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Large outgoing solutions to supercritical wave equations. (English) Zbl 1415.35199

Summary: We prove the existence of global solutions to the energy-supercritical wave equation in \(\mathbb R^{3+1}\) \[ u_{tt}-\Delta u \pm| u|^Nu=0,\, u(0) = u_0,\, u_t(0)=u_1,\, 4<N<\infty \] for a large class of radially symmetric finite-energy initial data.
Functions in this class are characterized as being outgoing under the linear flow – for a specific meaning of “outgoing” defined below.
In particular, we construct global solutions for initial data with large (even infinite) critical Sobolev, Besov, Lebesgue, and Lorentz norms and several other large critical norms.

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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