# zbMATH — the first resource for mathematics

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.

##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35L15 Initial value problems for second-order hyperbolic equations
Full Text:
##### References:
 [1] DOI: 10.1007/BF01212349 · Zbl 0573.76029 · doi:10.1007/BF01212349 [2] Christ M., Ann. Inst Henri Poincaré [3] DOI: 10.1016/0022-1236(92)90044-J · Zbl 0813.35054 · doi:10.1016/0022-1236(92)90044-J [4] DOI: 10.1006/jfan.1995.1119 · Zbl 0849.35064 · doi:10.1006/jfan.1995.1119 [5] DOI: 10.2307/1971427 · Zbl 0736.35067 · doi:10.2307/1971427 [6] DOI: 10.1002/cpa.3160450604 · Zbl 0785.35065 · doi:10.1002/cpa.3160450604 [7] DOI: 10.4310/MRL.1994.v1.n2.a9 · Zbl 0841.35067 · doi:10.4310/MRL.1994.v1.n2.a9 [8] DOI: 10.1353/ajm.1998.0039 · Zbl 0922.35028 · doi:10.1353/ajm.1998.0039 [9] DOI: 10.1002/cpa.3160460902 · Zbl 0803.35095 · doi:10.1002/cpa.3160460902 [10] Lebeau G., Hommage Pascal Laubin. Bull. Soc. Roy. Liége 70 pp 267– [11] Lebeau G., Bull. Soc. Math. France 133 pp 145– [12] Shatah J., Courant Lecture Notes in Mathematics 2, in: Geometric Wave Equations (1998) · Zbl 0993.35001 [13] Sogge C. D., Monographs in Analysis II, in: Lectures on Nonlinear Wave Equations (1995) [14] Struwe M., Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 pp 495–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.