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Regularity of solutions of critical and subcritical nonlinear wave equations. (English) Zbl 0831.35108
This paper reconsiders the problem of smoothness of the solutions of the nonlinear wave equation \(\square u + f(u) = 0\) in space dimension \(n\), with \(f\) suitably smooth and satisfying a power bound at infinity with power \(p \leq (n + 2)/(n - 2)\), with initial data \((u(0), \partial_t u(0)) \in H^{1 + \mu} \oplus H^\mu\), for \(\mu \geq 0\). Making a systematic use of the known space time integrability properties of solutions of the wave equation (namely of the generalized Strichartz inequalities for that equation), the authors recover in a systematic way and slightly improve the results of P. Brenner and W. von Wahl [Math. Z. 176, 87-121 (1981; Zbl 0457.35059)] and L. V. Kapitanskij [J. Sov. Math. 162, No. 3, 2746-2777 (1992); translation from Zap. Nauchn. Semin. LOMI 182, 38-85 (1990; Zbl 0733.35109)], namely: smoothness is propagated for arbitrary \(\mu \geq 0\) in space dimension \(n \leq 9\), but only up to some critical \(n\)-dependent \(\mu_c \leq 2\) for \(n \geq 10\). In particular the existence of global so-called strong solutions (corresponding to \(\mu = 1)\) is proved in the previous range of \(p\) only for \(n \leq 21\). The method consists in successively estimating the natural set of norms at the level of smoothness \(\mu\) for increasing values of \(\mu\) through the integral equation by using the generalized Strichartz inequalities mentioned above together with Besov space estimates of the nonlinear term.
Reviewer: J.Ginibre (Orsay)

MSC:
35L70 Second-order nonlinear hyperbolic equations
35B65 Smoothness and regularity of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
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