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

Struwe, Michael

Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 15, No. 3, 495-513 (1988).

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)].

Reviewer: J.Brüning (Augsburg)

MSC:

35L70

Second-order nonlinear hyperbolic equations

Keywords:

nonlinear wave equation; unique global regular solution; rotational invariance

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