# Reaction-diffusion systems treated as dynamical systems

I wonder if there is a good reference on reaction-diffusion systems on $$\mathbb{R}^N$$, that treats them as dynamical systems.

I have the books of Alain Haraux – Systèmes dynamiques dissipatifs et applications, and Joel Smoller – Shock waves and reaction–diffusion equations, but I look for something newer and more detailed.

I'm interested in the stability of a steady state of a homogenous reaction-diffusion system with two or three interacting populations in a general domain $$\Omega\subseteq\mathbb{R}^N$$ with $$N=2$$ or $$N=3$$ with Robin boundary conditions. They have the form:

$$\begin{cases} \dfrac{\partial y_1}{\partial t}(t,x)-d_1\Delta y_1=F_1(y_1,y_2) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y_2}{\partial t}(t,x)-d_2\Delta y_2=F_2(y_1,y_2) ,\ (t,x)\in (0,T)\times\Omega \\ \dfrac{\partial y_1}{\partial \nu}(t,x)+b_1y_1(t,x)=\dfrac{\partial y_2}{\partial \nu}(t,x)+b_2y_2(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega \\ y_1(0,x)=y_{01}(x),\ y_2(0,x)=y_{02}(x)\ x\in\Omega\end{cases}$$

I heard from some authors in that field that the stationary problem associated with a reaction-diffusion system has exactly two positive solutions in general but they gave me only some intuitive proof of that thing. Maybe they utilize $$\omega$$-limits and some parabolic estimates, but I can't figure out how.

Here is the stationary system:

$$\begin{cases} -d_1\Delta Y_1=F_1(Y_1,Y_2) ,\ x\in\Omega\\ -d_2\Delta Y_2=F_2(Y_1,Y_2) ,\ x\in \Omega \\ \dfrac{\partial Y_1}{\partial \nu}(x)+b_1Y_1(x)=\dfrac{\partial Y_2}{\partial \nu}(x)+b_2Y_2(x)=0,\ x\in\partial\Omega\end{cases}$$

I can't figure this out even for a single population, that is why if:

$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-d\Delta y=F(y) ,\ (t,x)\in (0,T)\times\Omega\\ \dfrac{\partial y}{\partial \nu}(t,x)+by(t,x)=0,\ (t,x)\in (0,T)\times\partial\Omega\\ y(0,x)=y_{0}(x),\ x\in\Omega\end{cases}$$

then the stationary problem:

$$\begin{cases} -d\Delta Y=F(Y) ,\ x\in\times\Omega\\ \dfrac{\partial Y}{\partial \nu}(x)+bY(x)=0,\ x\in \partial\Omega\end{cases}$$

has exactly two positive solutions: $$Y=0$$ and other strictly positive?

I wonder if there is a book treating this type of problems in details.

However, you will not find there general recipes for your second question and that's why. A stationary state, say $$y_{0}(x)$$, can be homogeneous (given by constant functions $$y_{0}(x) \equiv y_{0})$$ and non-homogeneous ($$y_{0}(x)$$ is not constant). For Robin conditions the only homogeneous state is $$0$$ (I assume $$b \not = 0$$ and $$F(0)=0$$). To find non-homogeneous states you have to solve a nonlinear PDE that is impossible to do analytically. So, the only way is to provide qualitative analysis of the equation that shows at least the existence (or non-existence) of non-homogeneous states (and their number if we get lucky). This highly relies on the structure of the nonlineraity $$F$$. For example, when $$F$$ is Lipschitz with a small Lipschitz constant, then the zero will be usually globally stable and there cannot be non-homogeneous states. Increasing the Lipschitz constant you may get the pitchfork bifurcation (the appearance of two non-homogeneous stationary states in a neighbourhood of zero) and so on. So, the result you mentioned (on the exsitence and number of non-homogeneous states) must use some very specifity of $$F$$.