2202.06525
Nonlinear semigroup approach to Hamilton-Jacobi equations—A toy model
Liang Jin, Jun Yan, Kai Zhao
wronghigh confidenceCounterexample detected
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s main theorem (Theorem 1.1) asserts that the stationary Hamilton–Jacobi equation h(x, Du) + λ(x)u = c admits a solution if and only if c > c(H). However, in its own Example 3.1 the paper shows c(H) = 0 and explicitly notes that u ≡ 0 solves the equation at c = 0, i.e., at the threshold. This contradicts the stated “iff c > c(H)” claim. The model correctly identified this flaw and provided the same type of counterexample. On the other points (nonexistence for c < c(H), existence and multiplicity for c > c(H), and W^{1,∞} a priori bounds), the model’s assessment agrees with the paper’s results.
Referee report (LaTeX)
\textbf{Recommendation:} major revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The manuscript gives a useful nonlinear semigroup treatment of a toy model with sign-changing zero-order term, proving existence, multiplicity, and bounds above a critical value. However, the headline theorem is overstated: the claim that solutions exist if and only if c > c(H) conflicts with the paper’s own example exhibiting a solution at c = c(H). This central inconsistency must be corrected. With the main statement fixed and a couple of citations adjusted, the work would be suitable for a specialist audience.