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Global existence and large time behavior of positive solutions to a reaction diffusion system. (English) Zbl 1038.35031

The authors are interested in global existence and long-time behaviour of solutions of the system of reaction-diffusion equations \[ u_t-a \Delta u=-f(u)g(v),\quad v_t-c\Delta u-d\Delta v=f(u)g(v)\text{ in } \mathbb{R}_+\times\Omega \tag{1} \] 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 \[ u(0,x)=u_0(x), \;v(0,x)=v_0(x) \text{ in }\Omega.\tag{2} \] 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.

MSC:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35K45 Initial value problems for second-order parabolic systems
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