Nonlinear superposition and absorption of delta waves in one space dimension
The author deals with problems of the following types: let \(u^{\epsilon}\) be the solution of the semilinear strictly hyperbolic system \[ (*)\quad (\partial_ t+A(x,t)\partial_ x+B(x,t))u=f(x,t,u) \] with the initial data of the form \(g+h^{\epsilon}\), where g is ``classical'' and smooth \(h^{\epsilon}\) converge to a distribution \(\mu\). Then \(u^{\epsilon}\) converge to \(\bar u+\sigma\) in specified sense, where \(\bar u\) is a solution of (*), \(\bar u(t=0)=g\), and \(\sigma\) is a solution of (*) with \(f=0\), \(\sigma (t=0)=\mu\). This fact expresses a superposition principle: the singular part of the solution propagates linearly, the classical part propagates by the nonlinear equation. The distribution \(\mu\) is a singular measure for f sublinear or can be more singular for f bounded. If f satisfies condition of superlinear dissipation \(\lim_{| u_ j| \to \infty} f_ j(x,t,u)/u_ j=- \infty\) and \(sgn(u_ j)f_ j(x,t,u)\leq c(l+\sum | u_ i|)\) (the solution do not blow up in finite time), then \(u^{\epsilon}\to \bar u,\) i.e. the singular part is absorbed.
scientific article; zbMATH DE number 4080036