# zbMATH — the first resource for mathematics

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
cluster growth
Full Text: