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Products of distributions and singular travelling waves as solutions of advection-reaction equations. (English) Zbl 1284.35112
Summary: We restrict our attention to the advection-reaction equation \(u_t+[\phi (u)]_x=\psi (u)\), where \(\phi\) and \(\psi\) are entire functions. Conditions for the propagation of a distributional wave profile are presented and the wave speed is evaluated. As an example, we prove that, under certain conditions, the propagation of delta-waves in models ruled by the diffusionless Burgers-Fisher equation is possible and compute the speeds of propagation of these waves. In the same setting, the propagation of travelling waves with the shape of a \(C^1\)-function with one jump discontinuity is also studied. These results will be easily explained by our theory of distributional products and are based on a rigorous and consistent concept of a solution that we have already introduced in previous works.

35C07 Traveling wave solutions
35F25 Initial value problems for nonlinear first-order PDEs
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