Solutions of Kadomtsev–Petviashvili equation with power law nonlinearity in 1+3 dimensions

This paper studies the solution of the Kadomtsev–Petviasvili equation with power law nonlinearity in 1+3 dimensions. The Lie symmetry approach as well as the extended tanh‐function and G′/G methods are used to carry out the analysis. Subsequently, the soliton solution is obtained for this equation with power law nonlinearity. Both topological as well as non‐topological solitons are obtained for this equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Global Pointwise Decay Estimates for Defocusing Radial Nonlinear Wave Equations

We prove global pointwise decay estimates for a class of defocusing semilinear wave equations in n = 3 dimensions restricted to spherical symmetry. The technique is based on a conformal transformation and a suitable choice of the mapping adjusted to the nonlinearity. As a result we obtain a pointwise bound on the solutions for arbitrarily large Cauchy data, provided the solutions exist globally. The decay rates are identical with those for small data and hence seem to be optimal. A generalization beyond the spherical symmetry is suggested.     

Two dimensional exterior mixed problem for semilinear damped wave equations

We consider two dimensional exterior mixed problems for a semilinear damped wave equation with a power type nonlinearity p|u|. For compactly supported initial data, which have a small energy we shall derive global in time existence results in the case when the power of the nonlinearity satisfies 2<p<+∞. This generalizes a previous result of [J. Differential Equations 200 (2004) 53-68], which dealt with a radially symmetric solution.     

Global Existence and Blow-up for Semilinear Wave Equations with Variable Coefficients

The authors consider the critical exponent problem for the variable coefficients wave equation with a space dependent potential and source term. For sufficiently small data with compact support, if the power of nonlinearity is larger than the expected exponent, it is proved that there exists a global solution. Furthermore, the precise decay estimates for the energy, L₂ and L^p+1 norms of solutions are also established. In addition, the blow-up of the solutions is proved for arbitrary initial data with compact support when the power of nonlinearity is less than some constant.     

Mixed problem for a nonlinear hyperbolic equation in a domain unbounded with respect to space variables

We investigate the first mixed problem for a quasilinear hyperbolic equation of the second order with power nonlinearity in a domain unbounded with respect to the space variables. The case of arbitrarily many space variables is considered. We establish conditions for the existence and uniqueness of a solution of this problem independent of the behavior of the solution as |x| → + ∞. The indicated classes of existence and uniqueness are the spaces of locally integrable functions, and, furthermore, the dimension of the domain does not limit the order of nonlinearity. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 11, pp. 1523–1531, November, 2007.     

Boundary Value Problems for the q — Laplacian on 𝕊N

Nonlinear boundary value problems for the q–Laplacian in spaces of constant positive curvature are considered. The nonlinearity is of the form of a power. Existence and nonexistence of positive radial solutions in balls is established. It turns out that the situation differs considerably from the corresponding problems in the Euclidean space. Special attention is given to the critical case which has some consequences in the calculus of variation.     

太阳辐射时间序列的非线性检验

检验太阳辐射时间序列是否有非线性特征,对于分析、建模和预测太阳辐射量是重要、有益的.提出用基于替代数据的检验方法来检验太阳辐射时间序列是否存在非线性特征,并将数据序列的三阶矩作为检验统计量.选取了美国Montana州Dillon地区和Wyoming州Green Rivet地区每日总辐射量、Utah州Moab地区的每月日平均总辐射量时间序列作为检验对象.数值分析的统计结果表明所研究的日总辐射时间序列存在非线性,而每月日平均总辐射时间序列未检测出非线性.因而,对太阳辐射时间序列建模和预测之前,检验其是否有非线性特征是必要的.     

Lipschitz Stability in Inverse Source Problems for Singular Parabolic Equations

We consider 2-D Klein-Gordon equation with quadratic nonlinearity and prove Strichartz type dispersive estimates for the global solution with small initial data in the Sobolev space H ^1+?.     

On existence and nonexistence of global solutions of Cauchy–Goursat problem for nonlinear wave equations

We consider the Cauchy–Goursat initial characteristic problem for nonlinear wave equations with power nonlinearity. Depending on the power of nonlinearity and the parameter in an equation we investigate the problem on existence and nonexistence of global solutions of the Cauchy–Goursat problem. The question on local solvability of the problem is also considered.     

Determination of the blow-up rate for a critical semilinear wave equation

In this paper, we determine the blow-up rate for the semilinear wave equation with critical power nonlinearity related to the conformal invariance.Mathematics Subject classification (2000): 35L05, 35L67Membre de lInstitut Universitaire de France     

Asymptotics and analytic modes for the wave equation in similarity coordinates

We consider the radial wave equation in similarity coordinates within the semigroup formalism. It is known that the generator of the semigroup exhibits a continuum of eigenvalues and embedded in this continuum there exists a discrete set of eigenvalues with analytic eigenfunctions. Our results show that, for sufficiently regular data, the long-time behaviour of the solution is governed by the analytic eigenfunctions. The same techniques are applied to the linear stability problem for the fundamental self-similar solution χ _T of the wave equation with a focusing power nonlinearity. Analogous to the free wave equation, we show that the long-time behaviour (in similarity coordinates) of linear perturbations around χ _T is governed by analytic mode solutions. In particular, this yields a rigorous proof for the linear stability of χ _T with the sharp decay rate for the perturbations.      

Asymptotic distribution theory of statistical functionals: The compact derivative approach for robust estimators

Summary Derivatives of statistical functionals have been used to derive the asymptotic distributions ofL-,M- andR-estimators. This approach is often heuristic because the types of derivatives chosen have serious limitations. The Gateaux derivative is too weak and the Fréchet derivative is too strong. In between lies the compact derivative. This paper obtains strong results in a rigorous manner using the compact derivative onC ₀(R). This choice of space allows results for a broader class of functionals than previous choices, and the fact that is often tight provides the compact set required. A major result is the derivation of the compact derivative of the inverse c.d.f. when the range space is endowed with the uniform norm. It has applications to the asymptotic theory ofL-,M- andR-estimators. We illustrate the power of this result by applications toL-estimators in settings including the one sample problem, data grouped by quantiles, and censored survival time data.     

Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D

We consider the following Cauchy problem for weakly coupled systems of semilinear damped elastic waves with a power source nonlinearity in three dimensions: where with b² > a² > 0 and θ ∈ [0,1]. Our interests are some qualitative properties of solutions to the corresponding linear model with vanishing right‐hand side and the influence of the value of θ on the exponents p₁,p₂,p₃ in to get results for the global (in time) existence of small data solutions.     

Global Existence with Large Data for a Nonlinear Weakly Hyperbolic Equation

We prove a global existence result for a semilinear degenerate hyperbolic Cauchyproblem with large data, in the case ofa subcritical defocusing power nonlinearity, in 1 and 2 space dimensions.     

1-Soliton solution of the D(m, n) equation with generalized evolution

This paper obtains the 1-soliton solution of the nonlinear dispersive Drinfel’d-Sokolov equation with power law nonlinearity. In the first case the soliton solution is without the generalized evolution. The solitary wave ansatz method is used to carry out the integration. Subsequently, the He’s semi-inverse variational principle is used to integrate the equation with power law nonlinearity. Parametric conditions for the existence of envelope solitons are given.     

The Nonlinear Schrödinger Equation on Hyperbolic Space

In this article we study some aspects of dispersive and concentration phenomena for the Schrödinger equation posed on hyperbolic space  ⁿ , in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties: we prove that the dispersion inequality is valid, in a stronger form than the one on ? ⁿ . However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.     

A distance-based statistical analysis of fuzzy number-valued data

Real-life data associated with experimental outcomes are not always real-valued. In particular, opinions, perceptions, ratings, etc., are often assumed to be vague in nature, especially when they come from human valuations. Fuzzy numbers have extensively been considered to provide us with a convenient tool to express these vague data. In analyzing fuzzy data from a statistical perspective one finds two key obstacles, namely, the nonlinearity associated with the usual arithmetic with fuzzy data and the lack of suitable models and limit results for the distribution of fuzzy-valued statistics. These obstacles can be frequently bypassed by using an appropriate metric between fuzzy data, the notion of random fuzzy set and a bootstrapped central limit theorem for general space-valued random elements. This paper aims to review these ideas and a methodology for the statistical analysis of fuzzy number data which has been developed along the last years.     

A semi-discrete scheme for the stochastic nonlinear Schrödinger equation

Summary. We study the convergence of a semi-discretized version of a numerical scheme for a stochastic nonlinear Schrödinger equation. The nonlinear term is a power law and the noise is multiplicative with a Stratonovich product. Our scheme is implicit in the deterministic part of the equation as is usual for conservative equations. We also use an implicit discretization of the noise which is better suited to Stratonovich products. We consider a subcritical nonlinearity so that the energy can be used to obtain an a priori estimate. However, in the semi discrete case, no Ito formula is available and we have to use a discrete form of this tool. Also, in the course of the proof we need to introduce a cut-off of the diffusion coefficient, which allows to treat the nonlinearity. Then, we prove convergence by a compactness argument. Due to the presence of noise and to the implicit discretization of the noise, this is rather complicated and technical. We finally obtain convergence of the discrete solutions in various topologies. Mathematics Subject Classification (2000):35Q55, 60H15, 65M06, 65M12     

Exact solutions of semilinear radial wave equations in n dimensions

Exact solutions are derived for an n-dimensional radial wave equation with a general power nonlinearity. The method, which is applicable more generally to other nonlinear PDEs, involves an ansatz technique to solve a first-order PDE system of group-invariant variables given by group foliations of the wave equation, using the one-dimensional admitted point symmetry groups. (These groups comprise scalings and time translations, admitted for any nonlinearity power, in addition to space-time inversions admitted for a particular conformal nonlinearity power.) This is shown to yield not only group-invariant solutions as derived by standard symmetry reduction, but also other exact solutions of a more general form. In particular, solutions with interesting analytical behavior connected with blow-ups as well as static monopoles are obtained.     

Asymptotic behavior for the one‐dimensional pth power Newtonian fluid in unbounded domains

We consider the initial and initial‐boundary value problems for a one‐dimensional pth power Newtonian fluid in unbounded domains with general large initial data. We show that the specific volume and the temperature are bounded from below and above uniformly in time and space and that the global solution is asymptotically stable as the time tends to infinity. Copyright © 2015 John Wiley & Sons, Ltd.