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Smoothing estimates for null forms and applications. (English) Zbl 0909.35094
The authors continue their work [Commun. Pure Appl. Math. 46, 1221-1268 (1993; Zbl 0803.35095)]. The main results concern the estimates of a null form as a right-hand side of the considered nonlinear hyperbolic equation. The estimations are formulated for space-time integral type norms with exponent \(s>1/2\) and for \(1/2< s<1\) and with a weight function (for the details see the original text). For such forms, the authors prove that the initial value problem \[ -\partial^2_t\Phi^I+ \Delta\Phi^I+ \sum_{J,K} \Gamma^I_{J,K}(\Phi) Q_0(\Phi^J,\Phi^K)= 0\quad (I= 1,\dots,N)\tag{1} \] with the inhomogeneous data \(\Phi(0, x)= f_0(x)\), \(\partial_t\Phi(0, x)= f_1(x)\) is well posed for \(f_0\in H^{3/2+s}\), \(f_1\in H^{1/2+s}\), where \(\Gamma^I_{J,K}(\Phi)\) are real analytic functions in \(\Phi= (\Phi^1,\dots, \Phi^N)\). The quoted result is sharp for the equation (1).

MSC:
35L70 Second-order nonlinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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