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Classification of the radial solutions of the focusing, energy-critical wave equation. (English) Zbl 1308.35143

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.

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
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