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Analytical examples of diffusive waves generated by a traveling wave. (English) Zbl 1375.35578
Summary: We construct analytical solutions for a system composed of a reaction-diffusion equation coupled with a purely diffusive equation. The question is to know if the traveling wave solutions of the reaction-diffusion equation can generate a traveling wave for the diffusion equation. Our motivation comes from the calcic wave, generated after fertilization within the egg cell endoplasmic reticulum, and propagating within the egg cell. We consider both the monostable (Fisher-KPP type) and bistable cases. We use a piecewise linear reaction term so as to build explicit solutions, which leads us to compute exponential tails whose exponents are roots of second-, third-, or fourth-order polynomials. These raise conditions on the coefficients for existence of a traveling wave of the diffusion equation. The question of positivity and monotonicity is only partially answered.
MSC:
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35C07 Traveling wave solutions
35K57 Reaction-diffusion equations
92C37 Cell biology
34B60 Applications of boundary value problems involving ordinary differential equations
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