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Existence of solutions for the discrete coagulation-fragmentation model with diffusion. (English) Zbl 0892.35077
We consider the following infinite system of reaction-diffusion equations: \[ \begin{aligned}{\partial u_1 \over \partial t} = & d_1 \Delta u_1-u_1 \sum^\infty_{j=1} a_{1j} u_j+ \sum^\infty_{j=1} b_{1j} u_{1+j},\\ {\partial u_i \over\partial t} = & d_i \Delta u_i+ {1\over 2} \sum^{i-1}_{j=1} (a_{i-j,j} u_{i-j} u_j-b_{i-j,j} u_i) - u_i \sum^\infty_{j=1} a_{ij} u_j+ \sum_{j= 1}^\infty b_{ij}u_{i+j}, \quad i=2,3,\dots,\end{aligned} \] on \(\Omega_T= \Omega \times (0,T)\), subject to the initial condition \(u_i(0,x) =U_i(x)\) for \(x\in\Omega\), and Neumann boundary condition \({\partial u_i\over \partial \nu}=0\) on \(\partial \Omega\times (0,T)\), where \(\Omega \subset \mathbb{R}^n\) is a bounded domain with smooth boundary, \(\nu\) is an outward normal vector of \(\Omega\), and \(U_i\in L^\infty (\Omega)\), \(i=1,2, \dots\), are given nonnegative functions. The system is a generalization of the discrete coagulation-fragmentation model which describes the dynamics of cluster growth. We assume two physically meaningful growth conditions of the coagulation coefficients, namely \(a_{ij}\leq Aij\) for \(i,j\geq 1\), \(A>0\), and \(a_{ij} \leq A(i+j)\) for \(i,j\geq 1\), \(A>0\).
Solutions are constructed as limits of solutions of finite systems.

MSC:
35K57 Reaction-diffusion equations
35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions)
92E20 Classical flows, reactions, etc. in chemistry
35K20 Initial-boundary value problems for second-order parabolic equations
Keywords:
cluster growth
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