Global existence and large time behavior of positive solutions to a reaction diffusion system. (Q5933745)

The authors are interested in global existence and long-time behaviour of solutions of the system of reaction-diffusion equations NEWLINE\[NEWLINEu_t-a \Delta u=-f(u)g(v),\quad v_t-c\Delta u-d\Delta v=f(u)g(v)\text{ in } \mathbb{R}_+\times\Omega \tag{1}NEWLINE\]NEWLINE with homogeneous Neumann boundary conditions \(\frac{\partial u}{ \partial \nu}=\frac{\partial v}{\partial\nu}= 0\) on \(\mathbb{R}_+\times \partial \Omega\) and initial data NEWLINE\[NEWLINEu(0,x)=u_0(x), \;v(0,x)=v_0(x) \text{ in }\Omega.\tag{2}NEWLINE\]NEWLINE Here \(\Omega\) is a bounded regular domain in \(\mathbb{R}^N\), \(u_0\) and \(v_0\) are given nonnegative and bounded functions. The constants \(a,c\), and \(d\) are positive numbers. The authors prove the existence of bounded (global) solutions using some properties of the Neumann function for the heat equation posed in a bounded domain. When the spatial domain is \(\mathbb{R}^N\), the proof relies on well-known properties of the fundamental solution of the heat equation.