Duyckaerts, Thomas; Kenig, Carlos; Merle, Frank Classification of the radial solutions of the focusing, energy-critical wave equation. (English) Zbl 1308.35143 Camb. J. Math. 1, No. 1, 75-144 (2013). This paper states that radial solutions of equation \[ u_{tt}-\Delta u = u^5, \] in three space dimensions, with initial data in \(\dot H^1\times L^2({\mathbf R}^3)\) fall into three classes. In the first, \(t\mapsto(u,u_t)\) blows up in finite time in \(\dot H^1\times L^2({\mathbf R}^3)\). In the second, \((u,u_t)\) behaves, as \(t\uparrow T_+\) (with \(T_+<\infty\)), like \((v_0+\sum_{j=1}^J w_j, v_1)\), where \(v_0\) and \(v_1\) do not depend on \(t\), and the \(w_j\) have the form \(\iota_j\lambda_j^{-1/2}W(x/\lambda_j)\), where \(J\geq 1\), \(\iota_j=\pm1\), \(\lambda_j=\lambda_j(t)>0\), with \(\lambda_1\ll \lambda_2\ll\cdots\ll\lambda_J\ll T_+-t\), and \(W\) is a radial solution of \(\Delta W+W^5=0\). In the third class, there is a similar statement, where \(T_+=+\infty\), \(J\geq 0\), \(\lambda_j=\lambda_j(t)>0\), with \(\lambda_1\ll \lambda_2\ll\cdots\ll\lambda_J\ll t\) and \((v_0,v_1)\) now depends on \(t\) and has the form \((v_L,\partial_tv_L)\), \(v_L\) being a solution of the (linear) wave equation. Reviewer: Satyanad Kichenassamy (Reims) Cited in 4 ReviewsCited in 92 Documents MSC: 35L71 Second-order semilinear hyperbolic equations 35B44 Blow-up in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35L15 Initial value problems for second-order hyperbolic equations Keywords:semilinear wave equation; profile decomposition; soliton resolution PDFBibTeX XMLCite \textit{T. Duyckaerts} et al., Camb. J. Math. 1, No. 1, 75--144 (2013; Zbl 1308.35143) Full Text: DOI arXiv