{"id":"9dcd2949-2fbd-4f17-80a2-9b84c1af94b8","arxiv_id":"1908.06915","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves short-time existence, uniqueness and maximal L^q-regularity for the fractional porous medium equation on manifolds with conical singularities, and gives the decay rate of solutions near the conical tips.","lead":"This paper proves that the fractional porous medium equation has unique short-time solutions on spaces with conical singularities, with controlled behavior near the singular tips. It also develops a general method for showing R-sectoriality of fractional powers of non-invertible operators, which may apply to other nonlocal diffusion problems.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No fatal flaw identified; the proof hinges on the compactly written commutator estimate in Lemma 6.1, which should be independently checked.","rationale":"The reader's verdict of ACCEPT with moderate confidence is reasonable. The central argument is a careful chain of established cone-calculus and maximal-regularity tools, and the main new step—R-sectoriality of w(−Δ)^σ+c—is supported by the commutator lemma and a freezing argument that appear internally consistent. My independent check did not find a definite gap in the order estimates or in the parameter choices used in Theorem 6.2. The highest-risk point remains Lemma 6.1, whose proof is abbreviated and depends on several imported regularity statements from [38]; but the stated thresholds are compatible with the standard order 2σ−1 of the commutator, and the applications use the lemma in a regime consistent with those thresholds. No change to the verdict is therefore warranted, though an independent verification of Lemma 6.1 would materially strengthen confidence.","tokens_in":34878,"tokens_out":57884,"duration_ms":557107,"concrete_test":"Re-derive Lemma 6.1 in the model case A_s=c−Δ_s on R^{n+1} with a smooth compactly supported w, replacing the imported [38] lemmas with explicit Fourier multiplier estimates. Compute the symbol of [w,A_s^σ] and the norm of A_s^ρ [w,A_s^σ] A_s^{-ν} as an L^2 (or L^p) multiplier; verify the bound is finite precisely for ν>σ+η−1, ρ<η−1/2 (and for the ξ<γ+2η+1 branch, ρ<(ξ−γ)/2−1). If the admissible range is narrower, the Neumann series in equations (6.46)–(6.48) may not be summable and Theorem 1.3 would need restriction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After tracing the chain from Theorem 1.1 through Lemma 6.1 and Theorem 6.2 to Theorem 1.3, I found no concrete error that invalidates the main result. The genuinely load-bearing point is Lemma 6.1, exactly as the reader noted: the freezing-of-coefficients argument in Theorem 6.2 needs [w,(c−Δ_s)^σ] to be a bounded map from D((c−Δ_s)^ν) into D((c−Δ_s)^ρ) with the stated parameter ranges. If the order loss were larger, the Neumann-series inversion in (6.46)–(6.48) and the uniform R-bounds would fail, and with them maximal L^q regularity and Theorem 1.3. The proof of Lemma 6.1 is compressed: it invokes the first-order form of [Δ_s,w], the mapping property [Δ_s,w]A_s^{-η} from [38, Cor. 3.3], and then asserts absolute convergence of the integral for ρ<η−1/2 (or the ξ-dependent alternative). I could not exhibit a counterexample to the stated thresholds, and the order calculation ν>σ+η−1 matches the expected pseudodifferential order 2σ−1, so the concern is lack of explicit verification rather than a demonstrated contradiction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the fractional porous medium equation u'(t)+(-Delta)^sigma u^m(t)=0 on a compact manifold with isolated conical singularities. Its abstract contribution is a transfer theorem (Theorem 1.1) showing that R-sectoriality of an operator A+c0 is inherited, up to an additive shift, by the fractional power A^sigma. The concrete setup is the cone Laplacian on weighted Mellin-Sobolev spaces, for which Theorem 1.2 establishes sectoriality and describes the domain of the fractional Laplacian. The main application is Theorem 1.3: for a range of parameters, strictly positive initial data in a real interpolation space produce a unique short-time solution with maximal L^q-regularity, smoothness in time, and a power-type decay near the conical tip. The proof strategy is to linearize the equation, prove R-sectoriality for the frozen-coefficient operators w(-Delta_0)^sigma+c via a nonlocal freezing-of-coefficients method, and then apply the Clement-Li fixed-point scheme. The central analytic ingredient is Lemma 6.1, a commutator estimate for [w,(c-Delta_s)^sigma], which is used to make the Neumann-series inversion in Theorem 6.2 contractive.","tokens_in":35093,"tokens_out":21244,"duration_ms":212831,"significance":"If the proofs are correct, the paper gives the first maximal L^q-regularity and short-time well-posedness result for the fractional porous medium equation on conic manifolds, together with a concrete asymptotic rate x^{gamma+2sigma-(n+1)/2} for the Mellin-Sobolev part near the singularity. The transfer theorem for R-sectoriality of fractional powers is a useful abstract contribution in its own right. The extension of the freezing-of-coefficients method to nonlocal operators expressed as products of functions and fractional powers of local operators is a genuine methodological novelty. The paper is careful to place the result in the context of prior work on the local porous medium equation on conic manifolds and on hyperbolic space. On the other hand, the proof depends heavily on a chain of results from the same research group, especially [38, 41, 43], and the most load-bearing commutator estimate is presented in compressed form, so independent verification of that estimate is needed before the main theorem can be regarded as fully established.","major_comments":[{"comment":"Lemma 6.1 is the load-bearing commutator estimate: the freezing-of-coefficients construction in Theorem 6.2 uses it to make the Neumann series in (6.46)-(6.47) contractive and to obtain the uniform R-bounds after (6.49). The proof, however, is compressed at exactly the point where the parameter ranges are decided. The reduction to the first-order form of [Delta_s,w], the appeal to [38, Cor. 3.3] for the mapping property of [Delta_s,w]A_s^{-eta}, and the assertion that the resulting integral converges absolutely for rho<rho_0 are stated without the intermediate order-and-weight bookkeeping. Since the thresholds rho_0 = eta-1/2 or (xi-gamma)/2-1 feed directly into the invertibility of I-Q(lambda), I recommend expanding this proof, or alternatively quoting the precise statement from [38] that covers the weighted mapping into D(A_s^rho).","section":"Lemma 6.1 and Theorem 6.2"},{"comment":"The displayed condition q > sigma/(sigma+sigma_0) cannot be right as printed. Since sigma_0 is defined as max{0, ...}, the right-hand side is at most 1, while q in (1,infty), so the condition rules out nothing. To make the interval for gamma in (1.2) nonempty in the case mu_1 < (n+3)/2, one needs q > sigma/(sigma-sigma_0); this is precisely the inequality that allows gamma > (n+1)/2 + 2sigma/q - 2sigma to lie below -1+mu_1. As stated, the theorem asserts a hypothesis that is vacuous, and the proof does not supply the missing inequality. Please correct the denominator and verify that the corrected condition is used consistently in the proof.","section":"Theorem 1.3"},{"comment":"The embedding condition in (6.54), gamma+2sigma-2sigma/q > (n+2)/2, is inconsistent with the assumption in Theorem 1.3, which only gives gamma+2sigma-2sigma/q > (n+1)/2. It is also incompatible with (1.2) for n=1, where gamma<0 and (n+2)/2 = 3/2. If the intended threshold is (n+1)/2, then the statement and proof of Theorem 1.3 are consistent; as written, the proof invokes (6.54) with a hypothesis it does not have.","section":"Corollary 6.4 / (6.54)"}],"minor_comments":[{"comment":"There are several typographical errors that should be corrected before publication: 'well possed' in Section 2, 'notting' and 'enugh' in the proof of Theorem 6.2, and inconsistent spacing in displayed exponents such as 'csigma +epsilon'.","section":"Throughout"},{"comment":"The embedding arrow in (1.8) is easy to misread: the formula 'u0 in (...) <- union' appears to say the interpolation space is contained in the union, while the intended meaning is the opposite direction. Please clarify the notation.","section":"Equation (1.8)"},{"comment":"The step where the resolvent identity is extended from H^{0,gamma}_2(B) to H^{0,gamma}_p(B) by a Neumann-series continuation is clear in outline, but the role of the constants r0 and K0 could be made more explicit to help the reader follow the finite-step iteration.","section":"Section 5, proof of Theorem 5.1"}],"recommendation":"major_revision","confidential_remarks":"The core idea and the overall architecture of the proof are convincing, and I found no demonstrated contradiction in the main chain from Theorem 1.1 through Lemma 6.1 to Theorem 6.2 and Theorem 1.3. My recommendation of major revision rests on two concrete issues: the commutator estimate in Lemma 6.1 is load-bearing and needs a fuller verification, and the displayed inequalities in Theorem 1.3 and Corollary 6.4 contain apparent sign/threshold errors that affect the statement and proof of the main well-posedness theorem. If the authors can supply the expanded proof of Lemma 6.1 and correct the inequalities, I would support publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a new abstract R-sectoriality result for fractional powers of sectorial operators and uses it to establish short-time well-posedness of the fractional porous medium equation on conic manifolds, with explicit asymptotics near the singularities. That combination is new; the fractional PME on singular spaces had not been treated before. The main work is coherent and the result looks right.\n\nWhat the paper does well: Theorem 1.1 is a clean, self-contained contribution that should be reusable. The freezing-of-coefficients method for operators of the form wA^σ is a genuine extension, not just a re-run of the local argument. The chain from R-sectoriality to maximal L^q-regularity to the Clement-Li fixed-point scheme is logically sound, and the authors are explicit about the restrictions on σ and the interpolation parameters needed for the nonlinearity.\n\nThe soft spots are real but not disqualifying. The proof depends heavily on prior work by Roidos and Schrohe, especially the commutator machinery in [38]. The load-bearing Lemma 6.1 is compressed: it invokes several black-box multiplier lemmas and asserts the crucial order thresholds without full detail. If the order loss were larger, the Neumann-series inversion and uniform R-bounds would fail. The stress-test found no concrete counterexample, and the order calculations are consistent with the expected pseudodifferential order, so the concern is about verification, not a demonstrated flaw. The sharp domain formula also comes from a coauthor's preprint [30], which is acceptable but worth noting.\n\nNo circularity concern: Theorem 1.1 stands on its own, and the self-citations are the necessary foundations in a very niche area, not an echo chamber.\n\nThis paper is for researchers working on maximal regularity on singular spaces, fractional diffusion, and quasilinear parabolic equations. The main theorem is worth having, and the proof is probably right, but Lemma 6.1 should be expanded or independently checked before publication. Send it to a serious referee.","headline":"Genuine advance for fractional diffusion on conic manifolds, with a proof that hinges on a compressed commutator lemma; deserves serious peer review.","tokens_in":35638,"tokens_out":1929,"would_cite":true,"duration_ms":21917,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K59","35K65","35R01","35R11","76S05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the fractional porous medium equation on a manifold with conical singularities has a unique short-time classical solution with maximal $L^q$-regularity and an explicit power-law decay near the conical tips.","keywords":["fractional porous medium equation","conical singularities","fractional Laplacian","R-sectoriality","maximal Lq-regularity","Mellin-Sobolev spaces","freezing-of-coefficients","cone Laplacian"],"falsifier":"For the model cone of Section 4, take $w(x,y)=x^a$ and compute the norm of $[w,(c-\\Delta_\\wedge)^\\sigma]$ from $D((c-\\Delta_\\wedge)^\\nu)$ to $D((c-\\Delta_\\wedge)^\\rho)$; exhibiting one triple $\\sigma,\\nu,\\rho$ inside the ranges of Lemma 6.1 for which the commutator is unbounded, or showing the claimed $\\rho_0$ is too large, would break the Neumann-series inversion in (6.46)-(6.48) and with it Theorem 1.3.","tokens_in":34651,"feed_emoji":"📐","tokens_out":11724,"duration_ms":100439,"temperature":0.7,"pith_summary":"This paper proves that the fractional porous medium equation $u'(t)+(-\\Delta)^\\sigma u^m(t)=0$ has a unique classical solution for short time on a Riemannian manifold with isolated conical singularities, provided the fractional exponent is above a threshold set by the local geometry. It also proves a general transfer principle: the fractional power of an $R$-sectorial operator is again $R$-sectorial up to an additive constant, even when the operator is not invertible. These results matter because they bring maximal $L^q$-regularity theory to a nonlocal diffusion equation on singular spaces, where Fourier-based definitions of the fractional Laplacian are not available. A concrete payoff is the asymptotic statement that near each conical tip the Mellin-Sobolev part of the solution decays like $x^{\\gamma+2\\sigma-(n+1)/2}$, tying the rate of decay to the dimension, the fractional exponent, and the weight describing the geometry.","feed_headline":"Fractional porous medium flow has unique short-time solutions on cones","feed_subtitle":"Smooth-in-time solutions with explicit decay near every conical tip.","key_machinery":"The load-bearing mechanism is the commutator estimate of Lemma 6.1: for $w$ in a sufficiently regular Mellin-Sobolev space, the commutator $[w,(c-\\Delta_s)^\\sigma]$ maps $D((c-\\Delta_s)^\\nu)$ into $D((c-\\Delta_s)^\\rho)$ whenever $\\nu>\\sigma+\\eta-1$ and $\\rho<\\rho_0$. This makes the commutator lower-order in the fractional scale and lets the authors extend the freezing-of-coefficients method to the nonlocal operator $w(-\\Delta)^\\sigma$. The proof runs through the Dunford-type integral formula for fractional powers, $R$-bounded resolvent families, and a Neumann-series inversion for $wA^\\sigma+c+\\lambda$ that produces both a left and a right inverse with uniform $R$-bounds.","core_discovery":"The central claim is that the cone Laplacian $\\Delta_s$, realized as an unbounded operator in the weighted Mellin-Sobolev space $H^{s,\\gamma}_p(B)$ with domain $H^{s+2,\\gamma+2}_p(B)\\oplus C_\\omega$, is sectorial of angle arbitrarily close to $\\pi$. Consequently its fractional power $(-\\Delta_s)^\\sigma$ is a well-defined sectorial operator, with domain exactly $H^{s+2\\sigma,\\gamma+2\\sigma}_p(B)\\oplus C_\\omega$ when $\\gamma+2\\sigma-1$ avoids the indicial roots $\\pm\\mu_j$. For the nonlinear problem, the paper proves that for strictly positive initial data in the interpolation space $(H^{2\\sigma,\\gamma+2\\sigma}_p(B)\\oplus C_\\omega, H^{0,\\gamma}_p(B))_{1/q,q}$, there is a time $T>0$ and a unique solution in the maximal-regularity class $W^{1,q}(0,T;H^{0,\\gamma}_p(B))\\cap L^q(0,T;H^{2\\sigma,\\gamma+2\\sigma}_p(B)\\oplus C_\\omega)$, smooth in time on $(0,T)$ and continuous on $[0,T]$, with the stated power-law decay of its Mellin-Sobolev component near the singular tips.","pith_inferences":["The near-tip decay rate implies a borderline: when $\\gamma+2\\sigma-(n+1)/2>0$ the Mellin part collapses at the tip, whereas crossing zero would likely change the admissible boundary data; the paper does not single out this threshold.","The same commutator-based freezing argument could be tested on fractional powers defined through a heat semigroup, such as the fractional Laplacian on domains with cracks or edges, since the abstract Theorem 1.1 is framed for general $R$-sectorial operators.","A numerical experiment on a model cone, for example a cross-section that is a circle, could check the predicted exponent by comparing the solution's near-tip decay with $x^{\\gamma+2\\sigma-(n+1)/2}$; the paper performs no such check."],"forward_implications":["For every strictly positive initial datum in the stated interpolation space, the fractional porous medium equation has a unique short-time solution with maximal $L^q$-regularity, a property that typically unlocks further perturbation arguments.","Near each conical tip the Mellin-Sobolev part of the solution vanishes at the rate $x^{\\gamma+2\\sigma-(n+1)/2}$, so the asymptotic shape of the solution is explicitly tied to the geometry of the cross-section, the dimension, and the fractional exponent.","If the initial datum is smoother, the solution gains correspondingly higher Mellin-Sobolev regularity, and in all cases the solution is $C^\\infty$ in time on the open interval $(0,T)$.","The abstract part of the paper gives a reusable transfer statement: fractional powers of $R$-sectorial operators are $R$-sectorial up to an additive constant, so any existing $R$-sectoriality result for a linear operator automatically yields maximal regularity for its fractional powers."],"supporting_citations":[{"why":"Supplies the abstract quasilinear existence theorem used to obtain the short-time solution in Theorem 1.3.","marker":"[4]"},{"why":"Provides the transference principle that upgrades an $H^\\infty$-calculus bound to R-sectoriality in the proof of Theorem 6.3.","marker":"[5]"},{"why":"Gives the product theorem for non-commuting operators used to establish the bounded $H^\\infty$-calculus for $A^\\sigma B$.","marker":"[11]"},{"why":"Gives the result that R-sectorial operators of angle greater than $\\pi/2$ have maximal $L^q$-regularity on UMD spaces.","marker":"[23]"},{"why":"Supplies the perturbation theorem used to turn small commutator norms into R-sectoriality of $wA^\\sigma+c$.","marker":"[27]"},{"why":"Provides the R-sectoriality and resolvent-pole analysis for the cone Laplacian realization, the starting point for fractional powers.","marker":"[37]"},{"why":"The predecessor paper on the porous medium equation on conic manifolds supplies the multiplication, interpolation, and resolvent lemmas reused in Section 6.","marker":"[38]"},{"why":"Supplies the bounded $H^\\infty$-calculus for cone differential operators used in the higher-regularity R-sectoriality argument.","marker":"[41]"},{"why":"Supplies the functional calculus formulas and resolvent decay estimates that define the fractional powers and control the commutator integrals.","marker":"[50]"}],"fun_headline_variants":["Cone singularities tamed in fractional porous medium flow","Unique flows on conical manifolds for fractional diffusion","Fractional porous medium: regularity and decay near cone tips","Cone Laplacian powers yield well-posed porous medium equations","Maximal regularity for fractional porous medium on cones"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction relies on the commutator estimate in Lemma 6.1: swapping a coefficient function past the fractional Laplacian must cost strictly less than one derivative, because if that estimate were false or had a larger order loss, the Neumann-series inversion in Theorem 6.2 would not close and Theorem 1.3 would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Cone singularities tamed in fractional porous medium flow","Unique flows on conical manifolds for fractional diffusion","Fractional porous medium: regularity and decay near cone tips","Cone Laplacian powers yield well-posed porous medium equations","Maximal regularity for fractional porous medium on cones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000585,"raw_usage":{"total_tokens":2751,"prompt_tokens":946,"completion_tokens":1805,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":562,"completion_tokens_details":{"reasoning_tokens":1726}},"tokens_in":562,"tokens_out":1805,"duration_ms":13045,"temperature":1.0,"reasoning_tokens":1726,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:32:05.372146+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the model cone of Section 4, take $w(x,y)=x^a$ and compute the norm of $[w,(c-\\Delta_\\wedge)^\\sigma]$ from $D((c-\\Delta_\\wedge)^\\nu)$ to $D((c-\\Delta_\\wedge)^\\rho)$; exhibiting one triple $\\sigma,\\nu,\\rho$ inside the ranges of Lemma 6.1 for which the commutator is unbounded, or showing the claimed $\\rho_0$ is too large, would break the Neumann-series inversion in (6.46)-(6.48) and with it Theorem 1.3.","supporting_citations":[{"cited_title":"Cl´ ement, S","cited_arxiv_id":null,"evidence_quote":"Supplies the abstract quasilinear existence theorem used to obtain the short-time solution in Theorem 1.3."},{"cited_title":"Cl´ ement, J","cited_arxiv_id":null,"evidence_quote":"Provides the transference principle that upgrades an $H^\\infty$-calculus bound to R-sectoriality in the proof of Theorem 6.3."},{"cited_title":"Haller-Dintelmann, M","cited_arxiv_id":null,"evidence_quote":"Gives the product theorem for non-commuting operators used to establish the bounded $H^\\infty$-calculus for $A^\\sigma B$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the result that R-sectorial operators of angle greater than $\\pi/2$ have maximal $L^q$-regularity on UMD spaces."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the perturbation theorem used to turn small commutator norms into R-sectoriality of $wA^\\sigma+c$."},{"cited_title":"Roidos, E","cited_arxiv_id":null,"evidence_quote":"Provides the R-sectoriality and resolvent-pole analysis for the cone Laplacian realization, the starting point for fractional powers."},{"cited_title":"Roidos, E","cited_arxiv_id":null,"evidence_quote":"The predecessor paper on the porous medium equation on conic manifolds supplies the multiplication, interpolation, and resolvent lemmas reused in Section 6."},{"cited_title":"Schrohe, J","cited_arxiv_id":null,"evidence_quote":"Supplies the bounded $H^\\infty$-calculus for cone differential operators used in the higher-regularity R-sectoriality argument."}],"review_version":1}