Abstract: In this work, we investigate the propagation of shear horizontal (SH) waves in a nonlinear hyperelastic plate. We assume that the plate is made of heterogeneous, isotropic, and generalized neo-Hookean materials. The problem is examined with a perturbation method that balances the nonlinearity and dispersion in the analysis. Then, a nonlinear Schrödinger (NLS) equation is derived describing the nonlinear self-modulation of the waves. Using known solutions of an NLS equation, we found that the symmetric bright solitary SH waves will exist and propagate in this plate. Moreover, not only the effect of the heterogeneity, but also the effect of the nonlinearity on the deformation field is also considered for these waves. PubDate: 2019-03-21 DOI: 10.1007/s00033-019-1108-0

Abstract: This paper is concerned with the stability of traveling wave solutions for a spatially discrete SIS epidemic model. We investigate the problem by using the weighted energy method and comparison principles for the Cauchy problem and initial-boundary value problem of the lattice differential equations. Our main results show that any solution of the Cauchy problem for the SIS model converges exponentially to the traveling wave solution provided that the initial perturbation around the traveling wave solution belongs to a suitable weighted Banach space. PubDate: 2019-03-19 DOI: 10.1007/s00033-019-1107-1

Abstract: We propose a simple yet effective method to derive the three-dimensional temperature field induced by a steady point heat source in a trimaterial composed of an intermediate layer of finite thickness sandwiched between two semi-infinite media. The heat source can be located either in the upper semi-infinite medium or in the intermediate layer. The resulting analytical solutions remain valid for the electrostatic problem of a point electric charge in a trimaterial and are used to obtain the Coulomb force in this case. PubDate: 2019-03-18 DOI: 10.1007/s00033-019-1105-3

Abstract: In this paper, we study a one-dimensional dissipative system of piezoelectric beams with magnetic effect, based on the work of Morris and Özer (in: The proceedings of 52nd IEEE conference on decision & control, pp 3014–3019, 2013). Our main goal is to prove the system’s exponential stability independent of any relation between the coefficients using terms of feedback at the boundary and consequently prove their equivalence with the exact observability at the boundary. PubDate: 2019-03-16 DOI: 10.1007/s00033-019-1106-2

Abstract: This paper is concerned with the vacuum free boundary problem for the compressible spherically symmetric Navier–Stokes equations with an external force and degenerate viscosities in \(\mathbb {R}^{n}(n\ge 2)\) . When the initial data are a small perturbation of the stationary profile and the viscosity coefficients are proportional to \( \rho ^{\theta } \) with \(\theta \in {\left\{ \begin{array}{ll} (0,2(\gamma -1))\cap (0,\frac{\gamma }{2}]&{}n=2\\ (0,\frac{\gamma }{2}]&{}n\ge 3 \end{array}\right. }\) , a result on the global existence as well as sharper time decay rates of the weak solution is obtained which improves the one in Wei et al. (SIAM J Math Anal 40:869–904, 2008). The proof is based on some weighted energy estimates, and in our analysis, no smallness constraint is prescribed upon the derivatives of the initial data. It is also worth pointing out that our result covers the interesting case of the Saint-Venant shallow water model (i.e., \(\gamma =2\) and \(\theta =1\) ). PubDate: 2019-03-12 DOI: 10.1007/s00033-019-1101-7

Abstract: We consider the focusing \(L^2\) -supercritical fractional nonlinear Schrödinger equation $$\begin{aligned} i\partial _t u - (-\varDelta )^s u = - u ^\alpha u, \quad (t,x) \in \mathbb {R}^+ \times \mathbb {R}^d, \end{aligned}$$ where \(d\ge 2, \frac{d}{2d-1} \le s <1\) and \(\frac{4s}{d}<\alpha <\frac{4s}{d-2s}\) . By means of the localized virial estimate, we prove that the ground-state standing wave is strongly unstable by blowup. This result is a complement to a recent result of Peng–Shi (J Math Phys 59:011508, 2018) where the stability and instability of standing waves were studied in the \(L^2\) -subcritical and \(L^2\) -critical cases. PubDate: 2019-03-11 DOI: 10.1007/s00033-019-1104-4

Abstract: We consider the parabolic problem \(\mathbf{u}_{t}-\Delta \mathbf{u} = F(t, \mathbf{u})\) in \(\Omega \times (0,T)\) with homogeneous Dirichlet boundary conditions. The nonlinear term is given by $$\begin{aligned} F(t, \mathbf{u})=(f_1(t) u_2^{p_1}, \ldots , f_m(t) u_1^{p_m}), \end{aligned}$$ where \(\mathbf{u}=(u_1, \ldots , u_m) \) , \( p_i \ge 1\) , and \( f_i \in C[0,\infty ),\) for \(i=1,\ldots ,m\) . The set of initial data is \(\left\{ \mathbf{u}_0=(u_{0,1},\ldots ,u_{0,m}) \in \right. \left. C_0(\Omega )^m; u_{i,0}\ge 0\right\} \) , where \(\Omega \) is an arbitrary domain (either bounded or unbounded) with smooth boundary. We determine conditions that guarantee either the global existence or the blowup in finite time of nonnegative solutions. These conditions are given in terms of the asymptotic behavior of \(\Vert \mathbf{S }(t) { \mathbf u}_0\Vert _{L^\infty (\Omega )}\) , where \((\mathbf{S }(t){ \mathbf u}_0)_{t\ge 0}\) is the heat semigroup on \(C_0(\Omega )^m\) . PubDate: 2019-03-09 DOI: 10.1007/s00033-019-1103-5

Abstract: We analyze the Bresse system with partial boundary dissipation. Our main result is to prove that these dissipative mechanisms are enough to stabilize exponentially the whole system provided the wave propagation speeds are equal. PubDate: 2019-03-07 DOI: 10.1007/s00033-019-1102-6

Abstract: In this paper, we study the rotating shallow water system with initial data containing vacuum, the viscosity of the system is degenerate, and the Coriolis force, the capillary force and the turbulent drag force from fiction are involved. When a positive mass is surrounding by a bounded vacuum region (isolated mass group), initially, we prove that any classical solutions to the initial-boundary-value problem and periodic problem will blow up in finite time. This shows that the global weak solutions obtained in Bresch and Desjardins (Commun Math Phys 238(1–2):211–223, 2003) cannot be a classical one as long as the initial data admit an isolated mass group. It also shows that compared with the smoothing properties provided by capillarity and drag terms, the degeneracy of viscosity plays a prominent role in the global regularity problem and leads to finite-time singularity of smooth solutions. PubDate: 2019-03-06 DOI: 10.1007/s00033-019-1093-3

Abstract: We discuss drift–diffusion models for charge carrier transport in organic semiconductor devices. The crucial feature in organic materials is the energetic disorder due to random alignment of molecules and the hopping transport of carriers between adjacent energetic sites. The former leads to statistical relations with Gauss–Fermi integrals, which describe the occupation of energy levels by electrons and holes. The latter gives rise to complicated mobility models with a strongly nonlinear dependence on temperature, density of carriers, and electric field strength. We present the state-of-the-art modeling of the transport processes and provide a first existence result for the stationary drift–diffusion model taking all of the peculiarities of organic materials into account. The existence proof is based on Schauder’s fixed-point theorem. PubDate: 2019-03-06 DOI: 10.1007/s00033-019-1089-z

Abstract: In this paper, we consider the following Kirchhoff problem $$\begin{aligned} -\left( a+b\int \limits _{\mathbb R^3} \nabla u ^2\mathrm{d}x\right) \Delta u+V( x )u=Q( x )u^{p},\quad x\in \mathbb {R}^3, \end{aligned}$$ where \(a,b>0\) , \(1<p<5\) , V(r) and Q(r) are bounded and positive functions. Assume that V(r) and Q(r) have the following expansions $$\begin{aligned} V(r)=1+\frac{d_1}{r^m}+O\left( \frac{1}{r^{m+\theta }}\right) ,\quad Q(r)=1+\frac{d_2}{r^n}+O\left( \frac{1}{r^{n+\kappa }}\right) , \end{aligned}$$ as \(r\rightarrow \infty \) , for some \(d_1>0,d_2\in \mathbb {R}\) , \(m,n>2\) and \(\theta ,\kappa >0\) . Infinitely many nonradial positive solutions are constructed either \(d_2>0\) and \(m<n\) or \(d_2<0\) . PubDate: 2019-03-05 DOI: 10.1007/s00033-019-1099-x

Abstract: Distributed-order fractional model of viscoelastic body is used to describe wave propagation in an infinite media. Existence and uniqueness of fundamental solution to the generalized Cauchy problem is obtained explicitly. The wave propagation speed is found to be related to the material properties at initial time. The fundamental solutions corresponding to four thermodynamically acceptable classes of linear fractional constitutive models and power-type distributed-order models are also obtained. PubDate: 2019-03-04 DOI: 10.1007/s00033-019-1097-z

Abstract: In this paper, the general equilibrium equations for a geometrically nonlinear version of the Timoshenko beam are derived from the energy functional. The particular case in which the shear and extensional stiffnesses are infinite, which correspond to the inextensible Euler beam model, is studied under a uniformly distributed load. All the global and local minimizers of the variational problem are characterized, and the relative monotonicity and regularity properties are established. PubDate: 2019-03-04 DOI: 10.1007/s00033-019-1098-y

Abstract: We are concerned with the following Schrödinger–Poisson systems $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+u+K(x)\phi (x)u=q(x)f(u), &{}\quad x\in {\mathbb {R}}^3, \\ -\Delta \phi =K(x)u^2, &{} \quad x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$ where f is asymptotically cubic, \(\lim _{ x \rightarrow \infty }K(x)=0\) and \(\lim _{ x \rightarrow \infty }q(x)=q_{\infty }>0\) . We establish the existence of bound state solutions to this problem by using the method developed in Szulkin and Weth [20, 21]. PubDate: 2019-03-02 DOI: 10.1007/s00033-019-1096-0

Abstract: This paper considers the chemotaxis–Navier–Stokes system with nonlinear diffusion and logistic-type degradation term $$\begin{aligned} {\left\{ \begin{array}{ll} n_t + u\cdot \nabla n = \nabla \cdot (D(n)\nabla n) - \nabla \cdot (n \chi (c) \nabla c) + \kappa n - \mu n^\alpha , &{}\quad x\in \Omega ,\ t>0, \\ c_t + u\cdot \nabla c = \Delta c - nf(c), &{}\quad x \in \Omega ,\ t>0, \\ u_t + (u\cdot \nabla )u = \Delta u + \nabla P + n\nabla \Phi + g, \quad \nabla \cdot u = 0, &{}\quad x \in \Omega ,\ t>0, \end{array}\right. } \end{aligned}$$ where \(\Omega \subset \mathbb {R}^3\) is a bounded smooth domain; \(D \ge 0\) is a given smooth function such that \(D_1 s^{m-1} \le D(s) \le D_2 s^{m-1}\) for all \(s\ge 0\) with some \(D_2 \ge D_1 > 0\) and some \(m > 0\) ; \(\chi ,f\) are given smooth functions satisfying $$\begin{aligned} { \left( \frac{f}{\chi } \right) ' >0, \quad \left( \frac{f}{\chi } \right) '' \le 0, \quad (\chi f)' \ge 0 \quad \text{ on } \ [0,\infty ); } \end{aligned}$$ \(\kappa \in \mathbb {R},\mu \ge 0,\alpha >1\) are constants. This paper shows existence of global weak solutions to the above system under the condition that $$\begin{aligned} m>\frac{2}{3},\quad \mu \ge 0 \quad \text{ and }\quad \alpha >1 \end{aligned}$$ hold or that $$\begin{aligned} m> 0, \quad \mu>0 \quad \text{ and } \quad \alpha > \frac{4}{3} \end{aligned}$$ hold. This result asserts that “strong” diffusion effect or “strong” logistic damping derives existence of global weak solutions even though the other effect is “weak” and can include previous works (Kurima and Mizukami in Nonlinear Anal Real World Appl, 2018. arXiv:1802.08807 [math.AP]; Lankeit in Math Models Methods Appl Sci 26:2071–2109, 2016; Winkler in Ann Inst H Poincaré Anal Non Linéaire 33:1329–1352, 2016; Zhang and Li in J Differ Equ 259:3730–3754, 2015). PubDate: 2019-02-21 DOI: 10.1007/s00033-019-1092-4

Abstract: We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip subject to a new deformation of “shearing”: the strip is built by translating a segment oriented in a constant direction along an unbounded curve in the plane. We locate the essential spectrum under the hypothesis that the projection of the tangent vector of the curve to the direction of the segment admits a (possibly unbounded) limit at infinity and state sufficient conditions which guarantee the existence of discrete eigenvalues. We justify the optimality of these conditions by establishing a spectral stability in opposite regimes. In particular, Hardy-type inequalities are derived in the regime of repulsive shearing. PubDate: 2019-02-20 DOI: 10.1007/s00033-019-1090-6

Abstract: An equilibrium problem for a 2D elastic body with thin inclusions and defects is analyzed. The presence of defects means that the problem is formulated in a non-smooth domain. The defects are characterized by a positive damage parameter. Nonlinear boundary conditions at the defect faces are imposed to prevent a mutual penetration between the faces. An existence of solutions is proved, and different formulations of the problem are proposed. We study an asymptotics of solutions with respect to the damage parameter and analyze the limit models. Moreover, we study the dependence of the solution on the rigidity parameter of the inclusions. In particular, passages to infinity and to zero of the rigidity parameter are investigated. PubDate: 2019-02-19 DOI: 10.1007/s00033-019-1091-5

Abstract: The feasibility of a realization method for the three-dimensional surface of an aircraft moving with hypersonic velocity is numerically investigated. An approximate mathematical method, which has the same accuracy at much lower computational cost, is used for calculating convective heat transfer over complex geometric shape of a vehicle. Unstructured grids are applied for numerical tests. The hypersonic aircraft computational aerodynamic predictions and comparisons with experimental data are performed. PubDate: 2019-02-19 DOI: 10.1007/s00033-019-1095-1

Abstract: In this paper, we consider an advection–diffusion equation, in one space dimension, whose diffusivity can be negative. Such equations arise in particular in the modeling of vehicular traffic flows or crowds dynamics, where a negative diffusivity simulates aggregation phenomena. We focus on traveling-wave solutions that connect two states whose diffusivity has different signs; under some geometric conditions, we prove the existence, uniqueness (in a suitable class of solutions avoiding plateaus) and sharpness of the corresponding profiles. Such results are then extended to the case of end states where the diffusivity is positive, but it becomes negative in some interval between them. Also the vanishing viscosity limit is considered. At last, we provide and discuss several examples of diffusivities that change sign and show that our conditions are satisfied for a large class of them in correspondence of real data. PubDate: 2019-02-19 DOI: 10.1007/s00033-019-1094-2

Abstract: We study a linear fluid–structure interaction problem between an incompressible, viscous 3D fluid flow, a 2D linearly elastic Koiter shell, and an elastic 1D net of curved rods. This problem is motivated by studying fluid–structure interaction between blood flow through coronary arteries treated with metallic mesh-like devices called stents. The flow is assumed to be laminar, modeled by the time-dependent Stokes equations, and the structure displacement is assumed to be small, modeled by a system of linear Koiter shell equations allowing displacement in all three spatial directions. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic coupling conditions describing continuity of velocity and balance of contact forces. The coupling conditions are evaluated along a linearized fluid–structure interface, which coincides with the fixed fluid domain boundary. No smallness on the structure velocity is assumed. We prove the existence of a weak solution to this linear fluid–composite structure interaction problem. This is the first result in the area of fluid–structure interaction that includes a 1D elastic mesh and takes into account structural displacements in all three spatial directions. Numerical simulations based on the finite element discretization of the coupled FSI problem are presented. PubDate: 2019-02-18 DOI: 10.1007/s00033-019-1087-1