Authors:Cameron L. Hall; Thomas Hudson; Patrick van Meurs Pages: 1 - 54 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: 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: 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: 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: 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: 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: 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: 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: 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:Abdelmouhcene Sengouga Abstract: By mean of generalized Fourier series and Parseval’s equality in weighted \(L^{2}\) -spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. The observability constants are explicitly given. Using the Hilbert Uniqueness Method we deduce the exact boundary controllability of the wave equation. PubDate: 2018-02-27 DOI: 10.1007/s10440-018-0166-1

Authors:Anselmo Torresblanca-Badillo; W. A. Zúñiga-Galindo Abstract: In this article we study certain ultradiffusion equations connected with energy landscapes of exponential type. These equations are connected with the \(p\) -adic models of complex systems introduced by Avetisov et al. We show that the fundamental solutions of these equations are transition density functions of Lévy processes with state space \(\mathbb{Q}_{p}^{n}\) , we also study some aspects of these processes including the first passage time problem. PubDate: 2018-02-22 DOI: 10.1007/s10440-018-0165-2

Authors:S. Mischler; Q. Weng Abstract: In order to describe the firing activity of a homogenous assembly of neurons, we consider time elapsed models, which give mathematical descriptions of the probability density of neurons structured by the distribution of times elapsed since the last discharge. Under general assumption on the firing rate and the delay distribution, we prove the uniqueness of the steady state and its nonlinear exponential stability in the weak connectivity regime. In other words, total asynchronous firing of neurons appears asymptotically in large time. The result generalizes some similar results obtained in Pakdaman et al. (Nonlinearity 23(1):55–75, 2010) and Pakdaman et al. (SIAM J. Appl. Math. 73(3):1260–1279, 2013) in the case without delay. Our approach uses the spectral analysis theory for semigroups in Banach spaces developed recently by the first author and collaborators. PubDate: 2018-02-08 DOI: 10.1007/s10440-018-0163-4

Authors:Zhipeng Cheng; Minbo Yang Abstract: We are going to study the standing waves for the generalized Choquard equation with potential: $$ -i\partial_{t} u-\Delta u+V(x)u=\bigl( x ^{-1}\ast u ^{p}\bigr) u ^{p-2}u,\quad \hbox{in } \mathbb{R}\times \mathbb{R}^{3}, $$ where \(V(x)\) is a real function, \(p>2\) is close to 2 and ∗ standards for the convolution in \(\mathbb{R}^{3}\) . The stability of the standing waves \(u(x)=e^{i\omega t}\varphi (x)\) is investigated under suitable assumptions on the potential and the frequency \(\omega \) . PubDate: 2018-02-06 DOI: 10.1007/s10440-018-0162-5

Authors:Trubee Davison Abstract: Given an iterated function system (IFS) on a complete and separable metric space \(Y\) , there exists a unique compact subset \(X \subseteq Y\) satisfying a fixed point relation with respect to the IFS. This subset is called the attractor set, or fractal set, associated to the IFS. The attractor set supports a specific Borel probability measure, called the Hutchinson measure, which itself satisfies a fixed point relation. P. Jorgensen generalized the Hutchinson measure to a projection-valued measure, under the assumption that the IFS does not have essential overlap (Jorgensen in Adv. Appl. Math. 34(3):561–590, 2005; Operator Theory, Operator Algebras, and Applications, pp. 13–26, 2006). In previous work, we developed an alternative approach to proving the existence of this projection-valued measure (Davison in Acta Appl. Math. 140(1):11–22, 2015; Acta Appl. Math. 140(1):23–25, 2015; Generalizing the Kantorovich metric to projection-valued measures: with an application to iterated function systems, 2015). The situation when the IFS exhibits essential overlap has been studied by Jorgensen and colleagues in Jorgenson et al. (J. Math. Phys. 48(8):083511, 35, 2007). We build off their work to generalize the Hutchinson measure to a positive-operator valued measure for an arbitrary IFS, that may exhibit essential overlap. This work hinges on using a generalized Kantorovich metric to define a distance between positive operator-valued measures. It is noteworthy to mention that this generalized metric, which we use in our previous work as well, was also introduced by R.F. Werner to study the position and momentum observables, which are central objects of study in the area of quantum theory (Werner in J. Quantum Inf. Comput. 4(6):546–562, 2004). We conclude with a discussion of Naimark’s dilation theorem with respect to this positive operator-valued measure, and at the beginning of the paper, we prove a metric space completion result regarding the classical Kantorovich metric. PubDate: 2018-02-05 DOI: 10.1007/s10440-018-0161-6

Authors:Guangwu Wang; Boling Guo Abstract: In this paper, we investigate the blow-up of the smooth solutions to a simplified Ericksen-Lesile system for compressible flows of nematic liquid crystals in different dimensional case. We prove that whether the smooth solution of the Cauchy problem or the initial-boundary problem to the nematic liquid crystal system will blow up in finite time. The main technique is the construction of the functional differential inequality. PubDate: 2018-01-26 DOI: 10.1007/s10440-018-0160-7

Authors:Z. Amiri; R. A. Kamyabi-Gol Abstract: Thresholding and compressed sensing in combination with both wavelet and shearlet transforms have been very successful in inpainting tasks. Recent results have demonstrated that shearlets outperform wavelets in the problem of image inpainting. In this paper, we provide a general framework for universal shearlet systems in high dimensions. This theoretical framework is used to analyze the recovery of missing data via \(\ell^{1} \) minimization in an abstract model situation. In addition, we set up a particular model inspired by seismic data and a box mask to model missing data. Finally, the results of numerical experiments comparing various inpainting methods are presented. PubDate: 2018-01-25 DOI: 10.1007/s10440-018-0159-0

Authors:Shapour Heidarkhani; Massimiliano Ferrara; Giuseppe Caristi; Amjad Salari Abstract: This article is concerned with the multiplicity results of the solutions for a Kirchhoff-type three-point boundary value problem. The method observed here is according to variational methods and critical point theory. In fact, through a consequence of the local minimum theorem due Bonanno and mountain pass theorem, we look into the existence results for our problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term, employing two consequences of the local minimum theorem due Bonanno we guarantee the existence of two solutions, and applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for the problem. PubDate: 2018-01-23 DOI: 10.1007/s10440-018-0157-2

Authors:Vincenzo Ambrosio Abstract: In this paper we deal with a one-dimensional free-boundary problem arising in the modeling of concrete carbonation process. More precisely, we investigate global existence, uniqueness and large-time behavior of weak solutions for the problem under consideration. We also obtain the existence of a weak solution when the measure of the initial domain vanishes. PubDate: 2018-01-19 DOI: 10.1007/s10440-018-0156-3

Authors:Huiling Lin Abstract: We first discuss some properties of the solution set of a pseudomonotone second-order cone linear complementarity problem (SOCLCP), and then analyse the limiting behavior of a sequence of strictly feasible solutions within a new wide neighborhood of the central trajectory for the pseudomonotone SOCLCP under assumptions of strict complementarity. Based on this, we derive four different characterizations of an error bound for the pseudomonotone SOCLCP. PubDate: 2018-01-18 DOI: 10.1007/s10440-018-0158-1