Algebraic decay to equilibrium for the Becker-Döring equations (Q2817445)

The subject of this paper are properties of the following infinite system of ODE's NEWLINE\[NEWLINE{d\over dt}c_i(t)=J_{i-1}(t)-J_i(t),\;i=2,3\cdotsNEWLINE\]NEWLINE NEWLINE\[NEWLINE{d\over dt}c_1(t)=J_1(t)-\sum_{i=1}^\infty J_i(t)\leqno(1)NEWLINE\]NEWLINE with NEWLINE\[NEWLINEJ_i(t)=a_ic_1(t)c_i(t)-b_{i+1}c_{i+1}(t),NEWLINE\]NEWLINE and \((a_i)\), \((b_i)\) are fixed positive sequences. The above system enters to the class of \textit{coagulation-fragmentation equations}. The authors are mainly interested in the solutions preserving in time the first moment: \(\sum_{i=1}^\infty ic_i(t)\equiv \rho>0\).NEWLINENEWLINEThe quilibrium solutions to the system \((1)\) are defined as certain sequences \((Q_i)\) satisfying the equation \(\sum_{i=1}^\infty iQ_i=\rho\).NEWLINENEWLINEWriting the solution of (1) in the form of perturbation of the equilibrium solution \(c_i=Q_i(1+h_i)\) and introducing the convenient Banach Spaces \(X_k\) with norms \(\|\cdot\|_{X_k}\), the authors formulate and prove the main theorem of this paper saying that the following estimate holds: NEWLINE\[NEWLINE\|h(t)\|_{X_{i+m}}\leq C_{k,m}(1+t)^{-(k-m-1)}\|h(0)\|_{X_{i+k}}\;\forall t>0.NEWLINE\]NEWLINE for certain constants \(C_{k,m}\), where \(h\) is related to \(h_i,\;i= 1,2\cdots\) via certain functional relation.NEWLINENEWLINEThe paper contains detailed proofs; 18 bibliography positions are given.