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Generalized solutions describing singularity interaction. (English) Zbl 1011.35093

The author considers initial value problems connected with nonlinear differential equations of the second degree, say L[u]=0, with initial data of the form u(x,0)=u 0 +u 1 θ(-x+a 1 )+u 2 θ(-x+a 2 ), where u 1 ,u 2 ,a 1 and a 2 stand for given real constants while θ(x) is the Heaviside unit step function, and gives a method to find an approximate weak solution valid in the asymptotic sense when certain parameter ε tends to zero. This approximation requires to replace u(x,t) by a function u ε (x,t) such that L[u ε ],η=O(ε) for any η(x)C 0 . The method is based on the identity

ω 1 x-a 1 εω 2 x-a 2 ε=θ(x-a 1 )B 1 Δa ε+θ(x-a 2 )B 2 Δa ε+O D (ε),

where ω j (z)C stands for any function such that its derivative belongs to the Schwartz space and lim z- ω j =0, lim z ω j =1, while B j (z) refer to two functions defined in terms of ω 1 and ω 2 . As to O D (ε), it denotes any distribution, say f, such that f,η=O(ε). The method is applied to the classical Hopf equation for which L[u]=u t +(u 2 ) x . An extension of the method to the case when x belongs to a two-dimensional space is also shown on a particular example. Remark that this second application contains some misprints.

MSC:
35L60Nonlinear first-order hyperbolic equations
35A21Propagation of singularities (PDE)