Minimal Travelling Wave Speed and Explicit Solutions in Monostable Reaction-Diffusion Equations

The paper rigorously proves minimality exchange for solvable monostable equations using the Hadeler–Rothe variational principle and the Lucia–Muratov–Novaga integrability criteria, deriving cl=2√(A−B) and cnl=A/√B and establishing exchange at A=2B, with precise orientations under L=1 and under monotonicity hypotheses (Theorems 5 and 8) . The model’s solution reaches the same conclusions via a phase–plane/graph-ODE comparison (a calibrated no-crossing inequality) and an eigen-slope analysis; its A<2B nonexistence proof for c<cnl is crisp and correct, while the A>2B part relies on standard continuation/transversality heuristics rather than the paper’s variational bound but still arrives at cl as the minimal speed under the stated assumptions. Net: same results, different methods; the paper’s argument is fully rigorous, while the model’s is essentially correct but less formal on the A>2B side.