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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 $u(x,0)= u_0+u_1\theta (-x+a_1)+u_2 \theta(-x+ a_2)$, where $u_1,u_2,a_1$ and $a_2$ stand for given real constants while $\theta (x)$ is the Heaviside unit step function, and gives a method to find an approximate weak solution valid in the asymptotic sense when certain parameter $\varepsilon$ tends to zero. This approximation requires to replace $u(x,t)$ by a function $u_\varepsilon(x,t)$ such that $\langle L[u_\varepsilon], \eta\rangle=O (\varepsilon)$ for any $\eta(x)\in C^\infty_0$. The method is based on the identity $$\omega_1\left({x-a_1\over \varepsilon}\right) \omega_2\left({x-a_2 \over\varepsilon} \right)=\theta (x-a_1)B_1\left({\Delta a\over \varepsilon} \right)+\theta (x-a_2)B_2\left({\Delta a\over \varepsilon} \right)+ O_D (\varepsilon), $$ where $\omega_j(z)\in C^\infty$ 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 $B_j(z)$ refer to two functions defined in terms of $\omega_1$ and $\omega_2$. As to $O_D (\varepsilon)$, it denotes any distribution, say $f$, such that $\langle f,\eta \rangle= O(\varepsilon)$. 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.

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