Mass conserving global solutions for the nonlinear collision-induced fragmentation model with a singular kernel

The paper proves global existence of mass-conserving solutions for collision-induced fragmentation with singular kernels, and uniqueness for the pure singular rate (ν=0), via truncation, contraction on weighted spaces, Arzelà–Ascoli compactness, and a Laplace/weighted-moment stability argument. These arguments are broadly sound under the stated growth/singularity assumptions on C and F, but two modeling assumptions crucial in the proofs are not stated in the hypotheses: (i) the mass-preserving identity for F, namely ∫_0^y x F(x,y|z) dx = y, used to prove mass conservation and moment bounds; and (ii) the use of a uniform bound on the fragment number ∫_0^y F(x,y|z) dx ≤ N in bounding N_{n,0}, which is not implied by H4 (F ≤ k2 y^{−β}) and, if intended, should be stated explicitly. See the model definition (1.1)–(1.2) and the use of ∫_0^y xF = y in the mass identity M1=M2 and subsequent finiteness checks (lines around (3.81)–(3.83)) , and the estimate for N_{n,0} using “N−1” (3.49)–(3.51) . The candidate solution gets existence via uniform-in-truncation bounds on an exponential moment and a negative moment and then compactness. This approach is plausible, but it makes a critical, incorrect monotonicity claim about negative-moment weights: it asserts that if g0 ∈ Ω_{.,r0} with r0 ≤ σ, one may choose r* > σ and still have g0 ∈ Ω_{.,r*} because x^{-r*} ≤ x^{-r0} on (0,1); the inequality is reversed for 0<x<1, so integrability at order r0 does not imply integrability at the stronger order r*>r0. Hence the claimed bootstrapping in the small-x regime is invalid. The paper’s uniqueness for ν=0 uses a weight exp(λx)+x^{−θ} with θ+σ<1 (and θ ≤ r, inherited from the initial data space) , whereas the candidate imposes the stricter r>σ condition and closes by Grönwall—correct under stronger data, but less general. Overall: the paper needs to state all kernel constraints it uses; the candidate solution contains a substantive error on small-size integrability and over-claims its scope.