Authors:Cameron L. Hall; Thomas Hudson; Patrick van Meurs Pages: 1 - 54 Abstract: Abstract This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. This setting covers, for instance, pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge in characterising the boundary behaviour is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via \(\Gamma \) -convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0119-0 Issue No:Vol. 153, No. 1 (2018)

Authors:S. Palau; J. C. Pardo Pages: 55 - 79 Abstract: Abstract In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0120-7 Issue No:Vol. 153, No. 1 (2018)

Authors:Francisco Marcellán; Misael Marriaga; Teresa E. Pérez; Miguel A. Piñar Pages: 81 - 100 Abstract: Abstract We consider Koornwinder’s method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder’s construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0121-6 Issue No:Vol. 153, No. 1 (2018)

Authors:Laurent Gosse Pages: 101 - 124 Abstract: Abstract Well-balanced schemes, nowadays mostly developed for both hyperbolic and kinetic equations, are extended in order to handle linear parabolic equations, too. By considering the variational solution of the resulting stationary boundary-value problem, a simple criterion of uniqueness is singled out: the \(C^{1}\) regularity at all knots of the computational grid. Being easy to convert into a finite-difference scheme, a well-balanced discretization is deduced by defining the discrete time-derivative as the defect of \(C^{1}\) regularity at each node. This meets with schemes formerly introduced in the literature relying on so-called ℒ-spline interpolation of discrete values. Various monotonicity, consistency and asymptotic-preserving properties are established, especially in the under-resolved vanishing viscosity limit. Practical experiments illustrate the outcome of such numerical methods. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0122-5 Issue No:Vol. 153, No. 1 (2018)

Authors:Mircea Sofonea; Flavius Pătrulescu; Ahmad Ramadan Pages: 125 - 146 Abstract: We consider a mathematical model which describes the sliding frictional contact between a viscoplastic body and an obstacle, the so-called foundation. The process is quasistatic, the material’s behavior is described with a viscoplastic constitutive law with internal state variable and the contact is modelled with normal compliance and unilateral constraint. The wear of the contact surfaces is taken into account, and is modelled with a version of Archard’s law. We derive a mixed variational formulation of the problem which involve implicit history-dependent operators. Then, we prove the unique weak solvability of the contact model. The proof is based on a fixed point argument proved in Sofonea et al. (Commun. Pure Appl. Anal. 7:645–658, 2008), combined with a recent abstract existence and uniqueness result for mixed variational problems, obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2014). PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0123-4 Issue No:Vol. 153, No. 1 (2018)

Authors:Huashui Zhan Pages: 147 - 161 Abstract: Abstract This paper is mainly about the infiltration equation $$ {u_{t}}= \operatorname{div} \bigl(a(x) u ^{\alpha }{ \vert { \nabla u} \vert ^{p-2}}\nabla u\bigr),\quad (x,t) \in \Omega \times (0,T), $$ where \(p>1\) , \(\alpha >0\) , \(a(x)\in C^{1}(\overline{\Omega })\) , \(a(x)\geq 0\) with \(a(x) _{x\in \partial \Omega }=0\) . If there is a constant \(\beta \) such that \(\int_{\Omega }a^{-\beta }(x)dx\leq c\) , \(p>1+\frac{1}{\beta }\) , then the weak solution is smooth enough to define the trace on the boundary, the stability of the weak solutions can be proved as usual. Meanwhile, if for any \(\beta >\frac{1}{p-1}\) , \(\int_{\Omega }a^{-\beta }(x)dxdt=\infty \) , then the weak solution lacks the regularity to define the trace on the boundary. The main innovation of this paper is to introduce a new kind of the weak solutions. By these new definitions of the weak solutions, one can study the stability of the weak solutions without any boundary value condition. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0124-3 Issue No:Vol. 153, No. 1 (2018)

Authors:B. Al Qurashi; A. H. Kara; H. Akca Pages: 163 - 169 Abstract: Abstract We analyse the symmetry, invariance properties and conservation laws of the partial differential equations (pdes) and minimization problems (variational functionals) that arise in the analyses of some noise removal algorithms. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0125-2 Issue No:Vol. 153, No. 1 (2018)

Authors:Gelson C. G. dos Santos; Giovany M. Figueiredo; Leandro S. Tavares Pages: 171 - 187 Abstract: Abstract In the present paper, we study the existence of solutions for some nonlocal problems involving the \(p(x)\) -Laplacian operator. The approach is based on a new sub-supersolution method. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0126-1 Issue No:Vol. 153, No. 1 (2018)

Authors:Baoquan Yuan; Xiaokui Zhao Pages: 189 - 195 Abstract: Abstract The blow-up of smooth solution to the isentropic compressible Navier-Stokes-Poisson (NSP) system on \(\mathbb{R}^{d}\) is studied in this paper. We obtain that if the initial density is compactly supported, the spherically symmetric smooth solution to the NSP system on \(\mathbb{R}^{d}\ (d\geq 2)\) blows up in finite time. In the case \(d=1\) , if \(2\mu +\lambda >0\) , then the NSP system only exits a zero smooth solution on ℝ for the compactly supported initial density. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0127-0 Issue No:Vol. 153, No. 1 (2018)

Authors:Jie Zhao; Chunlai Mu; Liangchen Wang; Deqin Zhou Pages: 197 - 220 Abstract: Abstract In this paper, we consider the initial-boundary value problem of the two-species chemotaxis Keller-Segel model $$\begin{aligned} \textstyle\begin{cases} u_{t}=\Delta u-\chi_{1}\nabla \cdot (u\nabla w), &x\in \varOmega , \ t>0, \\ v_{t}=\Delta v-\chi_{2}\nabla \cdot (v\nabla w), &x\in \varOmega , \ t>0, \\ 0=\Delta w-\gamma w+\alpha_{1}u+\alpha_{2}v, &x\in \varOmega , \ t>0, \end{cases}\displaystyle \end{aligned}$$ where the parameters \(\chi_{1}\) , \(\chi_{2}\) , \(\alpha_{1}\) , \(\alpha_{2}\) , \(\gamma \) are positive constants, \(\varOmega \subset \mathbb{R}^{2}\) is a bounded domain with smooth boundary. We obtain the results for finite time blow-up and global bounded as follows: (1) For any fixed \(x_{0}\in \varOmega \) , if \(\chi_{1}\alpha_{2}= \chi_{2}\alpha_{1}\) , \(\int_{\varOmega }(u_{0}+v_{0}) x-x_{0} ^{2}dx\) is sufficiently small, and \(\int_{\varOmega }(u_{0}+v_{0})dx>\frac{8\pi ( \chi_{1}\alpha_{1}+\chi_{2}\alpha_{2})}{\chi_{1}\alpha_{1}\chi_{2} \alpha_{2}}\) , then the nonradial solution of the two-species Keller-Segel model blows up in finite time. Moreover, if \(\varOmega \) is a convex domain, we find a lower bound for the blow-up time; (2) If \(\ u_{0}\ _{L^{1}(\varOmega )}\) and \(\ v_{0}\ _{L^{1}( \varOmega )}\) lie below some thresholds, respectively, then the solution exists globally and remains bounded. PubDate: 2018-02-01 DOI: 10.1007/s10440-017-0128-z Issue No:Vol. 153, No. 1 (2018)

Authors:J. López-Salazar; G. Pérez-Villalón Pages: 73 - 82 Abstract: Abstract Given a sequence of data \(\{ y_{n} \} _{n \in \mathbb{Z}}\) with polynomial growth and an odd number \(d\) , Schoenberg proved that there exists a unique cardinal spline \(f\) of degree \(d\) with polynomial growth such that \(f ( n ) =y_{n}\) for all \(n\in \mathbb{Z}\) . In this work, we show that this result also holds if we consider weighted average data \(f\ast h ( n ) =y_{n}\) , whenever the average function \(h\) satisfies some light conditions. In particular, the interpolation result is valid if we consider cell-average data \(\int_{n-a}^{n+a}f ( x ) dx=y_{n}\) with \(0< a\leq 1/2\) . The case of even degree \(d\) is also studied. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0112-7 Issue No:Vol. 152, No. 1 (2017)

Authors:Xiaojun Cui; Jian Cheng Pages: 93 - 110 Abstract: Abstract On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold there are two kinds of functions: Busemann functions with respect to rays and barrier functions with respect to lines (if there exists at least one). In this paper we collect some known properties on Busemann functions and introduce some new fundamental properties on barrier functions. Based on these properties of barrier functions, we could define some relations on the set of lines and thus classify them. With the equivalence relation we introduced, we present a generalization of a rigidity conjecture. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0114-5 Issue No:Vol. 152, No. 1 (2017)

Authors:Salih Djilali; Tarik Mohammed Touaoula; Sofiane El-Hadi Miri Pages: 171 - 194 Abstract: Abstract We consider an age structured heroin epidemic model, in a population divided into three sub-populations: \(S\) the susceptible individuals, \(U_{1}\) the drug users and \(U_{2}\) the drug users under treatment, interacting as follows: $$ \left \{ \textstyle\begin{array}{l} S'=A-\mu S-F ( S,U_{1} ) , \\ U_{1}'=F ( S,U_{1} ) - ( \mu +\delta_{1}+p ) U_{1}+\int_{0}^{\infty }k ( a ) U_{2} ( t,a ) da, \\ \frac{\partial U_{2}}{\partial t}+\frac{\partial U_{2}}{\partial a}=- ( \mu +\delta_{2}+k ( a ) ) U_{2}. \end{array}\displaystyle \right . $$ Our main contribution consists in considering a nonlinear incidence function \(F(S,U_{1})\) in its very general form. Global dynamics of the obtained problem is analyzed. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0117-2 Issue No:Vol. 152, No. 1 (2017)

Authors:Sabrine Arfaoui; Anouar Ben Mabrouk Abstract: Abstract In the present paper, new classes of wavelet functions are developed in the framework of Clifford analysis. Firstly, some classes of orthogonal polynomials are provided based on two-parameters weight functions generalizing the well known Jacobi and Gegenbauer classes when relaxing the parameters. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula as well as Fourier-Plancherel rule have been proved. PubDate: 2017-12-28 DOI: 10.1007/s10440-017-0150-1

Authors:Samuel Nordmann; Benoît Perthame; Cécile Taing Abstract: Abstract We study a mathematical model describing the growth process of a population structured by age and a phenotypical trait, subject to aging, competition between individuals and rare mutations. Our goals are to describe the asymptotic behavior of the solution to a renewal type equation, and then to derive properties that illustrate the adaptive dynamics of such a population. We begin with a simplified model by discarding the effect of mutations, which allows us to introduce the main ideas and state the full result. Then we discuss the general model and its limitations. Our approach uses the eigenelements of a formal limiting operator, that depend on the structuring variables of the model and define an effective fitness. Then we introduce a new method which reduces the convergence proof to entropy estimates rather than estimates on the constrained Hamilton-Jacobi equation. Numerical tests illustrate the theory and show the selection of a fittest trait according to the effective fitness. For the problem with mutations, an unusual Hamiltonian arises with an exponential growth, for which we prove existence of a global viscosity solution, using an uncommon a priori estimate and a new uniqueness result. PubDate: 2017-12-18 DOI: 10.1007/s10440-017-0151-0

Authors:Xin Zeng; Xianguo Geng Abstract: Abstract Based on the characteristic polynomial of Lax matrix for the hierarchy of coupled Toda lattices associated with a \(3\times3\) discrete matrix spectral problem, we introduce a trigonal curve with two infinite points, from which we establish the associated Dubrovin-type equations. The asymptotic properties of the meromorphic function and the Baker-Akhiezer function are studied near two infinite points on the trigonal curve. Finite-band solutions of the entire hierarchy of coupled Toda lattices are obtained in terms of the Riemann theta function. PubDate: 2017-12-14 DOI: 10.1007/s10440-017-0133-2

Authors:Yanyan Zhang; Yu Zhang Abstract: Abstract By the vanishing viscosity approach, a class of non-strictly hyperbolic systems of conservation laws that contain the equations of geometrical optics as a prototype are studied. The existence, uniqueness and stability of solutions involving delta shock waves and generalized vacuum states are discussed completely. PubDate: 2017-12-08 DOI: 10.1007/s10440-017-0149-7

Authors:Séverine Bernard; Catherine Cabuzel; Silvère Paul Nuiro; Alain Pietrus Abstract: Abstract This paper deals with variational inclusions of the form \(0 \in K-f(x)\) where \(f : \mathbb{R}^{n} \rightarrow \mathbb{R} ^{m}\) is a semismooth function and \(K\) is a nonempty closed convex cone in \(\mathbb{R}^{m}\) . We show that the previous problem can be solved by a Newton-type method using the Clarke generalized Jacobian of \(f\) . The results obtained in this paper extend those obtained by Robinson in the famous paper (Robinson in Numer. Math. 19:341–347, 1972). We provide a semilocal method with a superlinear convergence that is new in the context of semismooth functions. Finally, numerical results are also given to illustrate the convergence. PubDate: 2017-12-05 DOI: 10.1007/s10440-017-0146-x

Authors:Zuomao Yan; Fangxia Lu Abstract: Abstract In this paper, a new class of fractional impulsive stochastic partial integro-differential control systems with state-dependent delay and their optimal controls in a Hilbert space is studied. We firstly prove an existence result of mild solutions for the control systems by using stochastic analysis, analytic \(\alpha \) -resolvent operator, fractional powers of closed operators and suitable fixed point theorems. Then we derive the existence conditions of optimal pairs to the fractional impulsive stochastic control systems. Finally, an example is given to illustrate the effectiveness of our main results. PubDate: 2017-12-04 DOI: 10.1007/s10440-017-0145-y

Authors:Ya-Lun Tsai Abstract: Abstract In “Counting central configurations at the bifurcation points,” we proposed an algorithm to rigorously count central configurations in some cases that involve one parameter. Here, we improve our algorithm to consider three harder cases: the planar \((3+1)\) -body problem with two equal masses; the planar 4-body problem with two pairs of equal masses which have an axis of symmetry containing one pair of them; the spatial 5-body problem with three equal masses at the vertices of an equilateral triangle and two equal masses on the line passing through the center of the triangle and being perpendicular to the plane containing it. While all three problems have been studied in two parameter cases, numerical observations suggest new results at some points on the bifurcation curves. Applying the improved version of our algorithm, we count at those bifurcation points. As a result, for the \((3+1)\) -body problem, we identify three points on the bifurcation curve where there are 8 central configurations, which adds to the known results of \(8,9,10\) ones. For our 4-body case, at the bifurcation points, there are 3 concave central configurations, which adds to the known results of \(2,4\) ones. For our 5-body case, at the bifurcation point, there is 1 concave central configuration, which adds to the known results of \(0,2\) ones. PubDate: 2017-12-04 DOI: 10.1007/s10440-017-0147-9