Hartmut Pecher

Nonlinear small data scattering for the wave and Klein-Gordon equation

On etudie loperateur diffusion qui appartient a la paire dequations U tt +Au+f(u)=0 et U tt +Au=0. A represente loperateur -∑ j=1 n ∂ 2 /∂x j 2 +m 2 , m∈R et f(u)=λ|U|ρ −1 u, ou λ∈R, ρ>1

Scattering for semilinear wave equations with small data in three space dimensions

On considere la theorie de la diffusion pour des equations donde semilineaires du type u tt -△u=F(u) a 3 dimensions despace, ou F(u)=λ|u| P-1 u ou F(u)=λ|u| P , λ∈R, p>1

Time dependent nonlinear Schrödinger equations

We prove the existence of global classical solutions of the Cauchy-problem for nonlinear Schrödinger equations u′+iAu+f(|u|2)u or u′+Au+if(|u|2)u respectively. We need various growth conditions on the nonlinearity f and some restrictions on the admissible space dimension n.

Ein nichtlinearer Interpolationssatz und seine Anwendung auf nichtlineare Wellengleichungen.

fiir alle Funktionen ueHarc~H JP mit einer gewissen stetigen Funktion co gegeben ist. Hierbei bezeichnet H kp ffir kelR, l__<p<oe die Sobolevr~iume gebrochener Ordnung (vgl. Def. 1.1). Wesentlich hierbei ist die Tatsache, dab b nicht ganz zu sein braucht. Das Resultat folgt durch Interpolation mit Hilfe der von Peetre eingefiihrten K-Methode (vgl. hierzu etwa [1]) zwischen den sich aus den Sobolevschen EinbettungssStzen ergebenden Absch/itzungen flit die H tblqund die Htbl+lq-Norm von f (u) (Theorem 1.6). Ftir friJhere Resultate in dieser Richtung vgl. [7]. Im zweiten Teil der Arbeit wird dieses Resultat dann dazu verwendet, um das Cauchy-Problem ffir nichtlineare Wellengleichungen der Form

LP-Abschtzungen und klassische Lsungen fr nichtlineare Wellengleichungen II

This paper deals with the global classical solvability of the Cauchy problem for nonlinear wave equations of higher order of the following form where Δ denotes the Laplacian, ζ a real constant, m a positive integer, and f the nonlinearity. Using results of the first part [6] concerning estimations for the kernel of the associated integral equation a priori estimates for certain LP-norms of the local regular solution are established for higher space dimensions and for a class of nonlinearities satisfying certain growth and positivity conditions. This leads to completely regular solutions by means of the Sobolev embedding theorems.

Time Decay for Nonlinear Wave Equations in Two Space Dimensions.

The Cauchy Problem for the equation utt−Δu+|u|p−1u=0 (x∈ℝ2, t>0, ρ>1) is studied. Smooth Cauchy data is prescribed, and no smallness condition is imposed. For ρ>5, it is shown that the maximum amplitude of such a wave decays at the expected rate t−1/2 as t→∞. For 1+√8<ρ≦5, the maximum amplitude still decays, but at a slower rate. These results are then used to demonstrate the existence of the scattering operator when ρ>ρo, where ρo is the root of the cubic equation ρ3-2ρ2-7ρ-8=0; thus ρo≅4.15.

Global smooth solutions to a class of semilinear wave equaiions with strong nonlinearities

We consider the Cauchy problem for semilinear wave equationsutt−Δu=g(u) in 3+1 dimensions with smooth but possibly large data. Ifg isC2,α and bounded from above everywhere and from below for negative arguments the existence of a global classical solution is shown. If moreoverg is nonpositive and vanishes at least of order 2+∈ at the origin and if the data decay sufficiently rapidly at infinity the scattering operator exists.

Das Verhalten der Ableitungen globaler Lösungen nichtlinearer Wellengleichungen für grosse Zeiten

This note deals with the behavior of global classical solutions of a certain type of nonlinear wave-equations when t goes to infinity. Starting with results of Morawetz-Strauss [2] and of the author [3] about the time-decay rates for the spatial supremum of the soluion we estimate all its Sobolev-norms. Our main tools are estimates of the LP-decay rates for homogeneous wave-equations contained in a paper of von Wahl [5].