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

Abstract

Let (∂t, + Λ(x, t)∂x)y(x, t) - F(x, t, y(x, t)), y(x, 0) = A(x) (1) be a semilinear hyperbolic system, where Λ is a real diagonal matrix and a mapping y → F(x, t, y) is in CM(ℂn) with uniform bounds for (x, t) ∈ K ⊂⊂ ℝ2. OBERGUGGENBERGER [6] has constructed a generalized solution to (1) when A is an arbitrary generalized function and F has a bounded gradient with respect to y for (x, t) ∈ K ⊂⊂ ℝ2. The above system, in the case when the gradient of the nonlinear term F with respect to y is not bounded, is the subject of this paper. F is substituted by Fh(Ξ) which has a bounded gradient with respect to y for every fixed (φ, ε) and converges pointwise to F as ε → 0. A generalized solution to (∂t + Λ(x, t)∂x)y(x, t)) = Fh(Ξ)(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.