## Singular measure traveling waves in an epidemiological model with continuous phenotypes

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- by Quentin Griette PDF
- Trans. Amer. Math. Soc.
**371**(2019), 4411-4458 Request permission

## Abstract:

We consider the reaction-diffusion equation \begin{equation*} u_t=u_{xx}+\mu \left (\int _\Omega M(y,z)u(t,x,z)dz-u\right ) + u\left (a(y)-\int _\Omega K(y,z) u(t,x,z)dz\right ) , \end{equation*} where $u=u(t,x,y)$ stands for the density of a theoretical population with a spatial ($x\in \mathbb R$) and phenotypic ($y\in \Omega \subset \mathbb {R}^n$) structure, $M(y,z)$ is a mutation kernel acting on the phenotypic space, $a(y)$ is a fitness function, and $K(y,z)$ is a competition kernel. Using a vanishing viscosity method, we construct measure-valued traveling waves for this equation and present particular cases where singular traveling waves do exist. We determine that the speed of the constructed traveling waves is the expected spreading speed $c^*:=2\sqrt {-\lambda _1}$, where $\lambda _1$ is the principal eigenvalue of the linearized equation. As far as we know, this is the first construction of a measure-valued traveling wave for a reaction-diffusion equation.## References

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## Additional Information

**Quentin Griette**- Affiliation: IMAG, Université de Montpellier, 163 rue Auguste Broussonnet, 34090 Montpellier, France
- Address at time of publication: IMB, Université de Bordeaux, 351 cours de la Libération, 33800 Talence, France
- MR Author ID: 1152578
- Email: quentin.griette@math.u-bordeaux.fr
- Received by editor(s): October 5, 2017
- Received by editor(s) in revised form: June 1, 2018, and June 27, 2018
- Published electronically: November 13, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 4411-4458 - MSC (2010): Primary 35R09; Secondary 35C07, 35D30, 35Q92, 92D30, 92D15
- DOI: https://doi.org/10.1090/tran/7700
- MathSciNet review: 3917227