On the mass conserving solutions to the singular kernel coagulation with multi-fragmentation
Existence and uniqueness of weak solutions to the coagulation-fragmentation equation \begin{align*} \partial_t f(t,x) & = \frac{1}{2} \int_0^x K(x-y,y) f(t,x-y) f(t,y)\ dy - \int_0^\infty K(x,y) f(t,x) f(t,y)\ dy \\ & \qquad + \int_x^\infty S(y) b(x,y) f(t,y)\ dy - S(x) f(t,x), \qquad (t,x)\in (0,\infty)^2, \\ f(0) & = f_0, \qquad x\in (0,\infty), \end{align*} are studied when the coagulation kernel features a possible singularity for small sizes and has linear growth for large sizes: \[ 0 \le K(x,y) = K(y,x) \le k \frac{(1+x+y)^\lambda}{(xy)^\sigma}, \qquad \sigma\ge 0, \ \lambda-\sigma\in [0,1], \] while the overall fragmentation rate \(S\) and the daughter distribution \(b\) satisfy \begin{align*} S(x) \le s_0 x^\alpha, & \quad x b(x,y) \le \bar{b}, \\ \int_0^y b(x,y)\ dx = y, & \quad \int_0^y x^{-2\sigma} b(x,y)\ dx \le \bar{N} y^{-2\sigma}, \end{align*} for some \(r\ge 1\), \(\alpha\in \big[0,\min{\{r,1+2\sigma\}}\big]\), \(s_0>0\), \(\bar{b}>0\), and \(\bar{N}>0\). Assuming that \(f_0\) is a non-negative function in \(\Upsilon_{r,\sigma} = L^1((0,\infty);(x^r+x^{-2\sigma})dx)\), the existence of a non-negative weak solution \(f\in C([0,T],\Upsilon_{r,\sigma})\) is claimed for some time \(T>0\), using the strong compactness approach in \(L^1(0,\infty)\) developed in [\textit{P. B. Dubovskiĭ} and \textit{P. B. Stewart}, Math. Methods Appl. Sci. 19, No. 7, 571--591 (1996; Zbl 0852.45016)]. A global existence result is obtained in Theorem~8.4.2 in [\textit{J. Banasiak} et al., Analytic methods for coagulation-fragmentation models. Volume I. Boca Raton, FL: CRC Press (2020; Zbl 1434.82001)] under different integrability conditions on \(b\) by means of a weak compactness approach.
scientific article; zbMATH DE number 7642735
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