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  • Articles  (503)
  • PAPER CURRENT  (503)
  • ZAMP - Zeitschrift für Angewandte Mathematik und Physik  (503)
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  • Articles  (503)
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  • PAPER CURRENT  (503)
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  • 1
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    Finite-time blow-up in low-dimensional Keller–Segel systems with logistic-type superlinear degradation (2018)
    Springer
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    Publication Date: 2018-03-07
    Description: We consider radially symmetric solutions of the Keller–Segel system with generalized logistic source given by $$\begin{aligned} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) + \lambda u - \mu u^\kappa , \\ 0 = \Delta v - v + u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ under homogeneous Neumann boundary conditions in the ball \(\Omega =B_R(0) \subset \mathbb {R}^n\) for \(n\ge 3\) and \(R〉0\) , where \(\lambda \in \mathbb {R}, \mu 〉0\) and \(\kappa 〉1\) . Under the assumption that $$\begin{aligned} \kappa 〈 \left\{ \begin{array}{ll} \frac{7}{6} &{}\quad \text {if } n\in \{3,4\}, \\ 1+ \frac{1}{2(n-1)} &{}\quad \text {if } n \ge 5, \end{array} \right. \end{aligned}$$ a condition on the initial data is derived which is seen to be sufficient to ensure the occurrence of finite-time blow-up for the corresponding solution of ( \(\star \) ). Moreover, this criterion is shown to be mild enough so as to allow for the conclusion that in fact any positive continuous radial function on \(\overline{\Omega }\) is the limit in \(L^1(\Omega )\) of a sequence \((u_{0k})_{k\in \mathbb {N}}\) of continuous radial initial data which are such that for each \(k\in \mathbb {N}\) the associated initial-boundary value problem for ( \(\star \) ) exhibits a finite-time explosion phenomenon in the above sense. In particular, this apparently provides the first rigorous detection of blow-up in a superlinearly dampened but otherwise essentially original Keller–Segel system in the physically relevant three-dimensional case.
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    Electronic ISSN: 1420-9039
    Topics: Mathematics , Physics
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  • 2
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    Exact solution for a two-phase Stefan problem with variable latent heat and a convective boundary condition at the fixed face (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: Recently, in Tarzia (Thermal Sci 21A:1–11, 2017 ) for the classical two-phase Lamé–Clapeyron–Stefan problem an equivalence between the temperature and convective boundary conditions at the fixed face under a certain restriction was obtained. Motivated by this article we study the two-phase Stefan problem for a semi-infinite material with a latent heat defined as a power function of the position and a convective boundary condition at the fixed face. An exact solution is constructed using Kummer functions in case that an inequality for the convective transfer coefficient is satisfied generalizing recent works for the corresponding one-phase free boundary problem. We also consider the limit to our problem when that coefficient goes to infinity obtaining a new free boundary problem, which has been recently studied in Zhou et al. (J Eng Math 2017 . https://doi.org/10.1007/s10665-017-9921-y ).
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  • 3
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    Monostable traveling waves for a time-periodic and delayed nonlocal reaction–diffusion equation (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: This paper is concerned with a time-periodic and delayed nonlocal reaction–diffusion population model with monostable nonlinearity. Under quasi-monotone or non-quasi-monotone assumptions, it is known that there exists a critical wave speed \(c_*〉0\) such that a periodic traveling wave exists if and only if the wave speed is above \(c_*\) . In this paper, we first prove the uniqueness of non-critical periodic traveling waves regardless of whether the model is quasi-monotone or not. Further, in the quasi-monotone case, we establish the exponential stability of non-critical periodic traveling fronts. Finally, we illustrate the main results by discussing two types of death and birth functions arising from population biology.
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  • 4
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    Homogenization of Winkler–Steklov spectral conditions in three-dimensional linear elasticity (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: We consider a homogenization Winkler–Steklov spectral problem that consists of the elasticity equations for a three-dimensional homogeneous anisotropic elastic body which has a plane part of the surface subject to alternating boundary conditions on small regions periodically placed along the plane. These conditions are of the Dirichlet type and of the Winkler–Steklov type, the latter containing the spectral parameter. The rest of the boundary of the body is fixed, and the period and size of the regions, where the spectral parameter arises, are of order \(\varepsilon \) . For fixed \(\varepsilon \) , the problem has a discrete spectrum, and we address the asymptotic behavior of the eigenvalues \(\{\beta _k^\varepsilon \}_{k=1}^{\infty }\) as \(\varepsilon \rightarrow 0\) . We show that \(\beta _k^\varepsilon =O(\varepsilon ^{-1})\) for each fixed k , and we observe a common limit point for all the rescaled eigenvalues \(\varepsilon \beta _k^\varepsilon \) while we make it evident that, although the periodicity of the structure only affects the boundary conditions, a band-gap structure of the spectrum is inherited asymptotically. Also, we provide the asymptotic behavior for certain “groups” of eigenmodes.
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  • 5
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    Asymptotic profile of global solutions to the generalized double dispersion equation via the nonlinear term (2018)
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    Publication Date: 2018-03-06
    Description: In this paper, we investigate the initial value problem for the generalized double dispersion equation in \({\mathbb {R}}^n\) . Weighted decay estimate and asymptotic profile of global solutions are established for \(n\ge 3 \) . The global existence result was already proved by Kawashima and the first author in Kawashima and Wang (Anal Appl 13:233–254, 2015 ). Here, we show that the nonlinear term plays an important role in this asymptotic profile.
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  • 6
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    A class of fractional differential hemivariational inequalities with application to contact problem (2018)
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    Publication Date: 2018-03-06
    Description: In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin–Voigt law.
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  • 7
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    On the stabilization of viscoelastic laminated beams with interfacial slip (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: In this paper, we consider a viscoelastic laminated beam model. This structure is given by two identical uniform layers on top of each other, taking into account that an adhesive of small thickness is bonding the two surfaces and produces an interfacial slip. We use viscoelastic damping with general assumptions on the relaxation function and establish explicit energy decay result from which we can recover the optimal exponential and polynomial rates. Our result generalizes the earlier related results in the literature.
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  • 8
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    Finiteness of corner vortices (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: Till date, the sequence of vortices present in the solid corners of steady internal viscous incompressible flows was thought to be infinite. However, the already existing and most recent geometric theories on incompressible viscous flows that express vortical structures in terms of critical points in bounded domains indicate a strong opposition to this notion of infiniteness. In this study, we endeavor to bridge the gap between the two opposing stream of thoughts by diagnosing the assumptions of the existing theorems on such vortices. We provide our own set of proofs for establishing the finiteness of the sequence of corner vortices by making use of the continuum hypothesis and Kolmogorov scale, which guarantee a nonzero scale for the smallest vortex structure possible in incompressible viscous flows. We point out that the notion of infiniteness resulting from discrete self-similarity of the vortex structures is not physically feasible. Making use of some elementary concepts of mathematical analysis and our own construction of diametric disks, we conclude that the sequence of corner vortices is finite.
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  • 9
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    Blowup for biharmonic Schrödinger equation with critical nonlinearity (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: We consider the minimizers for the biharmonic nonlinear Schrödinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int \limits _{\mathbb {R}^d} |\Delta u(x)|^2 \mathrm{d}x + \int \limits _{\mathbb {R}^d} V(x) |u(x)|^2 \mathrm{d}x - a \int \limits _{\mathbb {R}^d} |u(x)|^{q} \mathrm{d}x \end{aligned}$$ with the mass constraint \(\int |u|^2=1\) . We focus on the special power \(q=2(1+4/d)\) , which makes the nonlinear term \(\int |u|^q\) scales similarly to the biharmonic term \(\int |\Delta u|^2\) . Our main results are the existence and blowup behavior of the minimizers when a tends to a critical value \(a^*\) , which is the optimal constant in a Gagliardo–Nirenberg interpolation inequality.
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  • 10
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    Asymptotic stability of shear-flow solutions to incompressible viscous free boundary problems with and without surface tension (2018)
    Springer
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    Publication Date: 2018-03-06
    Description: This paper concerns the dynamics of a layer of incompressible viscous fluid lying above a rigid plane and with an upper boundary given by a free surface. The fluid is subject to a constant external force with a horizontal component, which arises in modeling the motion of such a fluid down an inclined plane, after a coordinate change. We consider the problem both with and without surface tension for horizontally periodic flows. This problem gives rise to shear-flow equilibrium solutions, and the main thrust of this paper is to study the asymptotic stability of the equilibria in certain parameter regimes. We prove that there exists a parameter regime in which sufficiently small perturbations of the equilibrium at time \(t=0\) give rise to global-in-time solutions that return to equilibrium exponentially in the case with surface tension and almost exponentially in the case without surface tension. We also establish a vanishing surface tension limit, which connects the solutions with and without surface tension.
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