Threshold phenomenon for the quintic wave equation in three dimensions (Q2447596)

The paper is a continuation of the works by \textit{J. Krieger} and \textit{W. Schlag} [Am. J. Math. 129, No. 3, 843--913 (2007; Zbl 1219.35144)], \textit{T. Duyckaerts} et al. [Geom. Funct. Anal. 22, No. 3, 639--698 (2012; Zbl 1258.35148), Camb. J. Math. 1, No. 1, 75--144 (2013; Zbl 1308.35143)], wherein the long-time dynamics for radial data of solutions of the energy-critical focusing nonlinear wave equation \[ \partial_t - \Delta_x = u^5 \] supplemented by the initial data \[ u[0] = (u,u_t)_{t=0} = (u_0,u_1) \] on the Minkowski space \(\mathbb R^{3+1}\) is studied in the case of \(u[0] \in H^1\times L^2(\mathbb R^3)\) of arbitrary energy, \[ E = \int_{\mathbb R^3}\left(\frac{1}{2}|\nabla_{t,x}|^2 - \frac{1}{6}|u|^6\right)dx. \] In this paper the authors return to the point of view of Krieger and Schlag [loc. cit.] in order to establish a description of all possible dynamics with data near \((W,0)\), where \(W\) is the ground state stationary solution \(W(x) = (1 + |x|^2/3)^{-1/2}\). The following main result holds: Fix \(R > 1\). There exists an \(\varepsilon_* = \varepsilon_*(R) > 0\) with the following property. Consider all pairs of radial functions \((f_1,f_2)\) supported in \(B(0,R)\) with \(\|f_1\|_{H^3} + \|f_2\|_{H^2} < \varepsilon_*\). Denote by \(\Sigma\) the hypersurface constructed in the above-mentioned paper, parametrized by such pairs \((f_1,f_2)\) satisfying the condition \((k_0f_1 + f_2,g_0)\). Here \(-k^2_0\) is a unique negative eigenvalue of \(Hg_0 = -k^2_0\), where \(H = -\Delta - 5W^4\). Pick initial data \(v[0] \in \Sigma\) with \[ v(0,\cdot) = f_1 + h(f_1,f_2)g_0, \quad v_t(0,\cdot) = f_2. \] Then the following holds: If \(\varepsilon_* > \delta_0 > 0\), then initial data \[ \tilde u(0,\cdot) = W + f_1 +(h(f_1,f_2) + \delta_0)g_0, \quad \tilde u_t(0,\cdot) = f_2 \] lead to a solutions blowing up in finite positive time. If \(-\varepsilon_* < \delta_0 < 0\), then the initial data \[ \tilde u(0,\cdot) = W + f_1 + (h(f_1,f_2) + \delta_0)g_0, \quad \tilde u_t(0,\cdot) = f_2 \] lead to solutions existing globally in forward time and scattering to zero in the energy space.