Sign-changing Coefficient Research Articles

Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equationx″(t)+p(t)x(t−τ(t))=0,t≥swhere τ:R→[0,+∞), p:R→R are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case p(t)≤0,t∈R.

Adaptive edge finite element method and numerical design for metasurface cloak

Metasurfaces are metamaterials with two-dimensional (2D) or planar structures. Compared to traditional metamaterials, metasurfaces have the advantage of being lightweight, easy to control and simple to design. This paper outlines the phase matching principle and constitutive parameter theory of electromagnetic (EM) metasurfaces, as well as the constitutive parameter analysis and model equations for metasurface cloaks. Due to the singularity of the metasurface cloak, an Adaptive Edge Finite Element Method (AEFEM) is developed in this paper. In addition, the convergence theory of the AEFEM for the time-harmonic Maxwell's equations with one sign-changing coefficient is established under the T-coercivity assumption. Numerical simulations of the arc-shaped metasurface cloak have confirmed the accuracy of our model and numerical theory. Furthermore, we have used our numerical method to design a triangular-shaped metasurface cloak with fewer singularities.

Optimal conditions for the maximum principle for second-order periodic problems

We provide alternate necessary and/or sufficient conditions on the sign-changing coefficient p(t) for the maximum principle for the second-order periodic problem \(u''=p(t) u + q(t) \) to hold, i.e., for nonnegative \(q\) to yield a nonpositive periodic solution \(u\). See also https://ejde.math.txstate.edu/special/02/h2/abstr.html

Scattering resonances in unbounded transmission problems with sign-changing coefficient

Abstract It is well known that classical optical cavities can exhibit localized phenomena associated with scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ‘quasimodes to resonances’ argument from the black box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, e.g. made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and shows that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial’s properties, scattering resonances situated close to the real axis are associated with surface plasmons. Examples for several metamaterial cavities are provided.

On the approximation of electromagnetic fields by edge finite elements—Part 4: analysis of the model with one sign-changing coefficient

In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set \(\varOmega \) of \(\mathbb {R}^3\), and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale \((\varvec{H}^{\texttt {s}}(\varOmega ))_{\texttt {s}>0}\). With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived.

A novel series of mechanical metamaterials with sign-changing coefficient of thermal expansion and their parameter analysis

Mechanical metamaterials show great application prospects in many fields such as aerospace, biomedical engineering, energy and transportation. The development of mechanical metamaterials with multiple peculiar properties is an important prerequisite for the development of intelligent sensors with excellent performance and the realization of multifunctional integration of structures. In this paper, a series of novel mechanical metamaterials are designed by adding rhomboid negative thermal expansion units into re-entrant hexagonal honeycombs, which have different signs of the coefficient of thermal expansion when the temperature rises or falls, and the analytical formula of their equivalent thermal expansion coefficient are derived. Parameter analysis and numerical simulations show that the metamaterial can realize the sign-changing properties of thermal expansion coefficient without entering the range of large geometric deformation under specific constituting materials and geometric parameters. In addition, this design idea is extended to three-dimensional form.

Nonlinear Helmholtz equations with sign-changing diffusion coefficient

Nonlinear Helmholtz equations with sign-changing diffusion coefficient

Positive solutions of a nonlinear algebraic system with sign-changing coefficient matrix

Existence of positive solutions for the nonlinear algebraic system x=lambda GF ( x ) has been extensively studied when the ntimes n coefficient matrix G is positive or nonnegative. However, to the best of our knowledge, few results have been obtained when the coefficient matrix changes sign. In this case, some commonly applied analysis methods such as the cone theory, the Krein–Rutman theorem, the monotone iterative techniques, and so on cannot be directly applied. In this note, we prove the existence of positive solutions for the above nonlinear algebraic system with sign-changing coefficient matrix taking the advantages of the classical Brouwer fixed point theorem combined with a decomposition condition on the coefficient matrix. We provide an example in solving a second-order difference equation with periodic boundary conditions to illustrate the applications of the results.

A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment

This paper is concerned with a prey-predator model with sign-changing intrinsic growth rate in heterogeneous time-periodic environment, where the prey species lives in the whole space but the predator species lives in a region enclosed by a free boundary. It is shown that the results for the case of the non-periodic environment remain true in time-periodic environment. In fact, we first establish a similar spreading-vanishing dichotomy, which implies that if the predator species could spread successfully, then the two species will coexist, and this is certainly for the situation that the predation is relatively weak. Furthermore, some criteria are also obtained for spreading and vanishing. At last, some rough estimates of the asymptotic spreading speed are given if spreading occurs.

A Reaction-Diffusion-Advection Equation with a Free Boundary and Sign-Changing Coefficient

In this paper we investigate a reaction-diffusion-advection equation with a free boundary and sign-changing coefficient. The main objective is to understand the influence of the advection term on the long time behavior of the solutions. More precisely, we prove a spreading-vanishing dichotomy result, namely the species either successfully spreads to infinity as t??$t\rightarrow\infty$ and survives in the new environment, or it fails to establish and dies out in the long run. When spreading occurs, the spreading speed of the expanding front is affected by the advection. In this situation, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. Furthermore, when the environment is asymptotically homogeneous at infinity, these two bounds coincide.

A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment

This paper concerns a diffusive logistic equation with a free boundary and sign-changing intrinsic growth rate in heterogeneous time-periodic environment, in which the variable intrinsic growth rate may be “very negative” in a “suitable large region” (see conditions (H1), (H2), (4.3)). Such a model can be used to describe the spreading of a new or invasive species, with the free boundary representing the expanding front. In the case of higher space dimensions with radial symmetry and when the intrinsic growth rate has a positive lower bound, this problem has been studied by Du, Guo & Peng [11]. They established a spreading–vanishing dichotomy, the sharp criteria for spreading and vanishing and estimate of the asymptotic spreading speed. In the present paper, we show that the above results are retained for our problem.

The diffusive logistic equation with a free boundary and sign-changing coefficient

This short paper concerns a diffusive logistic equation with a free boundary and sign-changing coefficient, which is formulated to study the spread of an invasive species, where the free boundary represents the expanding front. A spreading–vanishing dichotomy is derived, namely the species either successfully spreads to the right-half-space as time t→∞ and survives (persists) in the new environment, or it fails to establish itself and will extinct in the long run. The sharp criteria for spreading and vanishing are also obtained. When spreading happens, we estimate the asymptotic spreading speed of the free boundary.

Multiple positive solutions to a class of modified nonlinear Schrödinger equations

This paper is concerned with the existence and multiplicity of positive solutions of the following equation−u″+u−2((|u|2)″)u=a(x)|u|p−2u+λk(x)u where u∈H1(R). We will prove that the modified term 2((|u|2)″)u is helpful to find multiple positive solutions in the case of the nonlinearity with sign changing coefficient a(x). Surprisingly, we do not need a sign condition ∫RNa(x)e1pdx<0, which has been proven to be a necessary condition to the existence of positive solutions for semilinear elliptic equations with sign changing nonlinearity (Alama and Tarantello (1993) [1]).

Existence and uniqueness of periodic solutions for a kind of high-order [formula omitted]-Laplacian Duffing differential equation with sign-changing coefficient ahead of linear term

In this paper, by using the continuation theorem of coincidence degree theory, we study a kind of high-order p -Laplacian differential equation as follows: ( φ p ( y ( m ) ( t ) ) ) ( m ) + β ( t ) y ′ ( t ) + g ( t , y ( t ) ) = e ( t ) . Some new results on the existence and uniqueness of periodic solutions are obtained. The interesting thing is that the coefficient β ( t ) is allowed to change sign, which could be achieved infrequently in previous papers. But, the methods to estimate a priori bounds of periodic solutions are different from the corresponding ones used in the past.

Periodic solutions for a kind of high-order [formula omitted]-Laplacian differential equation with sign-changing coefficient ahead of the non-linear term

In this paper, by using the theory of Fourier series, Bernoulli number theory and the continuation theorem of coincidence degree theory, we study a kind of high-order p -Laplacian differential equation as follows: ( φ p ( y ( m ) ( t ) ) ) ( m ) = f ( y ( t ) ) y ′ ( t ) + h ( y ( t ) ) + β ( t ) g ( y ( t − τ ( t ) ) ) + e ( t ) . Some new results on the existence of periodic solutions are obtained. The interesting thing is that the coefficient β ( t ) is allowed to change sign. But, the methods used to estimate a priori bounds of periodic solutions are different from the corresponding ones used in the past.

On eigenvalue problems with Robin type boundary conditions having indefinite coefficients

A Robin type boundary condition with a sign-changing coefficient is treated. First, the associated linear elliptic eigenvalue problem is studied, where the existence of a principal eigenvalue is discussed by the use of a variational approach. Second, the associated semilinear elliptic boundary value problem of logistic type is studied and the one parameter-dependent structure of positive solutions is investigated, where results obtained are due to the construction of suitable super- and subsolutions by using the principal positive eigenfunctions of the linear eigenvalue problem.