x INTRODUCTION such that limt↑1 (u(t),∂tu(t)) ˙ 1 ×L2 = ∞. (Type I blow-up, or ODE blow-up). Also, W , which solves ΔW +W 5 = 0 and is independent of time, is a global in time solution, which does not scatter. (If a solution u scatters, |x|≤1 |∇u(x, t)|2dx t→∞ −→ 0. This clearly fails for W ). Moreover, Krieger-Schlag-Tataru [71], Krieger-Schlag [69] have constructed type II blow-up solutions, i.e., solutions with T+ ∞, and sup 0tT+ (u(t),∂tu(t)) ˙ 1 ×L2 ∞, which are radial. Also, Donninger-Krieger [29] have constructed radial, global in time solutions, bounded in ˙ 1 × L2, which do not scatter to either a linear solution or to W . In the rest of this monograph, we will study the focusing case and call it (NLW) and its energy E. Here, the analog of (+) is (++) Ground State Conjecture (For critical focusing problems): There exists a “ground state”, whose energy is a threshold for global existence and scattering. In 2006-09, Frank Merle and I developed a program to attack critical dispersive problems and establish (+) and, for the first time (++) in focusing problems. We call this the “concentration-compactness/rigidity theorem method”, which was partly inspired by the earlier elliptic problems. The method gives a “road map” to attack both (+) and for the first time (++). The “road map” has already found an enormous range of applicability, to previously intractable problems, in work of many researchers. See for instance [64], [65], [66], [67], [23], [24], [25],[26],[27],[28],[6], [70] and many others. The result of Kenig-Merle [62], establishing the ground state conjecture (++) for (NLW), using the “concentration-compactness/rigidity theorem method” is: Theorem 2.6. If E(u0,u1) E(W, 0) then i) If ∇u0 ∇W , global existence, scattering. ii) If ∇u0 ∇W , T+, |T−| ∞. iii) The case ∇u0 = ∇W is impossible. The “concentration-compactness/rigidity theorem” method, as well as the proof of Theorem 2.6 above are discussed in detail in Chapters 2 and 3 of this monograph. In Chapters 4 and 5 of this monograph, we study solutions of (NLW) with the “com- pactness property”, an important class of “non-dispersive” solutions. In proving Theorem 2.6, a rigidity theorem for solutions with the compactness property and further size restrictions is crucial. In Chapter 4 we study solutions with the com- pactness property and no further size restriction. The main results are collected in Theorem 4.77. The main conjecture here is the “rigidity conjecture” for solutions with the compactness property, namely that they are all solitary waves, i.e., Lorentz transforms of stationary solutions (stationary solutions solve the elliptic equation ΔQ + Q5 = 0). This conjecture was established in [32] in the radial case, and in [35] under a non-degeneracy assumption. These results are dealt with in Chapter 5. They comprise Theorem 4.17, Theorem 4.18 and Theorem 5.6. In Chapter 4 we also give an extension (Theorem 4.4) of i) in Theorem 2.6, which was proved in [31] and uses the rigidity results of Chapter 5. In Chapter 6, we begin the systematic study of type II blow-up solutions and more generally, of extended type II solutions, namely non-zero solutions u for which sup0tT + (u) (u(t),∂tu(t)) ˙ 1 ×L2 ∞.
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