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Semilinear hyperbolic systems and equations with singular initial data. (English) Zbl 0747.35020

The paper examines the existence and the qualitative behaviour of the weak limits \(\lim_{\varepsilon\to 0}u^ \varepsilon\) of the solutions to semilinear strictly hyperbolic systems in one space dimension \((\partial_ t+A(t,x)\partial_ x+B(t,x))u^ \varepsilon=f(u^ \varepsilon)\) and the \(3-D\) semilinear wave equation \(\square_ 3u^ \varepsilon=(\partial^ 2_ t-\sum^ 3_{j=1}\partial^ 2_{x_ j})u^ \varepsilon=f(u^ \varepsilon)\), \(\varepsilon>0\), with initial data converging weakly for \(\varepsilon\to 0\) to distributions of type sum of derivatives of order \(\leq k\) of Dirac mesures.
The main result for systems is in the cases of a sublinear nonlinear term \(f(u)\). It proposes an optimal link between the type of the growth of \(f(u)\) when \(| u|\to\infty\) and the strength of the singularity of the initial data in order \(\lim_{\varepsilon\to 0}u^ \varepsilon\) to exist, extending in particular previous results of M. Oberguggenberger [Math. Ann. 274, 599-607 (1986; Zbl 0597.35012)] and J. Rauch, M. Reed [J. Funct. Anal. 73, 152-178 (1987; Zbl 0661.35058)].
For the multidimensional case we are able to prove a result on weak limits for the 3-\(D\) semilinear wave equation with sublinear nonlinear term, using essentially the fact that we have \(L^ 1(\mathbb{R}^ 3_ x)\to L^ 1(\mathbb{R}^ 3_ x)\) estimates for the fundamental solution of \(\square_ 3\) and it is supported in the cone \(\{(t,x):t\geq 0,\;t=| x|\}\).
Reviewer: T.Granchev

MSC:

35L60 First-order nonlinear hyperbolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35L45 Initial value problems for first-order hyperbolic systems
35L05 Wave equation
35L70 Second-order nonlinear hyperbolic equations
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References:

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