The author is concerned with some nonlinear propagation phenomena for reaction-advection-diffusion equation in the periodic framework. The author deals with travelling wave solution of the equation
\[ u_t=\nabla \cdot (A(z) \nabla u) +q(z)\cdot \nabla u+f(z,u) , \quad t\in \mathbb R,\;z\in \Omega, \]
propagating with speed \(c\). The author considers three types of nonlinearities.

1.

“Combustion” nonlinearity: A nonlinearity \(f=f(x,y,u) \) defined in \(\overline{\Omega }\times\mathbb R\), is called a function “combustion” nonlinearity if:
\[ \begin{aligned} &\begin{cases} f\text{ is globally Lipschitz-continuous in }\overline{\Omega}\times\mathbb R;\\ \forall (x,y) \in \overline{\Omega },\;\forall s\in (-\infty,0] \cup [ 1,+\infty),\;f(s,x,y) =0; \\ \exists \rho \in (0,1),\;\forall (x,y) \in \overline{\Omega },\;\forall 1-\rho \leq s\leq s'\leq 1, f(x,y,s) \geq f(x,y,s');\end{cases}\tag{1}\\ &f\text{ is L-periodic with respect to }x;\tag{2}\\ &\begin{cases} \exists \theta \in (0,1),\;\forall (x,y) \in\overline{\Omega },\;\forall s\in [0,\theta],\;f(x,y,s) =0; \\ \forall s\in (0,1),\;\exists (x,y) \in \overline{\Omega }\text{ such that }f(x,y,s) >0. \end{cases}\tag{3} \end{aligned} \]
In this case of “combustion” nonlinearity, the author proves that the speed \(c\) exists and it is unique, while the front \(u\) is unique up to a translation in \(t.\)

2.

“ZFK” nonlinearity (Zeldovich-Frank-Kamenetskii): If \(f\) satisfies (1) and (2), and instead of (3) we suppose that
\[ \begin{cases}\exists \delta >0, \text{ the restriction of \(f\) to }\overline{\Omega}\times[ 0,1] \text{ is of class }C^{1,\delta }; \\ \forall s\in (0,1),\;\exists (x,y) \in \overline{\Omega}\text{ such that }f(x,y,s) >0,\end{cases} \]
then \(f\) is called “ZFK” nonlinearity.

3.

“KPP” nonlinearity (Kolmogorov-Petrovsky-Piskunov): If \(f\) is a “ZFK” nonlinearity that satisfies
\[ f_u'(x,y,0)=\lim_{u\to 0^+} \frac{f(x,y,u) }u>0 \]
with the additional assumption
\[ \forall (x,y,s) \in \overline{\Omega }\times (0,1),\;0<f(x,y,s) \leq f_u'(x,y,0) \times s, \]
then \(f\) is called “KPP” nonlinearity.

In the case of “ZFK” nonlinearity or “KPP” nonlinearity, the author proves that there exist a minimal speed of propagation \(c^*\). In this situation the author gives a min-max formula for \(c^*\) and apply this formula to prove a variational formula involving eigenvalue problems for the minimal speed \(c^*\) in the “KPP” nonlinearity case.