MEASURE VALUED SOLUTIONS OF ASYMPTOTICALLY HOMOGENEOUS SEMILINEAR HYPERBOLIC SYSTEMS IN ONE SPACE VARIABLE

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, (μnk)‡, 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. © 1990, Edinburgh Mathematical Society. All rights reserved.