nach oben

2019 | Buch

Kapitel lesen Erstes Kapitel lesen

New Tools for Nonlinear PDEs and Application

herausgegeben von: Prof. Dr. Marcello D'Abbicco, Prof. Marcelo Rempel Ebert, Prof. Vladimir Georgiev, Prof. Tohru Ozawa

Verlag: Springer International Publishing

Buchreihe : Trends in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book features a collection of papers devoted to recent results in nonlinear partial differential equations and applications. It presents an excellent source of information on the state-of-the-art, new methods, and trends in this topic and related areas. Most of the contributors presented their work during the sessions "Recent progress in evolution equations" and "Nonlinear PDEs" at the 12th ISAAC congress held in 2017 in Växjö, Sweden. Even if inspired by this event, this book is not merely a collection of proceedings, but a stand-alone project gathering original contributions from active researchers on the latest trends in nonlinear evolution PDEs.

Inhaltsverzeichnis

Frontmatter

On Effective PDEs of Quantum Physics

Abstract

The Hartree-Fock equation is a key effective equation of quantum physics. We review the standard derivation of this equation and its properties and present some recent results on its natural extensions – the density functional, Bogolubov-de Gennes and Hartree-Fock-Bogolubov equations. This paper is based on a talk given at ISAAC2017.

Ilias Chenn, I. M. Sigal

Critical Exponents for Differential Inequalities with Riemann-Liouville and Caputo Fractional Derivatives

Abstract

We find the critical exponents for global in time solutions to differential inequalities with power nonlinearities, supplemented by an initial data condition. The operator for which the differential inequality is studied contains a Caputo or Riemann-Liouville time derivative of fractional order and a sum of homogeneous spatial partial differential operators. In the special case of a fractional diffusive equation, the obtained critical exponents are sharp. In particular, global existence of small data solutions to the fractional diffusive equation with Caputo and Riemann-Liouville time derivative of order in (0, 1) and in (1, 2), holds for supercritical powers. The existence result for the superdiffusive case (α ∈ (1, 2)), which interpolates a semilinear heat equation and a semilinear wave equation, was recently obtained in the general setting by the author and his collaborators. We use a simple representation of Mittag-Leffler functions to show that global existence of small data solutions for supercritical powers also holds for to the subdiffusive equation with Caputo and Riemann-Liouville time derivative (α ∈ (0, 1)).

Marcello D’Abbicco

Weakly Coupled Systems of Semilinear Effectively Damped Waves with Different Time-Dependent Coefficients in the Dissipation Terms and Different Power Nonlinearities

Abstract

We study the global existence of small data solutions to the Cauchy problem for the coupled system of semilinear damped wave equations with different effective dissipation terms and different exponents of power nonlinearities. The data are supposed to belong to different classes of regularity. We will show the interaction of the exponents p and q on the one hand and on the other hand the interaction of the dissipation terms b ₁(t)u _t and b ₂(t)v _t.

Abdelhamid Mohammed Djaouti, Michael Reissig

Incompressible Limits for Generalisations to Symmetrisable Systems

Abstract

We shortly review the incompressible limit of the barotropic Euler system of gas dynamics, also known as low Mach number limit, and the quasineutral limit of a simplified Euler–Poisson system. Then we develop a general pseudodifferential framework which is able to cover both examples, called generalised symmetrisable systems. This framework can also handle incompressible limits. As an application, we then discuss a barotropic Euler–Poisson system.

Michael Dreher

The Critical Exponent for Evolution Models with Power Non-linearity

Abstract

In this note we derive L ^r − L ^q estimates for the solutions to the Cauchy problem

$$\displaystyle u_{tt} +(-\varDelta )^{\sigma } u = 0\,, \qquad t\geq 0, \ x\in {\mathbb {R}}^n, \qquad u(0,x)=0, \;\; u_t(0,x)=g(x), $$

with σ > 1. Moreover, we derived the critical index p _c(n) for the existence of global in time small data solutions to the associated semilinear Cauchy problem with power nonlinearity |u|^p, p > 1.

Marcelo Rempel Ebert, Linniker Monteiro Lourenço

Blow-Up or Global Existence for the Fractional Ginzburg-Landau Equation in Multi-dimensional Case

Abstract

The aim of this work is to give a complete picture concerning the asymptotic behaviour of the solutions to fractional Ginzburg-Landau equation. In previous works, we have shown global well-posedness for the past interval in the case where spatial dimension is less than or equal to 3. Moreover, we have also shown blow-up of solutions for the future interval in one dimensional case. In this work, we summarise the asymptotic behaviour in the case where spatial dimension is less than or equal to 3 by proving blow-up of solutions for a future time interval in multidimensional case. The result is obtained via ODE argument by exploiting a new weighted commutator estimate.

Luigi Forcella, Kazumasa Fujiwara, Vladimir Georgiev, Tohru Ozawa

Semilinear Damped Klein-Gordon Models with Time-Dependent Coefficients

Abstract

We consider the following Cauchy problem for a wave equation with time-dependent damping term b(t)u _t and mass term m(t)² u, and a power nonlinearity |u|^p:

$$\displaystyle \begin {cases} u_{tt}-\varDelta u+b(t)u_t+m^2(t)u=|u|{ }^p, & t\geq 0, \ x\in \mathbb R^n,\\ u(0,x)=f(x), \quad u_t(0,x)=g(x). \end {cases} $$

We discuss how the interplay between an effective time-dependent damping term and a time-dependent mass term influences the decay rate of the solution to the corresponding linear Cauchy problem, in the case in which the damping term is dominated by the mass term, i.e. liminf_t→∞ (m(t)∕b(t)) > 1∕4.

Then we use the obtained estimates of solutions to linear Cauchy problems to prove that a unique global in-time energy solution to the Cauchy problem with power nonlinearity |u|^p at the right-hand side of the equation exists for any p > 1, assuming small data in the energy space (f, g) ∈ H ¹ × L ².

Giovanni Girardi

Wave-Like Blow-Up for Semilinear Wave Equations with Scattering Damping and Negative Mass Term

Abstract

In this paper we establish blow-up results and lifespan estimates for semilinear wave equations with scattering damping and negative mass term for subcritical power, which are the same as that of the corresponding problem without mass term, and also the same as that of the corresponding problem without both damping and mass term. For this purpose, we have to use the comparison argument twice, due to the damping and mass term, in additional to a key multiplier. Finally, we get the desired results by an iteration argument.

Ning-An Lai, Nico Michele Schiavone, Hiroyuki Takamura

4D Semilinear Weakly Hyperbolic Wave Equations

Abstract

In this paper we exploit the 4Dimensional weakly hyperbolic equation

$$\displaystyle u_{tt}-a(t)\varDelta u=-b(t)|u|{ }^{p-1}u\,. $$

We establish a global existence of radial solutions in a subcritical range of p. This range depends on the zero of a(t) and b(t). In particular we deal with $a(t)=|t-t_0|{ }^{\lambda _1}$ and $b(t)=|t-t_0|{ }^{\lambda _2}$ with λ ₁, λ ₂ ≥ 0. In the case λ ₁ = 2 the radial assumption can be omitted.

Sandra Lucente

Smoothing and Strichartz Estimates to Perturbed Magnetic Klein-Gordon Equations in Exterior Domain and Some Applications

Abstract

This paper is based on the talk of the first author at ISAAC Congress 2017 at Växjö, Sweden. We deal with the smoothing and Strichartz estimates to magnetic Klein-Gordon equations with time-dependent perturbations in exterior domain. Also, the smoothing estimates are applied to establish a scattering of solutions for small perturbations.

Kiyoshi Mochizuki, Sojiro Murai

The Cauchy Problem for Dissipative Wave Equations with Weighted Nonlinear Terms

Abstract

The Cauchy problem for dissipative wave equations with weighted nonlinear terms is considered. The nonlinear terms are power type with a singularity at the origin of Coulomb type. The local and global solutions are shown in the energy class by the use of the Caffarelli-Kohn-Nirenberg inequality. The exponential type nonlinear terms are also considered in the critical two-spatial dimensions.

Makoto Nakamura, Hidemitsu Wadade

Global Existence Results for a Semilinear Wave Equation with Scale-Invariant Damping and Mass in Odd Space Dimension

Abstract

We consider a semilinear wave equation with scale-invariant damping and mass and power nonlinearity. For this model we prove some global (in time) existence results in odd spatial dimension n, under the assumption that the multiplicative constants μ and ν ², which appear in the coefficients of the damping and of the mass terms, respectively, satisfy an interplay condition which makes the model somehow “wave-like”. Combining these global existence results with a recently proved blow-up result, we will find as critical exponent for the considered model the largest between suitable shifts of the Strauss exponent and of Fujita exponent, respectively. Besides, the competition among these two kind of exponents shows how the interrelationship between μ and ν ² determines the possible transition from a “hyperbolic-like” to a “parabolic-like” model. Nevertheless, in the case n ≥ 3 we will restrict our considerations to the radial symmetric case.

Alessandro Palmieri

Wave Equations in Modulation Spaces–Decay Versus Loss of Regularity

Abstract

Recently the study of partial differential equations in modulation spaces gained some relevance. A well-known Cauchy problem is that for the wave equation, where several contributions exist concerning the local (in time) well-posedness. We refer to Bényi et al. (J Func Anal 246.2:366–384, 2007), Cordero and Nicola (J Math Anal Appl 353.2:583–591, 2009) and Reich (Modulation spaces and nonlinear partial differential equations. PhD thesis, TU Bergakademie Freiberg, 2017). By taking advantage of some tools and concepts from the theory of partial differential equations the authors provide some time-dependent estimates of the solution u = u(t, x) to the Cauchy problem of the free wave equation. The main result yields the possibility to consider more delicate problems concerning the wave equation in modulation spaces such as global (in time) well-posedness results.

Maximilian Reich, Michael Reissig

Titel: New Tools for Nonlinear PDEs and Application
herausgegeben von: Prof. Dr. Marcello D'Abbicco
Prof. Marcelo Rempel Ebert
Prof. Vladimir Georgiev
Prof. Tohru Ozawa
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-10937-0
Print ISBN: 978-3-030-10936-3
DOI: https://doi.org/10.1007/978-3-030-10937-0