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Abstract: Abstract Although the Poincaré normal form theory is not applicable, a predator–prey system having fully null linear part was proved to be degenerate of codimension 2 within the class of the GLV vector fields and unfolded versally within the GLV class. In this paper, we study the case that the nondegeneracy condition no longer holds, i.e., a quadratic term vanishes. We prove that the quadratic terms in the GLV normal form cannot be eliminated any more, showing that the vanished quadratic term substantially contributes to the degeneracy. We give its versal unfolding of codimension 3 within the GLV class, display all its bifurcations near the equilibrium, and see that the Hopf bifurcation and the heteroclinic bifurcation, which occur in the codimension 2 case, do not happen but two transcritical bifurcations at different equilibria may occur simultaneously, which is impossible in the codimension 2 case. PubDate: 2022-05-18

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Abstract: Abstract This paper focuses on a special 2D Boussinesq equation with partial dissipation, for which the velocity equation involves no dissipation and there is only damping in the horizontal component equation. Without buoyancy force, the corresponding vorticity equation is a 2D Euler-like equation with an extra Calderon–Zygmund-type term. Its stability is an open problem. Our results reveal that the buoyancy force exactly stabilizes the fluids by the coupling and interaction between the velocity and temperature. In addition, we prove the solution decays exponentially to zero in Sobolev norm. PubDate: 2022-05-17

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Abstract: Abstract In this paper, we show that the position and the derivative operators, \({{\hat{q}}}\) and \({{\hat{D}}}\) , can be treated as ladder operators connecting various vectors of two biorthonormal families, \({{{\mathcal {F}}}}_\varphi \) and \({{{\mathcal {F}}}}_\psi \) . In particular, the vectors in \({{{\mathcal {F}}}}_\varphi \) are essentially monomials in x, \(x^k\) , while those in \({{{\mathcal {F}}}}_\psi \) are weak derivatives of the Dirac delta distribution, \(\delta ^{(m)}(x)\) , times some normalization factor. We also show how bi-coherent states can be constructed for these \({{\hat{q}}}\) and \({{\hat{D}}}\) , both as convergent series of elements of \({{{\mathcal {F}}}}_\varphi \) and \({{{\mathcal {F}}}}_\psi \) , or using two different displacement-like operators acting on the two vacua of the framework. Our approach generalizes well- known results for ordinary coherent states. PubDate: 2022-05-16

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Abstract: Abstract This paper deals with a predator–prey model with time delay and constant prey harvesting. We investigate the effect of the time delay on the stability of the coexistence equilibrium and demonstrate that time delay can induce spatial patterns. Furthermore, a Hopf bifurcation occurs when the delay increases to a critical value. By applying normal form theory and the center manifold theorem, we develop the explicit formulae that determines the stability and direction of the bifurcating periodic solutions. Finally, we show how the initial condition affects the types of spatial patterns by numerical simulations. PubDate: 2022-05-16

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Abstract: Abstract Many real-world applications are modeled by Volterra integral–differential equations of the form $$\begin{aligned} u_{tt}-\Delta u +\int \limits _{\alpha }^{t}g(t-s)\Delta u(s)\, \mathrm{d}s = 0 \;\; \text{ in } \;\; \Omega \times (0,\infty ), \end{aligned}$$ where \(\Omega \) is a bounded domain of \({\mathbb {R}}^N\) and g is a memory kernel. Our main concern is with the concept of so-called creation time, the time \(\alpha \) where past history begins. Separately, the cases \(\alpha =-\infty \) (history) and \(\alpha =0\) (null history) were extensively studied in the literature. However, as far as we know, there is no unified approach with respect to the intermediate case \(-\infty< \alpha <0\) . Therefore we provide new stability results featuring (i) uniform and general stability when the creation time \(\alpha \) varies over full range \((-\infty ,0)\) and (ii) connection between the history and the null history cases by means of a rigorous backward ( \(\alpha \rightarrow -\infty \) ) and forward ( \(\alpha \rightarrow 0^-\) ) limit analysis. PubDate: 2022-05-13

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Abstract: Abstract This paper investigates the dynamics and pattern formation of a system modeling the interaction of mussels and algae in the water layer overlying the mussel bed, where the algae are the main food source for mussels, and the advection of algae is directed from the open sea toward the shore. For such a class of systems, we first provide a clear picture on the local dynamics of the semi-trivial steady state in terms of diffusion rate of algae and advection rate, and then the global dynamics is presented by persistence theory and global stability of the semi-trivial steady state. Finally, the existence of the positive steady state is also investigated. PubDate: 2022-05-12

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Abstract: Abstract We investigate the scattering of a plane wave by a spatial curve known as Viviani’s curve which has several applications in optics. We formulate the problem in terms of a three-dimensional Lippmann–Schwinger equation, with a boundary-wall potential, and solve it exactly. PubDate: 2022-05-07

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Abstract: Abstract We provide an analytic derivation of the bifurcation conditions for localized bulging in an inflated hyperelastic tube of arbitrary wall thickness and axisymmetric necking in a hyperelastic sheet under equibiaxial stretching. It has previously been shown numerically that the bifurcation condition for the former problem is equivalent to the vanishing of the Jacobian determinant of the internal pressure P and resultant axial force N, with each of them viewed as a function of the azimuthal stretch on the inner surface and the axial stretch. This equivalence is established here analytically. For the latter problem for which it has recently been shown that the bifurcation condition is not given by a Jacobian determinant equal to zero, we explain why this is the case and provide an alternative interpretation. PubDate: 2022-05-07

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Abstract: Abstract We establish the equivalence of free piston and delta shock, for the one-space-dimensional pressureless compressible Euler equations. The delta shock appearing in the singular Riemann problem is exactly the piston that may move freely forward or backward in a straight tube, driven by the pressureless Euler flows on two sides of it in the tube. This result not only helps to understand the physics of the somewhat mysterious delta shocks, but also provides a way to reduce the fluid–solid interaction problem, which consists of several initial-boundary value problems coupled with moving boundaries, to a simpler Cauchy problem. We show the equivalence from three different perspectives. The first one is from the sticky particles, and derives the ordinary differential equation (ODE) of the trajectory of the piston by a straightforward application of conservation law of momentum, which is physically simple and clear. The second one is to study a coupled initial-boundary value problem of pressureless Euler equations, with the piston as a moving boundary following the Newton’s second law. It depends on a concept of Radon measure solutions of initial-boundary value problems of the compressible Euler equations which enables us to calculate the force on the piston given by the flow. The last one is to solve directly the singular Riemann problem and obtain the ODE of delta shock by the generalized Rankine–Hugoniot conditions. All the three methods lead to the same ODE. PubDate: 2022-05-06

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Abstract: Abstract The isotropic thermoelasticity system of type-III, describing both the mechanical and the thermal behaviours of a body occupying a bounded domain with a Lipschitz boundary, is considered. The displacement vector and either the normal heat flux or the temperature are prescribed on the boundary. Both the theoretical and the numerical reconstructions of a time-dependent heat source from the knowledge of an additional weighted integral measurement of the temperature are investigated. It is shown that the appropriate type of measurement depends on the thermal boundary condition available, whilst the existence and uniqueness of a weak solution for exact data are also proved. For each of the two inverse source problems investigated herein, a numerical algorithm is also proposed and the convergence of these numerical schemes for exact data is proved. Four numerical examples with noisy measurements are implemented using the finite element method and thoroughly investigated to validate the convergence and stability of the proposed algorithms. PubDate: 2022-05-04

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Abstract: To study the impact of vector’s habitat expansion and mobility of hosts on the geographic spread of diseases, we propose a diffusive vector–host epidemic model with a finite growing domain and mainly focus on its asymptotic profile. In a situation that the domain grows uniformly and isotropically with growth ratio \(\rho \) , we employ the Lagrangian transformations to transform the model in the growing domain into the one in a fixed domain, along with dilution terms and time-dependent diffusion coefficients. We analyze the well posedness of the model and define the basic reproduction number \(\Re _0^\rho \) . Our results indicate that if \(\Re _0^\rho < 1\) , the disease-free equilibrium is globally asymptotically stable without any extra conditions, while the infected populations will eventually tend to the set formulated by the maximum and minimum solutions of the associated steady-state problem if \(\Re _0^\rho >1\) . The analysis is carried out by using the comparison principle, the theory of quasimonotone nondecreasing elliptic and parabolic system, and convergence of abstract asymptotic autonomous system. Comparing the results with those in a fixed domain, we confirm that the growth of domain brings negative influences on disease control. PubDate: 2022-05-03

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Abstract: Abstract In this paper, we derive a nonlocal theory for porous elastic materials in the context of Mindlin’s strain gradient model. The second gradient of deformation and the second gradient of volume fraction field are added to the set of independent constitutive variables by taking into account the nonlocal length scale parameters effect. The obtained system of equations is a coupling of a two hyperbolic equations with higher gradients terms due to the strain gradient length scale parameter l and the elastic nonlocal parameter \(\varpi \) . This poses some new mathematical difficulties due to the lack of regularity. Under quite general assumptions on nonlinear sources terms and based on nonlinear semigroups and the theory of monotone operators, we establish existence and uniqueness of weak and strong solutions to the one dimensional nonlinear problem. By an approach based on the Gearhart–Herbst–Prüss–Huang Theorem, we prove that the semigroup associated with the derived model is not analytic in general ( \(\varpi =0\) or not). A frictional damping for the elastic component, whose form depends on the elastic nonlocal parameter ( \(\varpi =0\) or not), is shown to lead to exponential stability at a rate of decay determined explicitly. Without frictional damping, the derived system can be exponentially stable only in the absence of body forces and under the condition of equal wave speeds. PubDate: 2022-04-30

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Abstract: Abstract The Flamant problem of a cubic quasicrystal half-plane is solved when its surface is loaded by normal and tangential concentrated forces. Its solution serves as a fundamental solution since its superposition forms other solutions. The Flamant problem is converted to an associated boundary value problem. The Fourier transform method is applied to solve the Flamant problem. Explicit expressions for the fundamental solution of the phonon and phason stresses are obtained. A comparison between the fundamental solutions of the Flamant problem for cubic quasicrystals and conventional cubic crystals is made. The influence of the presence of phason field on the phonon stress and deformation is shown in graph. PubDate: 2022-04-30

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Abstract: Abstract What today we call Truesdell rate of the Cauchy stress was originally introduced by Truesdell, as a frame-indifferent measure of stress rate for his theory of hypoelasticity. This work aims at showing that the Truesdell rate is in fact a more general concept, which arises from the application of Reynolds’ Transport Theorem. In the customary three-dimensional setting, the Transport Theorem can be formulated in terms of differential forms, i.e., three-forms for volume integrals and two-forms for surface integrals, in which case the only differential operator involved is the Lie derivative with respect to the velocity field of the continuum body. Alternatively, the Transport Theorem can be formulated in terms of the pseudo-scalar density associated with a three-form and the pseudo-vector flux density associated with a two-form. In this alternative approach, the differential operator involved is precisely the Truesdell rate. As an example, we shall show how the “convected derivative” of the theory of Electromagnetism is in fact a Truesdell rate. PubDate: 2022-04-28

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Abstract: Abstract In this paper we consider a transmission problem for the wave equations with past history and acoustic boundary conditions, involving two distinct domains connected through a common interface. The frictional dampings are only distributed in a small neighbourhood of the interface. Under some geometric conditions, we obtain the energy decays at the rate quantified by the solution to a certain nonlinear ODE dependent on the damping terms. Our method is based on using appropriate weighted multipliers to establish the necessary observability inequality that allows to obtain the energy estimate. PubDate: 2022-04-27

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Abstract: Abstract We use complex variable methods to derive an analytical solution to the problem in plane elasticity associated with a circular elastic inhomogeneity with an eccentric interphase layer when the matrix is subjected to uniform remote in-plane stresses and the interphase layer undergoes uniform in-plane eigenstrains. The complex coefficients appearing in all three pairs of analytic functions characterizing the elastic fields in the composite are uniquely determined by solving two decoupled sets of linear algebraic equations obtained by enforcing the continuity conditions of tractions and displacements across the two perfect circular interfaces with the aid of analytic continuation. A simple analytical solution is also derived when the circular inhomogeneity becomes a traction-free hole and the interphase layer and the matrix have equal shear modulus but distinct Poisson’s ratios. The non-uniform mean stress inside the circular inhomogeneity and the hoop stress along the edge of the circular hole are calculated and illustrated graphically. PubDate: 2022-04-27

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Abstract: Abstract The aim of this paper is to investigate the existence of uniform random attractor for a non-autonomous stochastic strongly damped wave equation driven by multiplicative noise defined on \({{\mathbb {R}}}^{N}\) ( \(1\le N\le 3\) ). First, we prove the equation can generate a non-autonomous random dynamical system (NRDS), which is continuous in both phase space \(H^1({\mathbb {R}}^{N})\times L^2({\mathbb {R}}^{N})\) and symbol space. Then, we derive the uniform estimates of solutions for the equation and obtain a uniform random absorbing set with respect to the symbols. Finally, we get the uniformly asymptotic compactness of the NRDS by using the method of tail estimates and obtain the existence of a uniform random attractor for the dynamical system. Furthermore, we can also obtain that the uniform random attractor with respect to the deterministic non-autonomous symbols belonging to hull space coincides with the uniform random attractor with respect to initial time belonging to \({\mathbb {R}}\) . PubDate: 2022-04-27

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Abstract: Abstract This paper is devoted to studying the following Sobolev critical Schrödinger systems: $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda _1 u= u ^{2^*-2}u+\mu _1 u ^{p-2}u+\beta r_1 u ^{r_1-2}u v ^{r_2}\; &{}\hbox {in}\quad \mathbb {R}^N, \\ -\Delta v+\lambda _2 v= v ^{2^*-2}v+\mu _2 v ^{q-2}v+\beta r_2 u ^{r_1} v ^{r_2-2}v\; &{}\hbox {in}\quad \mathbb {R}^N, \\ \int \limits _{\mathbb {R}^N} u^2=a^2\quad \hbox {and}\quad \int \limits _{\mathbb {R}^N} v^2=b^2,\quad u,v\in H^1({\mathbb {R}}^N). \end{array}\right. } \end{aligned}$$ Here, \(N\ge 3\) , \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent, \(r_1, r_2>1\) , \(p, q, r_1+r_2\in (2+\frac{4}{N},2^*]\) , \(a,b,\mu _1,\mu _2,\beta >0\) are positive constants and \(\lambda _1,\lambda _2\in \mathbb {R}\) will arise as Lagrange multipliers. Any (u, v) solving such systems (for some \(\lambda _1,\lambda _2\) ) is called the normalized solution in the literature, where the normalization is settled in \(L^2(\mathbb {R}^N)\) . By revealing the basic behavior of the mountain-pass energy \(c_{\beta }(a,b)\) when \(\beta >0\) is sufficiently large, we firstly show that if \(p,q,r_1+r_2<2^*\) the existence of positive normalized solution. Then, for the case of \(p=q=r_1+r_2=2^*\) , we may obtain the nonexistence of positive normalized solution. PubDate: 2022-04-27

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Abstract: Abstract We consider the following Kirchhoff-type equation of the form $$\begin{aligned} -\left( a+b\int \limits _{\,\,\,\mathbb {R}^3} \nabla u ^2\right) \Delta u+(1+\mu g(x))u=\lambda \left( \frac{1}{ x ^{\alpha }}* u ^p\right) u ^{p-2}u+ u ^4u,\quad x\in \mathbb {R}^3\end{aligned}$$ where \(a>0, b\ge 0\) are constants, \(\lambda , \mu \) are positive parameters, \(\alpha \in (0,3), p\in \left( 2, 6-\alpha \right) \) and \(g\in C(\mathbb {R}^3)\) satisfies some conditions. By the mountain pass theorem, we establish the existence of ground state solutions. Besides, the concentration of ground state solutions is also described as \(\mu \rightarrow \infty \) . PubDate: 2022-04-27

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Abstract: Abstract In this work, we consider the spatial decay for high-order parabolic (and combined with a hyperbolic) equation in a semi-infinite cylinder. We prove a Phragmén-Lindelöf alternative function and, by means of some appropriate inequalities, we show that the decay is of the type of the square of the distance to the bounded end face of the cylinder. The thermoelastic case is also considered when the heat conduction is modeled using a high-order parabolic equation. Though the arguments are similar to others usually applied, we obtain new relevant results by selecting appropriate functions never considered before. PubDate: 2022-04-27