Authors:M. L. Santos; A. D. S. Campelo; D. S. Almeida Júnior Pages: 1 - 26 Abstract: Abstract In this work we are considering the porous elastic system with porous elastic dissipation and with elastic dissipation. Our main result is to show that the corresponding semigroup is exponentially stable if and only if the wave speeds of the system are equal. In the case of lack of exponential stability we show that the solution decays polynomially and we prove that the rate of decay is optimal. It is worth noting that the result obtained here is different from all existing in the literature for porous elastic materials, where the sum of the two slow decay processes determine a process that decay exponentially. Numerical experiments using finite differences are given to confirm our analytical results. Our numerical results are qualitatively in agreement with the corresponding results from dynamical in infinite dimensional. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0100-y Issue No:Vol. 151, No. 1 (2017)

Authors:Roberto Castelli Pages: 27 - 52 Abstract: 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: 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: 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:A. M. Grundland; V. Lamothe Pages: 89 - 119 Abstract: Abstract The generalized method of characteristics is used to obtain rank-2 solutions of the classical equations of hydrodynamics in ( \(3+1\) ) dimensions describing the motion of a fluid medium in the presence of gravitational and Coriolis forces. We determine the necessary and sufficient conditions which guarantee the existence of solutions expressed in terms of Riemann invariants for an inhomogeneous quasilinear system of partial differential equations. The paper contains a detailed exposition of the theory of simple wave solutions and a presentation of the main tool used to study the Cauchy problem. A systematic use is made of the generalized method of characteristics in order to generate several classes of wave solutions written in terms of Riemann invariants. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0104-7 Issue No:Vol. 151, No. 1 (2017)

Authors:Huiling Li; Yang Zhang Pages: 121 - 148 Abstract: 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:Cong Nhan Le; Xuan Truong Le Pages: 149 - 169 Abstract: Abstract The main goal of this work is to study an initial boundary value problem for a quasilinear parabolic equation with logarithmic source term. By using the potential well method and a logarithmic Sobolev inequality, we obtain results of existence or nonexistence of global weak solutions. In addition, we also provided sufficient conditions for the large time decay of global weak solutions and for the finite time blow-up of weak solutions. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0106-5 Issue No:Vol. 151, No. 1 (2017)

Authors:Claudianor O. Alves; Ailton R. da Silva Pages: 171 - 198 Abstract: Abstract The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems $$ - \mbox{div} \bigl(\epsilon^{2}\phi\bigl(\epsilon \nabla u \bigr)\nabla u \bigr) + V(x)\phi\bigl(\vert u\vert\bigr)u = f(u)\quad\mbox{in } \mathbb{R}^{N}, $$ where \(\epsilon\) is a positive parameter, \(N\geq2\) , \(V\) , \(f\) are continuous functions satisfying some technical conditions and \(\phi\) is a \(C^{1}\) -function. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0107-4 Issue No:Vol. 151, No. 1 (2017)

Authors:Kuo-Shou Chiu Pages: 199 - 226 Abstract: Abstract In this paper we introduce an impulsive cellular neural network models with piecewise alternately advanced and retarded argument. The model with the advanced argument is system with strong anticipation. Some sufficient conditions are established for the existence and global exponential stability of a unique periodic solution. The approaches are based on employing Banach’s fixed point theorem and a new integral inequality of Gronwall type with impulses and deviating arguments. The criteria given are easily verifiable, possess many adjustable parameters, and depend on impulses and piecewise constant argument deviations, which provides flexibility for the design and analysis of cellular neural network models. Several numerical examples and simulations are also given to show the feasibility and effectiveness of our results. PubDate: 2017-10-01 DOI: 10.1007/s10440-017-0108-3 Issue No:Vol. 151, No. 1 (2017)

Authors:Gelson C. G. dos Santos; Giovany M. Figueiredo; Leandro S. Tavares 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: 2017-09-20 DOI: 10.1007/s10440-017-0126-1

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: 2017-09-07 DOI: 10.1007/s10440-017-0125-2

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: 2017-09-07 DOI: 10.1007/s10440-017-0119-0

Authors:S. Palau; J. C. Pardo 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: 2017-08-31 DOI: 10.1007/s10440-017-0120-7

Authors:Mircea Sofonea; Flavius Pătrulescu; Ahmad Ramadan 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: 2017-08-28 DOI: 10.1007/s10440-017-0123-4

Authors:Laurent Gosse 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: 2017-08-25 DOI: 10.1007/s10440-017-0122-5

Authors:Francisco Marcellán; Misael Marriaga; Teresa E. Pérez; Miguel A. Piñar 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: 2017-08-24 DOI: 10.1007/s10440-017-0121-6

Authors:Huashui Zhan 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: 2017-08-24 DOI: 10.1007/s10440-017-0124-3

Authors:Jae-Myoung Kim Abstract: Abstract We present an interior regularity condition for suitable weak solutions with respect to the magnetic pressure under the class of scaling invariance. PubDate: 2017-08-14 DOI: 10.1007/s10440-017-0113-6

Authors:Ewa Zadrzyńska; Wojciech M. Zaja̧czkowski Abstract: 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: 2017-08-03 DOI: 10.1007/s10440-017-0116-3

Authors:Roelof Bruggeman; Ferdinand Verhulst Abstract: 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: 2017-08-02 DOI: 10.1007/s10440-017-0115-4