of the problem's geometry, the proposed method is highly accurate and very economical. [ABSTRACT FROM AUTHOR]

The mass-in-mass (MiM) lattice consists of an infinite chain of identical beads that are both nonlinearly coupled to their nearest neighbors and linearly coupled to a distinct resonator particle; it serves as a prototypical model of wave propagation in granular crystals and metamaterials. We study traveling waves in an MiM lattice whose bead interaction is governed by the cubic Fermi–Pasta–Ulam–Tsingou potential and whose resonator mass is small compared to the bead mass. Excluding a countable number of "antiresonance" resonator masses accumulating at 0, we prove the existence of nanopteron traveling waves in this small mass limit. The profiles of these waves consist of the superposition of an exponentially localized core and a small amplitude periodic oscillation that itself is a traveling wave profile for the lattice. Our arguments use functional analytic techniques originally developed by Beale for a capillary–gravity water wave problem and recently employed in a number of related nanopteron constructions in diatomic Fermi–Pasta–Ulam–Tsingou lattices. [ABSTRACT FROM AUTHOR]

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues. [ABSTRACT FROM AUTHOR]

It is shown that the Ornstein–Uhlenbeck operator perturbed by a multipolar inverse square potential A Φ , G + V = Δ - ∇ Φ · ∇ + G · ∇ + ∑ i = 1 n c | x - a i | 2 with suitable domain generates a quasi-contractive and positive analytic C 0 -semigroup on the weighted space L 2 (R N , d μ) , where d μ = exp (- Φ (x)) d x , Φ ∈ C 2 (R N , R) , G ∈ C 1 (R N , R N) , c > 0 , and a 1 , ... , a n ∈ R N . The proofs are based on an L 2 -weighted Hardy inequality and bilinear form techniques. [ABSTRACT FROM AUTHOR]

To deal with permanent deformations and residual stresses, we consider a morphoelastic model for the scar formation as the result of wound healing after a skin trauma. Next to the mechanical components such as strain and displacements, the model accounts for biological constituents such as the concentration of signaling molecules, the cellular densities of fibroblasts and myofibroblasts, and the density of collagen. Here we present stability constraints for the one-dimensional counterpart of this morphoelastic model, for both the continuous and (semi-) discrete problem. We show that the truncation error between these eigenvalues associated with the continuous and semi-discrete problem is of order O (h 2) . Next we perform numerical validation to these constraints and provide a biological interpretation of the (in)stability. For the mechanical part of the model, the results show the components reach equilibria in a (non) monotonic way, depending on the value of the viscosity. The results show that the parameters of the chemical part of the model need to meet the stability constraint, depending on the decay rate of the signaling molecules, to avoid unrealistic results. [ABSTRACT FROM AUTHOR]

We study propagation direction of the traveling wave for the diffusive Lotka–Volterra competition system with bistable nonlinearity in a periodic habitat. By directly proving the strong stability of two semitrivial equilibria, we establish a new and sharper result on the existence of traveling wave. Using the method of upper and lower solutions, we provide two comparison theorems concerning the direction of traveling wave propagation. Several explicit sufficient conditions on the determination of the speed sign are established. In addition, an interval estimation of the bistable-wave speed reveals the relations among the bistable speed and the spreading speeds of two monostable subsystems. Biologically, our idea and insight provide an effective approach to find or control the direction of wave propagation for a system in heterogeneous environments. [ABSTRACT FROM AUTHOR]

A very important population threshold quantity is the target reproduction number, which is a measure of control effort required for a target prevention, intervention or control. This concept, as a generalization of type reproduction number, was first introduced in Shuai et al. (J Math Biol 67:1067–1082, 2013) for nonnegative matrices with immediate applications to compartmental population models of ordinary differential equations. The current paper is devoted to the study of all target reproduction numbers for reaction-diffusion population models with compartmental structure. It turns out that the target reproduction number can be regarded as the basic reproduction number of a modified system, where the state of newborn individuals is limited to the target control set and the offspring from the non-target set is regarded as a part of the transition. In other words, the target reproduction number can be interpreted as the expected number of offspring in a specific target set that a primary newborn individual of the same set would produce during its lifetime. We also characterize the target reproduction number so that it can be easily computed numerically for reaction-diffusion models. At the end, we demonstrate our theoretical observations using two examples. [ABSTRACT FROM AUTHOR]

We solve second order parabolic equations with nonsmooth coefficients and initial data in suitable uniform spaces. We also show the smoothing effect of the corresponding analytic semigroup depending on the integrability properties of the coefficients. Robustness with respect to perturbations in the coefficients is also obtained. [ABSTRACT FROM AUTHOR]

A class of coupled cell-bulk ODE-PDE models is formulated and analyzed in a two-dimensional domain, which is relevant to studying quorum-sensing behavior on thin substrates. In this model, spatially segregated dynamically active signaling cells of a common small radius $$\epsilon \ll 1$$ are coupled through a passive bulk diffusion field. For this coupled system, the method of matched asymptotic expansions is used to construct steady-state solutions and to formulate a spectral problem that characterizes the linear stability properties of the steady-state solutions, with the aim of predicting whether temporal oscillations can be triggered by the cell-bulk coupling. Phase diagrams in parameter space where such collective oscillations can occur, as obtained from our linear stability analysis, are illustrated for two specific choices of the intracellular kinetics. In the limit of very large bulk diffusion, it is shown that solutions to the ODE-PDE cell-bulk system can be approximated by a finite-dimensional dynamical system. This limiting system is studied both analytically, using a linear stability analysis and, globally, using numerical bifurcation software. For one illustrative example of the theory, it is shown that when the number of cells exceeds some critical number, i.e., when a quorum is attained, the passive bulk diffusion field can trigger oscillations through a Hopf bifurcation that would otherwise not occur without the coupling. Moreover, for two specific models for the intracellular dynamics, we show that there are rather wide regions in parameter space where these triggered oscillations are synchronous in nature. Unless the bulk diffusivity is asymptotically large, it is shown that a diffusion-sensing behavior is possible whereby more clustered spatial configurations of cells inside the domain lead to larger regions in parameter space where synchronous collective oscillations between the small cells can occur. Finally, the linear stability analysis for these cell-bulk models is shown to be qualitatively rather similar to the linear stability analysis of localized spot patterns for activator-inhibitor reaction-diffusion systems in the limit of long-range inhibition and short-range activation. [ABSTRACT FROM AUTHOR]