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 book 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.

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.

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 book of ALAIN HARAUX - SYSTEMES 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.

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 book 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.

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 book of ALAIN HARAUX - SYSTEMES DYNAMIQUES DISSIPATIFS ET APPLICATIONS, and 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.

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 book of ALAIN HARAUX - SYSTEMES 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.