Reaction-diffusion systems treated as dynamical systems

Question

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.

Answer 1

For your basic question (topic name) the classical reference is

Henry D. Geometric Theory of Semilinear Parabolic Equations. SpringerVerlag, 1981.

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 $F$.

For general approaches, I would also recommend the paper

Ni W. M., Tang M. (2005). Turing patterns in the Lengyel-Epstein system for the CIMA reaction. Transactions of the American Mathematical Society, 357(10), 3953-3969,

where with the aid of topological methods and a priori estimates for certain parameters it is proved the uniqueness of the zero stationary state or the existence of non-homogeneous states.