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Semilinear hyperbolic systems in one space dimension with strongly singular initial data. (English) Zbl 0886.35090

Summary: Interactions of singularities in semilinear hyperbolic partial differential equations in \(\mathbb{R}^2\) are studied. Consider a simple nonlinear system of three equations with derivatives of Dirac delta functions as initial data. As the microlocal linear theory prescribes, the initial singularities propagate along forward bicharacteristics. But there are also anomalous singularities created when these characteristics intersect. Their regularity satisfies the following “sum law”: the “strength” of the anomalous singularity equals the sum of the “strengths” of the incoming singularities. Hence the solution to the system becomes more singular as time progresses.

MSC:

35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35A20 Analyticity in context of PDEs
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