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A reaction-diffusion system arising in modelling man-environment diseases. (English) Zbl 0704.35069

The system that under consideration: \[ (*)\quad \partial v/\partial t=- a_ 1v_ 1+\Delta v_ 1;\quad \partial v_ 2/\partial t=-a_ 2v_ 2+g(v_ 1),\quad (t,x)\in (0,\infty)\times \Omega; \]
\[ \partial v_ 1/\partial n+\beta v_ 1=\int_{\Omega}K(x,x')v_ 2(t,x')dx',\quad (t,x)\in (0,\infty)\times \Gamma;\quad v_ 1(0,x)=v^ 0_ 1(x),\quad v_ 2(0,x)=v^ 0_ 2(x), \] where \(\Omega\) is a bounded domain \({\mathbb{R}}^ n\) \((n=2,3)\) with boundary \(\Gamma\) and \(\partial /\partial n\) denotes the outward normal derivative. Two sets of regularity conditions will be used for the parameters \((a_ 1,a_ 2,\Omega,\beta,g,K)\) determining (*):
(A1) \(a_ 1\in L^{\infty}(\Omega)\), \(a_ 1(x)\geq \alpha_ 1\) a.e. for some \(\alpha_ 1>0\), \(\Gamma\) is \(C^{0,1}\)-smooth, \(\beta \in L^{\infty}(P)\), \(\beta\geq 0\), \(K\in L^{\infty}(\Gamma \times \Omega);\)
(A2) \(a_ 1\in C^ 1({\bar \Omega})\), \(a_ 1(x)\geq \alpha_ 1>0\), \(x\in \Omega\), \(a_ 2\in C({\bar \Omega})\), \(a_ 2\geq 0\), \(\Gamma\) is \(C^{2,1}\)-smooth, \(\beta \in C^ 1(P)\), \(\beta\geq 0\), g Lipschitz continuous from \(L^ 1(\Omega)\) to \(L^ 1(\Omega)\), \(K\in L^{\infty}(\Gamma \times \Omega)\), \(K_ x\in L^{\infty}(\Gamma \times \Omega).\)
After the explanation of the motivation for studying (*), the authors show the existence and uniqueness of (*) and give conditions which guarantee that the solution operator associated with the system generates a strongly continuous semigroup of nonlinear operators on the space of \(L^ 1(\Omega)\times L^ 1(\Omega)\), devote to some aspects of the linearized model containing density result which is essential for the proof of positivity of the solution semigroup and continue to investigate qualitative properties of the solutions of (*) and in particular, they show monotonicity as well as concavity properties.
Reviewer: Y.Ebihara

MSC:

35K57 Reaction-diffusion equations
47H20 Semigroups of nonlinear operators
35B99 Qualitative properties of solutions to partial differential equations
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