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ANALYSIS & PDE Volume 6 No. 1 2013 msp Analysis & PDE msp.org/apde EDITORS EDITOR-IN-CHIEF Maciej Zworski [email protected] University of California Berkeley, USA BOARD OF EDITORS Michael Aizenman Princeton University, USA Nicolas Burq Université Paris-Sud 11, France [email protected] [email protected] Luis A. Caffarelli University of Texas, USA Sun-Yung Alice Chang Princeton University, USA [email protected] [email protected] Michael Christ University of California, Berkeley, USA Charles Fefferman Princeton University, USA [email protected] [email protected] Ursula Hamenstaedt Universität Bonn, Germany Nigel Higson Pennsylvania State Univesity, USA [email protected] [email protected] Vaughan Jones University of California, Berkeley, USA Herbert Koch Universität Bonn, Germany [email protected] [email protected] Izabella Laba University of British Columbia, Canada Gilles Lebeau Université de Nice Sophia Antipolis, France [email protected] [email protected] László Lempert Purdue University, USA Richard B. Melrose Massachussets Institute of Technology, USA [email protected] [email protected] Frank Merle Université de Cergy-Pontoise, France William Minicozzi II Johns Hopkins University, USA [email protected] [email protected] Werner Müller Universität Bonn, Germany Yuval Peres University of California, Berkeley, USA [email protected] [email protected] Gilles Pisier Texas A&M University, and Paris 6 Tristan Rivière ETH, Switzerland [email protected] [email protected] Igor Rodnianski Princeton University, USA Wilhelm Schlag University of Chicago, USA [email protected] [email protected] Sylvia Serfaty New York University, USA Yum-Tong Siu Harvard University, USA [email protected] [email protected] Terence Tao University of California, Los Angeles, USA Michael E. Taylor Univ. of North Carolina, Chapel Hill, USA [email protected] [email protected] Gunther Uhlmann University of Washington, USA András Vasy Stanford University, USA [email protected] [email protected] Dan Virgil Voiculescu University of California, Berkeley, USA Steven Zelditch Northwestern University, USA [email protected] [email protected] PRODUCTION [email protected] Silvio Levy, Scientific Editor See inside back cover or msp.org/apde for submission instructions. The subscription price for 2013 is US $160/year for the electronic version, and $310/year (C$35, if shipping outside the US) for print and electronic. Subscriptions, requests for back issues from the last three years and changes of subscribers address should be sent to MSP. Analysis & PDE (ISSN 1948-206X electronic, 2157-5045 printed) at Mathematical Sciences Publishers, 798 Evans Hall #3840, c/o Uni- versity of California, Berkeley, CA 94720-3840, is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. APDE peer review and production are managed by EditFLOW® from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2013 Mathematical Sciences Publishers ANALYSIS AND PDE Vol. 6, No. 1, 2013 dx.doi.org/10.2140/apde.2013.6.1 msp SOME RESULTS ON SCATTERING FOR LOG-SUBCRITICAL AND LOG-SUPERCRITICAL NONLINEAR WAVE EQUATIONS HSI-WEI SHIH We consider two problems in the asymptotic behavior of semilinear second order wave equations. First, P 1 2 we consider the Hx × L x scattering theory for the energy log-subcritical wave equation 4 u D juj ug.juj/ in R1C3, where g has logarithmic growth at 0. We discuss the solution with general (respectively spherically symmetric) initial data in the logarithmically weighted (respectively lower regularity) Sobolev space. We also include some observation about scattering in the energy subcritical case. The second problem studied involves the energy log-supercritical wave equation D j j4 α C j j2 ≤ 4 u u u log .2 u / for 0 < α 3 1C3 P 1 P 2 1 in R . We prove the same results of global existence and .Hx \ Hx / × Hx scattering for this equation with a slightly higher power of the logarithm factor in the nonlinearity than that allowed in previous work by Tao (J. Hyperbolic Differ. Equ., 4:2 (2007), 259–265). 1. Introduction Consider the semilinear wave equation 2 3 u VD −@t u C 4u D f .u/ on R × R ; u.0; x/ D u0.x/; (1) @t u.0; x/ D u1.x/; where f is a complex-valued function. Let the potential function F V C ! R be a real-valued function such that 2FzN.z/ D f .z/; (2) D 2 P 1 \ f V R 1g with F.0/ 0 and u being the solution to (1) with initial data u0 Hx φ R3 F(φ/ dx < and 2 u1 2 L x . We can easily verify that the equation has conserved energy Z VD 1 j j2 C 1 jr j2 C E.u/.t/ 2 ut .t; x/ 2 u.t; x/ F.u.t; x// dx: (3) R3 MSC2010: 35L15. Keywords: scattering, log-subcritical, radial Sobolev inequality. 1 2 HSI-WEISHIH P 1 2 The main goal of the paper is to study the Hx × L x scattering theory for log-subcritical wave equations with finite energy initial data, where the energy is defined by (3). In this paper, the term log-subcritical wave equation refers to (1) with f defined by jzj4zg.jzj/; jzj 6D 0; f .z/ VD (4) 0; jzj D 0; where g V .0; 1/ ! R is smooth, nonincreasing, and satisfies 8−log x; 0 < x < 1 ; < 3 VD ∼ 1 ≤ g.x/ 1; 3 x < 1; (5) :1; x ≥ 1: We also prove global existence in the case of spherical symmetry for log-supercritical wave equations, by which we mean equations of the form 4 α 2 u D juj u log .2 C juj / (6) ≤ 4 ≤ In this paper, we will allow 0 < α 3 , extending the range 0 < α 1 allowed in[Tao 2007]. We also assume that the initial data is in the energy space, the set of data for which the energy (3) is finite. Remark 1.1. We can easily compute that the potential function of log-subcritical wave equations (1), (4), and (5) is 8− 1 jzj6log.jzj/ − 1 ; 0 < jzj < 1 ; <> 6 6 3 D ∼ 1 j j6 1 ≤ j j Fsub.z/ 6 z ; 3 z < 1; (7) > : 1 j j6 j j ≥ 6 z ; z 1; and the potential function of the log-supercritical wave equations (6) is 6 α 2 Fsup.z/ ∼ jzj log .2 C jzj /: (8) We quickly recall some common terminology associated to the scaling properties of (1). Consider f .z/ D jzjp−1z and let u be the solution of (1). By scaling, λ2=.1−p/u.t/λ; x/λ) is also a solution with 2=.1−p/ .1Cp/=.1−p/ initial data λ u0.t0/λ; x/λ) and λ u1.t0/λ; x/λ). Hence the scaling of u preserves the homogeneous Sobolev norm ku0kHP sc .R3/ C ku1kHP sc−1.R3/ if 3 2 4 sc VD − ; or equivalently p D 1 C : 2 p − 1 3 − 2sc p−1 P s Definition 1.2. For f .z/Djzj z and a given value s, we call (1) an Hx -critical (subcritical, supercritical) nonlinear wave equation if p equals (is less than, is greater than) 1C4=.3−2s/. In particular, when s D 1, we call (1) an energy critical (subcritical, supercritical) nonlinear wave equation if p D 5 (p < 5, p > 5). 4 The results of global existence and uniqueness for the energy-critical (u D juj u) and energy- p−1 subcritical (u D juj u, where p < 5) wave equations are already established by [Brenner and von Wahl 1981; Struwe 1988; Grillakis 1990; 1992; Shatah and Struwe 1993; 1994; Kapitanski 1994; Ginibre and Velo 1985]. It is natural to consider the decay of the solution, which we expect to behave linearly SCATTERINGTHEORYFORLOG-SUBCRITICALANDLOG-SUPERCRITICALNLW 3 as t ! ±∞. The decay estimate and scattering theory (see section 2 for definition) of critical wave equations are shown in[Bahouri and Shatah 1998]; see also[Bahouri and Gérard 1999; Ginibre and Velo 1989; Nakanishi 1999]. Hidano[2001] (see also[Ginibre and Velo 1987]), by the property of conformal 5 ≤ invariance, proved that the solutions for certain subcritical wave equations ( 2 < p 3) scatter in the weighted Sobolev space 6 VD X × Y , where 1 3 2 3 2 3 2 3 X VD Hx .R / \ fφ V jxjrφ 2 L x .R /g; Y VD L x .R / \ fφ V jxjφ 2 L x .R /g: P 1 2 1 However, for energy subcritical equations, the Hx × L x scattering theory still remains open. In this paper, we consider the solutions to the log-subcritical wave equations (1), (4), and (5) with finite energy initial data. The global existence result is established in [Grillakis 1990; 1992; Kapitanski 1994; Nakanishi P 1 2 1999]. We will prove that the solutions with a class of initial data scatter in Hx × L x . This class of data is contained in logarithmically weighted Sobolev spaces X1 × Y1, where P 1 3 γ 2 3 X1 VD Hx .R / \ fφ V log .1 C jxj/rφ 2 L x .R /g; (9) 2 3 γ 2 3 Y1 VD L x .R / \ fφ V log .1 C jxj)φ 2 L x .R /g 1 for some γ > 2 . For initial data in these spaces, we show that the potential energy of the solution decays logarithmically for all large times. After dividing the time interval suitably, this decay helps us to control the key spacetime norm k f .u/k 1 2 .
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