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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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[1] Dunford, N; Schwartz, J.T, “linear operators,” pt. 1, (), 298
[2] Oberguggenberger, M, Weak limits of solutions of semilinear hyperbolic systems, Math. ann., 274, 599-607, (1986) · Zbl 0597.35012
[3] Rauch, J; Reed, M, Propagation of singularities for semilinear hyperbolic equations in one space variable, Ann. of math., 111, 531-532, (1980) · Zbl 0432.35055
[4] Rauch, J; Reed, M, Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: creation and propagation, Comm. math. phys., 81, 203-207, (1981) · Zbl 0468.35064
[5] Rauch, J; Reed, M, Nonlinear microlocal analysis of semi-linear hyperbolic systems in one space dimension, Duke math. J., 49, 397-476, (1982)
[6] Stein, E, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton, NJ · Zbl 0207.13501
[7] Taylor, M, Pseudodifferential operators, (1981), Princeton Univ. Press Princeton, NJ · Zbl 0453.47026
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