an:04080036 Zbl 0661.35058 Rauch, Jeffrey; Reed, Michael C. Nonlinear superposition and absorption of delta waves in one space dimension EN J. Funct. Anal. 73, 152-178 (1987). 00153238 1987
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35L60 35L45 35B30 35B65 35B40 distribution initial data; semilinear strictly hyperbolic; classical; smooth; superposition principle; singular part; singular measure; superlinear dissipation 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. A.Doktor