Solvability of a doubly singular boundary value problem arising in front propagation for reaction-diffusion equations
Cristina Marcelli
TL;DR
This work addresses the solvability of a doubly singular boundary value problem $\dot z = c g(u) - f(u) - \dfrac{h(u)}{z^{\alpha}}$ with $z(0^+)=z(1^-)=0$ and $z>0$ on $(0,1)$, which arises in travelling-wave analyses of reaction-diffusion-convection equations governed by the $p$-Laplacian. The authors establish a sharp solvability criterion controlled by $h_{0,\alpha}=\lim_{u\to0^+} h(u)/u^{\alpha}$: if this limit is infinite there are no solutions for any speed $c$, while if finite there exists a threshold $c^*$ such that solvability holds iff $c\ge c^*$. They derive explicit bounds on $c^*$ in terms of integral quantities $G_0,F_0,H_0$ built from $g,f,h$, and prove existence for $c=c^*$ and $c>c^*$ via a combination of lower-upper solutions and compactness arguments, with uniqueness for $c\ge c^*$. The results thus connect the near-origin behavior of the reaction-diffusion-convection data to admissible travelling-wave speeds, providing a tool for understanding front propagation in degenerate parabolic models.
Abstract
The paper deals with the solvability of the following doubly singular boundary value problem \[\begin{cases} \dot z = c g(u)-f(u) -\dfrac{h(u)}{z^α}\\ z(0^+)=0, z(1^-)=0, \ z(u)>0 \text{ in } (0,1)\end{cases}\] naturally arising in the study of the existence and properties of travelling waves for reaction-diffusion-convection equations governed by the $p-$Laplacian operator. Here $c,α$ are real parameters, with $α>0$, and $f,g,h$ are continuous functions in $[0,1]$, with \[ h(0)=h(1), \quad h(u)>0 \text{ in } (0,1).\]