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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 $L\left[u\right]=0$, with initial data of the form $u\left(x,0\right)={u}_{0}+{u}_{1}\theta \left(-x+{a}_{1}\right)+{u}_{2}\theta \left(-x+{a}_{2}\right)$, where ${u}_{1},{u}_{2},{a}_{1}$ and ${a}_{2}$ stand for given real constants while $\theta \left(x\right)$ is the Heaviside unit step function, and gives a method to find an approximate weak solution valid in the asymptotic sense when certain parameter $\epsilon$ tends to zero. This approximation requires to replace $u\left(x,t\right)$ by a function ${u}_{\epsilon }\left(x,t\right)$ such that $〈L\left[{u}_{\epsilon }\right],\eta 〉=O\left(\epsilon \right)$ for any $\eta \left(x\right)\in {C}_{0}^{\infty }$. The method is based on the identity

${\omega }_{1}\left(\frac{x-{a}_{1}}{\epsilon }\right){\omega }_{2}\left(\frac{x-{a}_{2}}{\epsilon }\right)=\theta \left(x-{a}_{1}\right){B}_{1}\left(\frac{{\Delta }a}{\epsilon }\right)+\theta \left(x-{a}_{2}\right){B}_{2}\left(\frac{{\Delta }a}{\epsilon }\right)+{O}_{D}\left(\epsilon \right),$

where ${\omega }_{j}\left(z\right)\in {C}^{\infty }$ stands for any function such that its derivative belongs to the Schwartz space and ${lim}_{z\to -\infty }{\omega }_{j}=0$, ${lim}_{z\to \infty }{\omega }_{j}=1$, while ${B}_{j}\left(z\right)$ refer to two functions defined in terms of ${\omega }_{1}$ and ${\omega }_{2}$. As to ${O}_{D}\left(\epsilon \right)$, it denotes any distribution, say $f$, such that $〈f,\eta 〉=O\left(\epsilon \right)$. The method is applied to the classical Hopf equation for which $L\left[u\right]={u}_{t}+{\left({u}^{2}\right)}_{x}$. 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:
 35L60 Nonlinear first-order hyperbolic equations 35A21 Propagation of singularities (PDE)
##### Keywords:
initial value problems; Hopf equation