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Measure valued solutions of asymptotically homogeneous semilinear hyperbolic systems in one space variable

Published online by Cambridge University Press:  20 January 2009

F. Demengel  and
J. Rauch
F. Demengel
Affiliation:
Département de MathematiquesUniversité de Paris SudOrsay, 91405 Cedex, France
J. Rauch
Affiliation:
Department of MathematicsUniversity of MichiganAnn Arbor, MI 48109, U.S.A.
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Abstract

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We study systems which in characteristic coordinates have the form

where A is a k × k diagonal matrix with distinct real eigenvalues. The nonlinearity F is assumed to be asymptotically homogeneous in the sense, that it is a sum of two terms, one positively homogeneous of degree one in u and a second which is sublinear in u and vanishes when u = 0. In this case, F(t, x, u(t)) is meaningful provided that u(t) is a Radon measure, and, for Radon measure initial data there is a unique solution (Theorem 2.1).

The main result asserts that if μn is a sequence of initial data such that, in characteristic coordinates, the positive and negative parts of each component, , converge weakly to μ±, then the solutions coverge weakly and the limit has an interesting description given by a nonlinear superposition principle.

Simple weak converge of the initial data does not imply weak convergence of the solutions.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 33 , Issue 3 , October 1990 , pp. 443 - 460
Copyright
Copyright © Edinburgh Mathematical Society 1990

References

REFERENCES

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