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Nonlinear superposition and absorption of delta waves in one space dimension. (English) Zbl 0661.35058
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.
Reviewer: A.Doktor

##### MSC:
 35L60 First-order nonlinear hyperbolic equations 35L45 Initial value problems for first-order hyperbolic systems 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B65 Smoothness and regularity of solutions to PDEs 35B40 Asymptotic behavior of solutions to PDEs
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