Diffeomorphic shape evolution coupled with a reaction-diffusion PDE on a growth potential

The paper states and proves existence and uniqueness for the coupled PDE–ODE system (13)–(14) under V ↪ Cm+1_0, Lipschitz and bounded R,Q, Lipschitz dependence of A_φ, and a uniform bound on the frame field, culminating in Theorem 3; the proof proceeds via (i) well-posedness of the reaction–diffusion problem on a prescribed moving domain (Theorem 4), (ii) continuity bounds for p_φ and the yank j_φ, and (iii) a contraction mapping Γ_η on diffeomorphisms, followed by continuation to [0,T] . The candidate solution follows the same blueprint: Lax–Milgram for v, flow existence and Jacobian control, fixed-φ parabolic well-posedness, Lipschitz estimates, a small-time contraction (on the product space (φ,p) rather than on φ alone), and continuation. These steps match the paper’s lemmas/propositions (e.g., Lemma 2/Corollary 1/Theorem 4 for the parabolic part; Proposition 2/3 and Lemma 5/6 for Lipschitz estimates; Section 4 for the fixed-point/continuation) and yield the same conclusion . Minor differences are not substantive: the paper sometimes inserts a cutoff χ in j during estimates, while the model omits it; the paper packages the contraction on φ via Γ_η, whereas the model does it on (φ,p); and the paper handles coercivity by adding λ (Lions–Magenes) rather than invoking a uniform ellipticity phrasing for J_φ S_φ. None of these affect correctness.