Well-posedness for boundary value problems for coagulation-fragmentation equations

TL;DR

This paper proves well-posedness and analyzes the long-term behavior of solutions for a boundary value coagulation-fragmentation equation, including cases with singular kernels and boundary conditions, with implications for stability and regularity.

Contribution

It establishes well-posedness for bounded and singular kernels, and characterizes the asymptotic behavior of solutions under various boundary conditions.

Findings

Solutions converge exponentially fast to zero without fragmentation.

Solutions stabilize to equilibrium under detailed balance boundary conditions.

Finiteness of negative moments improves solution regularity.

Abstract

We investigate a coagulation-fragmentation equation with boundary data, establishing the well-posedness of the initial value problem when the coagulation kernels are bounded at zero and showing existence of solutions for the singular kernels relevant in the applications. We determine the large time asymptotic behavior of solutions, proving that solutions converge exponentially fast to zero in the absence of fragmentation and stabilize toward an equilibrium if the boundary value satisfies a detailed balance condition. Incidentally, we obtain an improvement in the regularity of solutions by showing the finiteness of negative moments for positive time.