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Entire functions of certain singular distributions and interaction of delta-waves in nonlinear conservation laws. (English) Zbl 1223.46041
Summary: With the help of our theory of multiplication of distributions it is possible to give a meaning in \(\mathcal D^\prime\) to the composition \(\phi \circ T\) , where \(\phi \) is an entire function and \(T\) belongs to a certain class of strongly singular distributions. As an application we are able to prove that, in our setting, the nonlinear conservation law \(u_t +[\phi (u)]_x = 0\) has solutions which propagate like solitary delta-waves with constant speed. The interaction of two delta-waves is also studied and the speed functions of these waves are evaluated. We conclude that such speed functions are always bounded and the collision of two such delta waves is impossible. These results are obtained with the help of a rigorous and consistent concept of solution we have already introduced in previous works.

MSC:
46F10 Operations with distributions and generalized functions
35F20 Nonlinear first-order PDEs
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