Soliton Resolution at Blow-Up for the Subconformal Nonlinear Wave Equation

RESUMO

We consider the nonlinear wave equation (NLW) with a superlinear pure power nonlinearity in the subconformal and conformal cases. Under some conditions on initial data, this equation is known to have solutions which blow up in finite time. Two questions are then relevant: (i) the classification of all possible blow-up behaviors; (ii) the construction of examples of blow-up solutions. As we will show, the situation is entirely settled in the one-dimensional case, where we fully solve the famous soliton resolution conjecture. Various extensions to higher dimensions and to perturbed versions of NLW are given, including the construction of a solution in 2-d, with a nearly pyramidal blow-up graph. Carrying out this program was made possible thanks to a synergy of techniques from the PDE theory, mathematical physics, and analysis, including ODE techniques, spectral theory, and energy methods. Throughout this presentation, we will insist on connections with the study of other types of PDEs, in particular in the parabolic case. Surprisingly enough, in spite of the difference between parabolic and hyperbolic equations at the linear level, the nonlinear nature brings in a strong unity-both in the results and in the methods-between these two important classes of PDEs.