## 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
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### References:

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