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Global regularity for a logarithmically supercritical defocusing nonlinear wave equation for spherically symmetric data. (English) Zbl 1124.35043
This paper deals with the Cauchy problem for the defocusing nonlinear wave equation \[ \square u= u^5\log(2+ u^2)\tag{1} \] in three spatial dimensions. Assuming the initial data to be spherically symmetric and \(C^\infty\) smooth it is proved the existence of a unique global smooth solution of (1). The solution \(u\) is supposed to be compactly supported in space for each fixed \(t\). When establishing his main result the author modifies the arguments of Ginibre, Soffer and Velo for the energy-critical equation \(\square u= u^5\). Moreover, he shows also that (1) is globally well-posed for radial initial data in \(H^2(\mathbb{R}^3)\times H^1(\mathbb{R}^3)\).
It is well-known that the solutions of the focusing supercritical equation \(\square u= -|u|^{p-1}u\), \(p> 1\) can blow up instantaneously from finite initial data, while the defocusing counterpart \(\square u= |u|^{p-1}u\), \(p> 1\) is highly unstable in the energy class.

35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
Full Text: DOI arXiv
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