×

Globally regular solutions to the \({\mathcal U}^ 5\) Klein-Gordon equation. (English) Zbl 0728.35072

The author studies the Cauchy problem for the nonlinear wave equation \(u_{tt}-\Delta_ u+u^ 5=0\) (*) in \({\mathbb{R}}^ 3\times {\mathbb{R}}_+\). If the exponent 5 is replaced by a number less than 5 then Jörgens proved the existence of a unique global regular solution. This was later proved by Rauch also for \(p=5\), under the assumption of small initial energy. In the present paper the smallness assumption is replaced by assuming rotational invariance for the initial data; then again global regularity holds. The difficulty with this problem can be expressed by saying that the exponent 5 is critical for certain relevant Sobolev imbeddings. But equation (*) acquires a crucial \({\mathbb{R}}_+\)-scaling invariance (sending u to \(R^{1/2}u(Rx,Rt))\) which is a decisive tool in the proof presented here.
Recently, M. Grillakis has removed also the assumption of rotational invariance [cf. Ann. Math., II. Ser. 132, 485-509 (1990)].

MSC:

35L70 Second-order nonlinear hyperbolic equations
PDF BibTeX XML Cite
Full Text: Numdam EuDML

References:

[1] K. Jörgens , Das Anfangswertproblem im Grossen für eine Klasse nichtlinearer Wellengleichungen , Math. Z. 77 ( 1961 ), pp. 295 - 308 . Article | MR 130462 | Zbl 0111.09105 · Zbl 0111.09105
[2] S.I. Pohožaev , Eigenfunctions of the equation \Delta u + \lambda f(u) = 0 . Doklady , 165 ( 1965 ), pp. 1408 - 1411 . Zbl 0141.30202 · Zbl 0141.30202
[3] J. Rauch , The u5 - Klein-Gordon equation, in Nonlinear PDE’s and applications , Ed. Brezis, Lions; Pitman Research Notes in Math . 53 ( 1981 ), pp. 335 - 364 . MR 631403 | Zbl 0473.35055 · Zbl 0473.35055
[4] P. Brenner - W. Von Wahl , Global classical solutions of nonlinear wave equations , Math. Z. 176 ( 1981 ), pp. 87 - 121 . MR 606174 | Zbl 0457.35059 · Zbl 0457.35059
[5] H. Pecher , Ein nichtlinear Interpolationsatz und seine Anwendung auf nichtlineare Wellengleichungen , Math. Z. 161 ( 1978 ), pp. 9 - 40 . Article | MR 499812 | Zbl 0384.35039 · Zbl 0384.35039
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.