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Generalized solution to a semilinear hyperbolic system with a non-Lipshitz nonlinearity

Authors:

Marko Nedeljkov Stevan Pilipović

Institution:

(1) Faculty of Science Institute of Mathematics, University of Novi Sad, Trg D. Obradovi cacute

a 4, Novi Sad, Yugoslavia

Abstract:

Let

$(\partial _t + \Lambda (x,t)\partial _x )y(x,t) = F(x,t,y(x,t)),y(x,0) = A(x)$

((1))

be a semilinear hyperbolic system, whereA is a real diagonal matrix and a mappingy rarr

F(x, t, y) is in $\mathcal{O}_M (\mathbb{C}^n )$ with uniform bounds for (x, t) isin

².Oberguggenberger 6] has constructed a generalized solution to (1) whenA is an arbitrary generalized function andF has a bounded gradient with respect toy for (x, t) isin

². The above system, in the case when the gradient of the nonlinear termF with respect toy is not bounded, is the subject of this paper. F is substituted byF _h() which has a bounded gradient with respect toy for every fixed ( phiv

) and converges pointwise toF as epsi

0. A generalized solution to

$(\partial _t + \Lambda (x,t)\partial _x )y(x,t) = F_{h(\varepsilon )} (x,t,y(x,t)),y(x,0) = A(x)$

((2))

is obtained. It is compared to a continuous solution to (1) (if it exists) and the coherence between them is proved.

Keywords:

1991 Mathematics Subject Classification" target="_blank">1991 Mathematics Subject Classification 35A05 35L60 46F10

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