Authors:Lance Nielsen Pages: 1 - 31 Abstract: We establish a comprehensive stability theory for Feynman’s operational calculus (informally, the forming of functions of several noncommuting operators) in the time-dependent setting. Indeed, the main theorem, Theorem 2, contains many of the current stability theorems for the operational calculus and allows the stability theory to be significantly extended. The assumptions needed for the main theorem, Theorem 2, are rather mild and fit in nicely with the current abstract theory of the operational calculus in the time-dependent setting. Moreover, Theorem 2 allows the use of arbitrary time-ordering measures, as long as the discrete parts of these measures are finitely supported. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0109-2 Issue No:Vol. 152, No. 1 (2017)

Authors:Silvia Bertoluzza; Valérie Perrier Pages: 33 - 56 Abstract: In this article we introduce a new mixed Lagrange–Hermite interpolating wavelet family on the interval, to deal with two types (Dirichlet and Neumann) of boundary conditions. As this construction is a slight modification of the interpolating wavelets on the interval of Donoho, it leads to fast decomposition, error estimates and norm equivalences. This new basis is then used in adaptive wavelet collocation schemes for the solution of one dimensional fourth order problems. Numerical tests conducted on the 1D Euler–Bernoulli beam problem, show the efficiency of the method. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0110-9 Issue No:Vol. 152, No. 1 (2017)

Authors:Wenbin Yang Pages: 57 - 72 Abstract: The paper is concerned with a predator-prey diffusive system subject to homogeneous Neumann boundary conditions, where the growth rate \((\frac{\alpha}{1+\beta v})\) of the predator population is nonlinear. We study the existence of equilibrium solutions and the long-term behavior of the solutions. The main tools used here include the super-sub solution method, the bifurcation theory and linearization method. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0111-8 Issue No:Vol. 152, No. 1 (2017)

Authors:J. López-Salazar; G. Pérez-Villalón Pages: 73 - 82 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:Jae-Myoung Kim Pages: 83 - 91 Abstract: We present an interior regularity condition for suitable weak solutions with respect to the magnetic pressure under the class of scaling invariance. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0113-6 Issue No:Vol. 152, No. 1 (2017)

Authors:Xiaojun Cui; Jian Cheng Pages: 93 - 110 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:Roelof Bruggeman; Ferdinand Verhulst Pages: 111 - 145 Abstract: A 4-particles chain with different masses represents a natural generalization of the classical Fermi-Pasta-Ulam chain. It is studied by identifying the mass ratios that produce prominent resonances. This is a technically complicated problem as we have to solve an inverse problem for the spectrum of the corresponding linearized equations of motion. In the case of such an inhomogeneous periodic chain with four particles each mass ratio determines a frequency ratio for the quadratic part of the Hamiltonian. Most prominent frequency ratios occur but not all. In general we find a one-dimensional variety of mass ratios for a given frequency ratio. A detailed study is presented of the resonance \(1:2:3\) . A small cubic term added to the Hamiltonian leads to a dynamical behaviour that shows a difference between the case that two opposite masses are equal and a striking difference with the classical case of four equal masses. For two equal masses and two different ones the normalized system is integrable and chaotic behaviour is small-scale. In the transition to four different masses we find a Hamiltonian-Hopf bifurcation of one of the normal modes leading to complex instability and Shilnikov-Devaney bifurcation. The other families of short-periodic solutions can be localized from the normal forms together with their stability characteristics. For illustration we use action simplices and examples of behaviour with time. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0115-4 Issue No:Vol. 152, No. 1 (2017)

Authors:Ewa Zadrzyńska; Wojciech M. Zaja̧czkowski Pages: 147 - 170 Abstract: In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove existence of 3d global strong solutions via stability. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0116-3 Issue No:Vol. 152, No. 1 (2017)

Authors:Salih Djilali; Tarik Mohammed Touaoula; Sofiane El-Hadi Miri Pages: 171 - 194 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:Yi-rong Jiang; Nan-jing Huang; Donal O’Regan Pages: 195 - 210 Abstract: The main purpose of this paper is to establish variational inequality theory in connection with demicontinuous \(\psi_{p}\) -dissipative maps in reflexive smooth Banach spaces by considering the convergence of approximants. As an application of this variational inequality theory, existence, uniqueness and convergence of approximants of positive weak solution for \(p\) -Laplacian elliptic inequalities are obtained under suitable conditions. PubDate: 2017-12-01 DOI: 10.1007/s10440-017-0118-1 Issue No:Vol. 152, No. 1 (2017)

Authors:Roberto Castelli Pages: 27 - 52 Abstract: In this paper a method to rigorously compute several non trivial solutions of the Gray-Scott reaction-diffusion system defined on a 2-dimensional bounded domain is presented. It is proved existence, within rigorous bounds, of non uniform patterns significantly far from being a perturbation of the homogenous states. As a result, a non local diagram of families that bifurcate from the homogenous states is depicted, also showing coexistence of multiple solutions at the same parameter values. Combining analytical estimates and rigorous computations, the solutions are sought as fixed points of a operator in a suitable Banach space. To address the curse of dimensionality, a variation of the existing technique is presented, necessary to enable successful computations in reasonable time. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0101-x Issue No:Vol. 151, No. 1 (2017)

Authors:Balázs Boros; Josef Hofbauer; Stefan Müller Pages: 53 - 80 Abstract: Chemical reaction networks with generalized mass-action kinetics lead to power-law dynamical systems. As a simple example, we consider the Lotka reactions with two chemical species and arbitrary power-law kinetics. We study existence, uniqueness, and stability of the positive equilibrium, in particular, we characterize its global asymptotic stability in terms of the kinetic orders. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0102-9 Issue No:Vol. 151, No. 1 (2017)

Authors:Wolfgang Bock; Torben Fattler; Ludwig Streit Pages: 81 - 88 Abstract: We prove that there exists a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion, \(\mu_{ {g,H}}\) , \(H\in (0,1)\) for \(dH < 1\) . The diffusion is constructed in the framework of Dirichlet forms in infinite dimensional (Gaussian) analysis. Moreover, the process is invariant under time translations. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0103-8 Issue No:Vol. 151, No. 1 (2017)

Authors:Huiling Li; Yang Zhang Pages: 121 - 148 Abstract: This paper concerns global existence and finite time blow-up behavior of positive solutions for a nonlinear reaction-diffusion system with different diffusion coefficients. By use of algebraic matrix theory and modern analytical theory, we extend results of Wang (Z. Angew. Math. Phys. 51:160–167, 2000) to a more general system. Furthermore, we give a complete answer to the open problem which was brought forward in Wang (Z. Angew. Math. Phys. 51:160–167, 2000). PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0105-6 Issue No:Vol. 151, No. 1 (2017)

Authors:Dongfeng Zhang; Junxiang Xu Abstract: In this paper we prove the existence of a Gevrey family of invariant curves for analytic area preserving mappings. The Gevrey smoothness is expressed by Gevrey index. We specifically obtain the Gevrey index of families of invariant curves which is related to the smoothness of area preserving mappings and the exponent of small divisors condition. Moreover, we obtain a Gevrey normal form of area preserving mappings in a neighborhood of the union of the invariant curves. PubDate: 2017-11-09 DOI: 10.1007/s10440-017-0131-4

Authors:Rostislav Vodák; Pavel Ženčák Abstract: In the paper we introduce a new approximation scheme for modelling a spherical cloud of cavitation bubbles based upon a model developed in (Wang and Brennen in J. Fluids Eng. 121(4):872–880, 1999) which consists of fully nonlinear continuum mixture equations coupled with the Rayleigh-Plesset equation for dynamics of the bubbles. We prove existence of a unique, local-in-time solution to the equations using the Banach fixed-point theorem which also provides us with the convergent scheme for a numerical simulation of the solution. We further demonstrate acquired numerical results. PubDate: 2017-11-07 DOI: 10.1007/s10440-017-0132-3

Authors:Weiren Zhao Abstract: In this paper, we prove that Besov regularity of the initial data can persist for a generalized drift-diffusion equation with pressure under a very weak condition on the drift velocity. In particular, the solution is Hölder continuous. PubDate: 2017-11-06 DOI: 10.1007/s10440-017-0134-1

Authors:Paulo M. de Carvalho-Neto; Renato Fehlberg Júnior Abstract: When addressing ordinary differential equations in infinite dimensional Banach spaces, an interesting question that arises concerns the existence (or non existence) of blowing up solutions in finite time. In this manuscript we discuss this question for the fractional differential equation \(cD_{t}^{\alpha}u=f(t,u)\) proving that when \(f\) is locally Lipschitz in the second variable, uniformly with respect to the first variable, however does not maps bounded sets into bounded sets, we can construct a maximal local solution that does not “blow up” in finite time. PubDate: 2017-11-06 DOI: 10.1007/s10440-017-0130-5

Authors:Farid Bozorgnia Abstract: We study a class of elliptic competition-diffusion systems of long range segregation models for two and more competing species. We prove the uniqueness result for positive solution of those elliptic and related parabolic systems when the coupling in the right hand side involves a non-local term of integral form. Moreover, alternate proofs of some known results, such as existence of solutions in the elliptic case and the limiting configuration are given. The free boundary condition in a particular setting is given. PubDate: 2017-11-03 DOI: 10.1007/s10440-017-0129-y

Authors:Baoquan Yuan; Xiaokui Zhao 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: 2017-10-09 DOI: 10.1007/s10440-017-0127-0