Smooth type II blow up solutions to the four dimensional energy critical wave equation

Abstract

We exhibit ${\mathcal{C}}^{\infty }$ type II blow up solutions to the focusing energy critical wave equation in dimension $N=4$. These solutions admit near blow up time a decomposiiton $u(t,x)=1/l(t)(Q+e(t))(x/l(t)}}$ with $|e(t),\pa_t e(t)|_{\dot{H}^1\times L^2}<<1$ where $Q$ is the extremizing profile of the Sobolev embedding ${\dot{H}}^1\subset L^{2^{\ast }}$, and a blow up speed $l(t)=(T-t)e^{-\sqrt{|\log (T-t)|}(1+o(1))}$ as $t\to T.$

48 pages, 9 figures

References in corpus (2)

• Formation of singularities for equivariant 2+1 dimensional wave maps into the two-sphere
• Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations

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• Blow up for the critical gKdV equation I: dynamics near the soliton
• The instability of Bourgain-Wang solutions for the L^2 critical NLS
• Center-stable manifold of the ground state in the energy space for the critical wave equation
• Blow up for critical wave equations on curved backgrounds