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Homogenization of the discrete diffusive coagulation-fragmentation equations in perforated domains. (English) Zbl 1402.35030

The discrete coagulation-fragmentation equation with diffusion is considered in a three-dimensional periodically perforated domain when the size of the holes converges to zero, the boundary of each hole bearing a source of monomers. More precisely, let \(\Omega\) be a bounded domain of \(\mathbb{R}^3\) with a smooth boundary \(\partial\Omega\) and consider a set \(T_\varepsilon\) of periodically distributed holes in \(\Omega\) which does not intersect \(\partial\Omega\), each hole having a volume of order \(\varepsilon^3\). The discrete coagulation-fragmentation equation with diffusion in \(\Omega_\varepsilon=\Omega\setminus\overline{T}_\varepsilon\) then reads \[ \begin{aligned} \partial_t u_1^\varepsilon - d_1 \Delta_x u_1^\varepsilon & = - u_1^\varepsilon \sum_{j=1}^\infty a_{1,j} u_j^\varepsilon + \sum_{j=1}^\infty B_{1+j} \beta_{1+j,1} u_{1+j}^\varepsilon \;\text{ in }\; (0,T)\times\Omega_\varepsilon, \\ \partial_\nu u_1^\varepsilon & = 0 \;\text{ on }\; (0,T)\times\partial\Omega\;, \\ \partial_\nu u_1^\varepsilon(t,x) & = \varepsilon \psi\left( t,x,\frac{x}{\varepsilon} \right) \;\text{ on }\; (0,T)\times\partial T_\varepsilon\;, \\ u_1^\varepsilon & = U_1 \;\text{ in }\; \Omega\;,\end{aligned} \] and, for \(i\geq 2\), \[ \begin{aligned} \partial_t u_i^\varepsilon - d_i \Delta_x u_i^\varepsilon & = - u_i^\varepsilon \sum_{j=1}^\infty a_{i,j} u_j^\varepsilon + \sum_{j=1}^\infty B_{i+j} \beta_{i+j,i} u_{i+j}^\varepsilon \cr & \qquad + \frac{1}{2} \sum_{j=1}^{i-1} a_{j,i-j} u_{i-j}^\varepsilon u_j^\varepsilon - B_i u_i^\varepsilon \;\text{ in }\; (0,T)\times\Omega_\varepsilon\;, \cr \partial_\nu u_1^\varepsilon & = 0 \;\text{ on }\; (0,T)\times\partial\Omega_\varepsilon\;, \cr u_i^\varepsilon & = 0 \;\text{ in }\; \Omega\;. \end{aligned} \] The source of monomers \(\psi\) belongs to \(C^1([0,T]\times\overline{\Omega}\times\mathbb{T}^3)\) and the limit as \(\varepsilon\to 0\) of the above system is identified with the help of the two-scale convergence of G. Nguetseng [SIAM J. Math. Anal. 20, No. 3, 608–623 (1989; Zbl 0688.35007)] and G. Allaire [SIAM J. Math. Anal. 23, No. 6, 1482–1518 (1992; Zbl 0770.35005)]. Classical assumptions on the diffusion coefficients \((d_i)\) and the coagulation and fragmentation rates \((a_{i,j})\), \((B_i)\), and \((\beta_{i+j,i})\) are needed to derive suitable estimates in order to perform the limit \(\varepsilon\to 0\).

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K51 Initial-boundary value problems for second-order parabolic systems
35K57 Reaction-diffusion equations
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References:

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