zbMATH — the first resource for mathematics

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

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.