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Abstract: Abstract Dynamics of the Fermi–Pasta–Ulam (FPU) system on a two-dimensional square lattice is considered in the limit of small-amplitude long-scale waves with slow transverse modulations. In the absence of transverse modulations, dynamics of such waves, even at an oblique angle with respect to the square lattice, is known to be described by the Korteweg–de Vries (KdV) equation. For the three basic directions (horizontal, vertical, and diagonal), we prove that the modulated waves are well described by the Kadomtsev–Petviashvili (KP-II) equation. The result was expected long ago but proving rigorous bounds on the approximation error turns out to be complicated due to the nonlocal terms of the KP-II equation and the vector structure of the FPU systems on two-dimensional lattices. We have obtained these error bounds by extending the local well-posedness result for the KP-II equation in Sobolev spaces and by controlling the error terms with energy estimates. The bounds are useful in the analysis of transverse stability of solitary and periodic waves in two-dimensional FPU systems due to many results available for the KP-II equation. PubDate: 2022-09-18

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Abstract: Abstract In this paper, we compute universal estimates of eigenvalues of a coupled system of elliptic differential equations in divergence form on a bounded domain in Euclidean space. As an application, we show an interesting case of rigidity inequalities of the eigenvalues of the Laplacian, more precisely, we consider a countable family of bounded domains in Gaussian shrinking soliton that makes the behavior of known estimates of the eigenvalues of the Laplacian invariant by a first-order perturbation of the Laplacian. We also address the Gaussian expanding soliton case in two different settings. We finish with the special case of divergence-free tensors which is closely related to the Cheng–Yau operator. PubDate: 2022-09-16

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Abstract: Abstract In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} \left( \varepsilon ^{2\alpha }a+\varepsilon ^{4\alpha -3}b\int _{{\mathbb {R}}^3} (-\Delta )^{\frac{\alpha }{2}} u ^2{{\mathrm{d}}}x\right) (-\Delta )^\alpha {u}+V(x)u = H(u-\beta )f(u) &{} \quad \text{ in }\,\,{\mathbb {R}}^3, \\ u\in H^\alpha ({\mathbb {R}}^3),\quad u>0 &{} \quad \text{ in }\,\, {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$ where \(\varepsilon ,\beta >0\) are small parameters, \(\alpha \in (\frac{3}{4},1)\) and a, b are positive constants, \((-\Delta )^{\alpha }\) is the fractional Laplacian operator, H is the Heaviside function, V is a positive continuous potential, and f is a superlinear continuous function with subcritical growth. By using minimax theorems together with the non-smooth theory, we obtain existence and concentration properties of positive solutions to this non-local system. PubDate: 2022-09-16

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Abstract: Abstract This paper deals with the higher dimension quasilinear parabolic–parabolic chemotaxis model involving a source term of logistic type \(u_{t}=\nabla \cdot \left( \phi (u)\nabla u\right) -\nabla \cdot \left( \psi (u)\nabla \upsilon \right) +g(x,u)\) , \(\tau \upsilon _{t}=\Delta \upsilon -\upsilon +u\) , in \(\ (x,t)\in \Omega \times (0,T)\) , subject to nonnegative initial data and homogeneous Neumann boundary condition, where \(\Omega \) is a smooth and bounded domain in \( {\mathbb {R}} ^{N}\) , \(N\ge 1\) and \(\psi ,\) \(\phi ,\) g are smooth, positive functions satisfying \(\nu s^{q}\le \psi \le \chi s^{q}\) , \(\phi \ge \sigma s^{p}\) , \(p,q\in {\mathbb {R}} \) , \(\nu ,\chi ,\sigma >0\) when \(s\ge s_{0}>1\) , \(g(x,s)\le \eta s^{k(x)}-\mu s^{m(x)}\) for \(s>0\) , \(\eta \ge 0\) , \(\mu >0\) constants and \( g(x,0)\ge 0\) , \(x\in \Omega \) , where k, m are measurable functions with \( 0\le k^{-}:=\underset{x\in \Omega }{ess\text { inf}}k\left( x\right) \le k(x)\le m^{+}\underset{x\in \Omega }{:=ess\text { sup }k(x)}<+\infty \) , \( 1<m^{-}:=\underset{x\in \Omega }{ess\text { inf}}m\left( x\right) \le m(x)\le m^{+}\underset{x\in \Omega }{:=ess\text { sup }m(x)}<+\infty \) . We extend the constant exponents \(k=\left\{ 0,1\right\} ,\) \(m>1\) which in logistic source term \(g(s)\le \eta s^{k}-\mu s^{m}\) for \(s>0\) , \(\eta \ge 0\) , \(\mu >0\) as variable exponents \(k(\cdot )\ge 0,\) \(m(\cdot )>1\) with \( k^{+}<m^{-}\) . It is proved that if PubDate: 2022-09-16

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Abstract: Abstract The electromagnetic wave diffraction from perfectly conducting truncated wedges is considered on the rigorous level in cylindrical coordinates. An analytical regularization method is developed to obtain mathematically accurate problem solutions. The solution method is based on the unknown field representation through the principal value Kontorovich–Lebedev integral and the eigenfunctions series. We analyze the scattering from the semi-infinite truncated wedge, which consists of two non-parallel and non-intersecting perfectly conducting and infinitely thin half-planes, and develop this technique for analysis of more complicated problems of wave diffraction from the truncated wedge of finite length. The problems are reduced to the infinite systems of linear algebraic equations (ISLAE) of the first kind. The convolution type operators and their inverse ones are used to reduce them to the ISLAE of the second kind applied to the analytical regularization procedure. Two versions of the procedure, such as left- and right-sides regularization, are considered. The developed technique is compared with the Wiener–Hopf method. The numerical examples of wave scattering from the truncated wedge, including its well-known geometries as the semi-plane and the slit in the infinite plane, are analyzed. PubDate: 2022-09-08

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Abstract: Abstract A continuum model intended to provide predictions for the response of a woven fabric that includes the effects of friction between fibers is proposed. Specifically, we consider a macroscopic formulation in which the fabric weave is composed of two orthogonal families of continuously distributed yarns. The elastic behavior of the planar fabric is characterized by a second-gradient formulation, incorporating the capacity of the fibers to resist a bending deformation. Particular care is devoted to modeling the action of preventing fiber overlapping through a potential energy barrier. The frictional sliding effect of warp threads interwoven with the weft yarns is introduced through a Rayleigh dissipative function that can be appropriately shaped to consider a Coulomb-type law. Spinning friction of yarns belonging to different families also is conceived when a relative rotation between fibers is present to generalize the dissipation phenomenon involved in the considered sheet. Numerical simulations of the proposed model are provided and discussed. PubDate: 2022-09-08

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Abstract: Abstract We construct nontrivial weak solutions to the 4-dimensional Boussinesq system with viscosity and with or without thermal conductivity in periodic domain \({\mathbb {T}}^{4}\) . Our proof is based on a convex integration scheme by building blocks of concentrated Mikado flow. The proof is also valid for dimensions \(d>4\) . PubDate: 2022-09-01

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Abstract: Abstract This work is concerned with finding nonlinear stability limits of a thermally stratified horizontal layer governed by the Darcy–Lapwood–Brinkman equations. Time periodic modulation is imposed on either the boundary temperatures or the gravitational field permeating the layer. Unlike the classical theory of porous media, thermodynamic non-equilibrium is assumed within the layer in the form of non-negligible temperature differences. The solid and fluid constituents may undergo their own temperature variations and the heat flow is studied using the two-temperature model. The generalized energy method is employed to determine the relevant limits that suffices the system stability. The non-equilibrium and modulational parameters are found to affect the stability limits that are based on a higher-order Galerkin solution. Moreover, it is established that the stability limits so obtained are unconditional in the Euclidean measure. PubDate: 2022-08-30

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Abstract: Abstract In this paper, the initial-boundary value problem for the two-dimensional viscous, compressible and heat-conducting magnetohydrodynamics (MHD) equations with vacuum is considered. We show that the strong solution exists globally in the time provided the viscosity coefficient \(\mu \) is suitably large, and there is no small restriction on the initial data. As a result, we extend the works given by Li and Shang [20] for the full compressible Navier–Stokes equations and by Liu [25] for the isentropic compressible MHD equations. PubDate: 2022-08-24

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Abstract: Abstract The paper considers mean-square continuous, wide-sense homogeneous, and isotropic random fields taking values in a linear space of polyadics. We find a set of such fields whose values are symmetric and positive-definite dyadics, and outline a strategy for their simulation. PubDate: 2022-08-22

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Abstract: Abstract We derive two consequences of the distributional form of the stress equilibrium condition while incorporating piecewise smooth stress and body force fields with singular concentrations on an interface. First, we obtain the local equilibrium conditions in the bulk and at the interface, the latter inclusive of interfacial stress and stress dipole fields. Second, we obtain the necessary and the sufficient conditions on the divergence-free non-smooth stress field for there to exist a stress function field such that the stress equilibrium is trivially satisfied. In doing so, we allow the domain to be non-contractible with mutually disjoint connected boundary components. Both derivations illustrate the utility of the theory of distributions in dealing with singular stress fields. PubDate: 2022-08-22

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Abstract: Abstract In this paper, we consider the stability of solutions to a class of cancer invasion model. This kind of model was first proposed by Chaplain and Lolas in 2005. In fact, they proposed two kinds of tumor invasion models in 2005 (Chaplain and Lolas in Math Models Methods Appl Sci 15(11):1685–1734, 2005) and 2006 (Chaplain and Lolas in Netw Heterog Media 1(3):399–439, 2006), respectively, which have similar structures, but their solutions have different properties. For convenience of distinction, we call them Model I and Model II, respectively. A common feature of the two models is that they both consider the remodeling of extracellular matrix. For the research on the stability of solutions of the two models, there are only the three papers (Hillen in Math Models Methods Appl Sci 23(01):165–198, 2013; Tao in SIAM J Math Anal 47(6): 4229–4250, 2015; Wang in J Differ Equ 260(9):6960–6988, 2016) on Model II by removing the remodeling of extracellular matrix, and the paper (Jin in Nonlinearity 33(10):5049–5079, 2020) on Model I. However, as far as we know, although the existence of solutions of this kind of model with porous medium diffusion has achieved fruitful results, there are no relevant results on stability even without considering the remodeling effect of extracellular matrix. In the present paper, we consider the stability of this kind of model with nonlinear diffusion, we find that the simultaneous emergence of haptotaxis, nonlinear diffusion and remodeling effect does bring essential difficulties to the study of stability, such as the method of constructing Lyapunov functional is no longer applicable. In this paper, we use some detailed analytical techniques to prove the global asymptotic stability of bounded solutions. PubDate: 2022-08-18

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Abstract: Abstract We consider a non-classical 2D mathematical equilibrium model describing a possible mechanical contact of a composite structure having a sharp-shaped edge. Nonlinearity of the model is caused by conditions of inequality type for a corresponding variational problem. The main feature of this basic model consists in its geometrical configuration, which determines non-convexity of the problem under consideration. Namely, the composite in its reference state touches a wedge-shaped rigid obstacle at a single contact point. On the basis of this model, we consider an induced family of problems depending on different functions of external loads. For a given set of functions, describing admissible external loads, we formulate an optimal control problem, where functions of external loads serve as a control. A cost functional is given with the help of an arbitrary weakly upper semicontinuous functional defined on the Sobolev space of feasible solutions. The solvability of the optimal control problem is proved. Furthermore, for a sequence of solutions corresponding to a maximizing sequence, a strong convergence in the corresponding Sobolev space is proven. PubDate: 2022-08-18

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Abstract: Abstract In this paper, we consider the attraction–repulsion chemotaxis system with nonlocal terms $$\begin{aligned} \left\{ \begin{array}{lll} u_t=\Delta u-\chi \nabla \cdot (u\nabla v)+\xi \nabla \cdot (u\nabla w)+f(u), &{}x\in \Omega , t>0, \\ \tau _1v_t=\Delta v-\alpha _1v+\beta _1u, &{} x\in \Omega , t>0, \\ \tau _2w_t=\Delta w-\alpha _2w+\beta _2u, &{} x\in \Omega , t>0 \end{array}\right. (*) \end{aligned}$$ under Neumann boundary conditions in a bounded domain with smooth boundary, where \(f(u)=u^{\sigma }(a_0-a_1u-a_2 \int \limits _{\Omega }u^{\beta })\) . We show that the system ( \(*\) ) possesses a unique global classical solution in three different cases (parabolic–elliptic–elliptic, fully parabolic, parabolic–parabolic–elliptic). Our results generalize and improve partial previously known ones. PubDate: 2022-08-18

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Abstract: Abstract In this paper, we consider the steady flow of a thermomicropolar fluid through a thin curvilinear channel. The flow is governed by the prescribed pressure drop between the channel’s ends. The heat exchange between the fluid inside the channel and the exterior medium is allowed through the upper curved wall, while the lower wall is insulated. Using asymptotic analysis with respect to the domain’s thickness, we compute the asymptotic approximation of the solution. The derived solution is obtained in explicit form, allowing us to clearly observe the effects of the curvature of the domain as well as the micropolarity of the fluid. A boundary layer analysis is provided for the microrotation. The proposed effective model is rigorously justified via error estimates in suitable norms. PubDate: 2022-08-15

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Abstract: Abstract Plane-strain motion of a flexural seismic metasurface in the form of a regular array of thin Kirchhoff plates attached to the surface of an elastic half-space is analysed. Two types of contact conditions, including simply supported plates and plates moving along horizontal rails are studied. Dispersion of time harmonic waves is investigated both asymptotically and numerically. A major effect of the contact conditions on metasurface behaviour is discovered. In particular, it is shown that frequency band gaps are not the feature of the array composed of simply supported plates. It is also demonstrated that the scaling laws, expressed through geometric and material problem parameters, drastically differ from each other for two considered setups. PubDate: 2022-08-15

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Abstract: Abstract We consider the motion of compressible Navier–Stokes fluids with the hard sphere pressure law around a rigid obstacle when the velocity and the density at infinity are nonzero. This kind of pressure model is largely employed in various physical and industrial applications. We prove the existence of weak solution to the system in the exterior domain. PubDate: 2022-08-15

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Abstract: Abstract We consider a free boundary problem with nonlocal diffusion and unbounded initial range, which can be used to model the propagation phenomenon of an invasion species whose habitat is the interval \((-\infty ,h(t))\) with h(t) representing the spreading front. Since the spatial scale is unbounded, a different method from the existing works about nonlocal diffusion problem with free boundary is employed to obtain the well-posedness. Then we prove that the species always spreads successfully, which is very different from the free boundary problem with bounded range. We also show that there is a finite spreading speed if and only if a threshold condition is satisfied by the kernel function. Moreover, the rate of accelerated spreading and accurate estimates on longtime behaviors of solution are derived. PubDate: 2022-08-15

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Abstract: Abstract Thermoelastic interactions in a linear, isotropic and homogeneous unbounded solid resulting from a continuous line heat source are investigated utilizing modified temperature-rate-dependent two-temperature thermoelasticity theory (MTRDTT, recently proposed by Shivay and Mukhopadhyay in J Heat Transf 142:4045241, 2019). By incorporating the temperature-rate terms of thermodynamic temperature and conductive temperature, the two-temperature relation is modified in this theory. The problem is studied with the unified version of two-temperature relation to compare the results for displacement, temperatures and stresses in the MTRDTT model with the corresponding results of the two-temperature Green-Lindsay (TTGL) model. To solve the problem, Laplace and Hankel transforms are employed. Explicit expressions for these field variables are obtained for the short-time approximation case. Further, the computational tool is used to graphically depict the analytical findings and compare the results obtained from both models. Some important observations about these models are highlighted. PubDate: 2022-08-15

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Abstract: Abstract In this article, a theoretical framework for problems involving fractional equations of hyperbolic type arising in the theory of viscoelasticity is presented. Based on the Galerkin method, a variational problem of the fractionary viscoelasticity is studied. An appropriate functional setting is introduced in order to establish the existence, uniqueness and a priori estimates for weak solutions. This framework is developed in close concordance with important physical quantities of the theory of viscoelasticity. PubDate: 2022-08-15