A class of singular first order differential equations with applications in reaction-diffusion

Ricardo Enguiça; Andrea Gavioli; Luís Sanchez

doi:10.3934/dcds.2013.33.173

Article Contents

2013, Volume 33, Issue 1: 173-191. Doi: 10.3934/dcds.2013.33.173

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A class of singular first order differential equations with applications in reaction-diffusion

1.
Area Departamental de Matemática, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Emídio Navarro, 1 - 1950-062 Lisboa
2.
Dipartimento di Matematica Pura ed Applicata, Univ. di Modena e Reggio Emilia, Via Campi, 213b, 41100 Modena
3.
Faculdade de Ciências da Universidade de Lisboa, CMAF, Avenida Professor Gama Pinto 2, 1649-003 Lisboa

Received: July 31, 2011

Revised: January 31, 2012

Published: December 2012

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Abstract

We study positive solutions $y(u)$ for the first order differential equation $$y'=q(c\,{y}^{\frac{1}{p}}-f(u))$$ where $c>0$ is a parameter, $p>1$ and $q>1$ are conjugate numbers and $f$ is a continuous function in $[0,1]$ such that $f(0)=0=f(1)$. We shall be particularly concerned with positive solutions $y(u)$ such that $y(0)=0=y(1)$. Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the $p$-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for $p=2$.

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Mathematics Subject Classification: Primary: 34B18, 34C37, 35K57.

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