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

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→F(x, t, y) is in$\mathcal{O}_M (\mathbb{C}^n )$ with uniform bounds for (x, t) ∈K ⊂⊂ ∝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) ∈K ⊂⊂ ∝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(ε) which has a bounded gradient with respect toy for every fixed (ϕ, ε) and converges pointwise toF as ε→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.