Weak limits of solutions to semilinear hyperbolic systems (Q1078748)

The paper deals with the Cauchy problem for the semilinear strictly hyperbolic first order system in diagonal form \[ (\partial_ t+\Lambda (x,t)\partial_ x)u(x,t)=F(x,t,u),\quad (x,t)\in R^ 2;\quad u(x,0)=a(x),\quad x\in R \] where \(u: R^ 2\to R^ n\), \(F: R^{n+2}\to R^ n\) is smooth and \(\Lambda\) is a smooth, real valued diagonal matrix. Define the so-called delta-net as \(\{\phi_{\epsilon}\}_{0<\epsilon \leq 1}\subset {\mathcal D}(R)\) with the following properties: supp (\(\phi\) \({}_{\epsilon})\to \{0\}\) as \(\epsilon\) \(\to 0\), \(\int \phi_{\epsilon}(x) dx=1\), \(\int | \phi_{\epsilon}(x)| dx\) is uniformly bounded for \(0<\epsilon \leq 1\); a are distributions with support at finitely many points. If F(x,t,.) and \(\nabla_ uF\) are uniformly bounded for any compact set, the smooth solutions with initial data \(a*\phi_{\epsilon}\) converge in \({\mathcal D}'(R^ 2)\) to \(v+w\), where \(v\in {\mathcal D}'(R^ 2)\) is the solution to \[ (\partial_ t+\Lambda (x,t)\partial_ x)v=0;\quad v(x,0)=a(x) \] and \(w\in {\mathcal C}^{\infty}(R^ 2)\) is the solution to \[ (\partial_ t+\Lambda (x,t)\partial_ x)w=F(x,t,w);\quad w(x,0)=0. \] An application to a nonlinear Klein-Gordon equation is given.