METHODS AND APPLICATIONS OF ANALYSIS. © 2000 International Press Vol. 7, No. 3, pp. 479-488, September 2000 005 TIME DECAY FOR THE NONLINEAR BEAM EQUATION* STEVEN P. LEVANDOSKYt AND WALTER A. STRAUSS* Abstract. We derive an analogue of Morawetz' radial identity for the case of a fourth-order wave equation. It follows that, for a nonlinear term with repelling sign, all the solutions decay to zero in a certain sense as t —>• 00. This is an initial step in the construction of a scattering theory for such nonlinear waves. Cathleen Morawetz wrote a short but seminal paper in 1968 on the nonlinear Klein-Gordon equation in which she proved her important Radial Identity and deduced the decay of the local energy of solutions. This identity subsequently had several other important consequences. It was the key ingredient in the proof that all states are scattering states for the Klein-Gordon equation with nonlinear terms with repelling signs, proved in several stages of increasing generality [MS][Br]. Analogues of it were also used in the scattering theory for nonlinear Schrodinger equations [LS][GV1], for nonlinear wave equations with zero mass [GV2], and for Hartree equations [GV3]. It was also recently used to prove the well-posedness of critical nonlinear Schrodinger equations [B][G]. A pseudodifferential generalization of it was also used to prove that the local energy of a solution of the linear wave equation satisfying the Dirichlet boundary condition in the exterior of a non-trapping obstacle decays to zero [MRS]. However, nothing of this sort is known for an equation of order four like (1) uu + A 2 u + f{u) = 0, the simplest nonlinear perturbation of the classical vibrating beam equation. A general theory of such a nonlinear beam equation was considered in two papers by the first author, including the following results, (i) The local well-posedness in H 2 (R n ) x L 2 (E n ) for f(u) = u + 0(\u\ p ) with 1 < p < 1 + ^^. (ii) Low energy scattering in iJ 2 (E n ) x L 2 (l n ) for f(u) = u+ {u^u with p > 1 + |. (iii) Stability and instability of solitary and standing waves for the case of a nonlinear term of attractive sign like f(u) = u- lu^u with 1< p < 1 + ^. In this note we consider the global scattering problem for the fourth-order non- linear beam equation. By a rather long but totally explicit calculation, we prove an exact analogue of Morawetz' Radial Identity. The integrability in time of certain quantities follows automatically. We conjecture that, for the nonlinear beam equation for f(u) = u+ \u\ p '~ 1 u with l + ^<p<l + ^j- , all states are scattering states. This note is a report of work in progress and at the end of it we discuss some supporting evidence for the scattering conjecture. 1. The identity. For equation (1) the energy E = [{l^t + liM'+m} dx * Received March 1, 2000. t Department of Mathematics, Stanford University, Stanford, CA 94305, U.S.A. (spl@math. stanford.edu). * Department of Mathematics, Lefschetz Center for Dynamical Systems, Brown University, Providence, RI 02912, U.S.A. (wstrauss@math.brown.edu). Supported in part by NSF grant 97- 03695. 479