{"id":"355b9c47-b7f4-4c87-8ff8-4108a50083cb","arxiv_id":"2411.14260","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For the porous fractional p-Laplacian problem with power source, the authors prove existence and uniqueness of weak-mild solutions and characterize global stabilization, finite-time extinction, and blow-up.","lead":"This paper proves existence, uniqueness, and long-term behavior of solutions to a fractional version of the porous medium equation, where the diffusion operator is the fractional p-Laplacian and the density nonlinearity is a power. It shows that, depending on parameters, solutions can stabilize to a steady state, disappear in finite time, or grow without bound.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.9 is false as stated: with h ≡ -1 and v0 ≡ 0 the discrete first step forces v1 < 0, so no nonnegative global solution exists; a sign hypothesis on h is missing.","rationale":"The reader's weakest assumption was the imported boundary regularity supplying v∞ ≥ c d^s. That is a plausible secondary concern, but the more decisive and internally checkable problem is the missing sign condition in Theorem 1.9. The paper's own Remark 3.2 states that nonnegativity of the discretized solution requires h and v0 nonnegative, yet Theorem 1.9 omits this requirement and its proof asserts nonnegativity without a supporting argument. The counterexample h ≡ -1, v0 ≡ 0 shows the theorem cannot be true as written: the first discrete step has a negative right-hand side, and comparison forces the first iterate to be negative. This is not a matter of an imported theorem being hard to verify; it is a concrete failure of a stated main result under hypotheses that are explicitly allowed. The theorem is used in the proof of Theorem 1.12 only through the homogeneous source ⌈v⌉^{q/m}, which is nonnegative on the nonnegative cone, so the stabilization result may still be correct after the statement of Theorem 1.9 is corrected. For that reason a conditional verdict is appropriate: the paper needs a revised hypothesis in Theorem 1.9 (and any other statement asserting global nonnegative solutions for general h), and a correspondingly adjusted proof. The boundary-regularity concern identified by the reader should also be clarified, but it is not the single most load-bearing issue; the sign gap is immediate, falsifiable, and located inside the paper's own argument.","tokens_in":25798,"tokens_out":45765,"duration_ms":425053,"concrete_test":"Run the paper's time-discretization (Section 3.1) with any bounded smooth domain Ω, s ∈ (0,1), p = 2, m = 2, h ≡ -1, v0 ≡ 0, T > 0, and Δt = T/N. The first discrete step requires β(v1) + Δt (-Δ)^s_2 v1 = -Δt. Comparing with w = 0 via Proposition 2.2 gives v1 ≤ 0 and v1 ≠ 0, so the discrete solution is negative after one step and cannot converge to a nonnegative weak-mild solution. This directly falsifies Theorem 1.9 as stated. Repeating the test with h(θ) = ⌈θ⌉^{q/m}, the nonnegative case used in Theorem 1.12, would show that the missing sign hypothesis is exactly what separates the false statement from the valid homogeneous application.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3.4, Theorem 1.9 claims that if p > q/m + 1 and 0 ≤ v0 ≤ C d(·,∂Ω)^s, then (Ph) has a nonnegative ∞-weak-mild solution for any h satisfying (Hq). No sign condition on h is imposed. The proof begins: 'By Theorem 3.1, for R > 0 and T > 0, there exists a nonnegative weak solution v of (Ph,R).' But Theorem 3.1 proves only existence of a weak solution; Remark 3.2 explicitly states that nonnegativity requires h and v0 to be nonnegative. The missing hypothesis is not cosmetic: take h(t,x,θ) ≡ -1 and v0 ≡ 0. The first elliptic step of the discretization in Theorem 3.1 is β(v1) + Δt (-Δ)^s_p v1 = -Δt. Since the zero function solves the same equation with right-hand side 0, the comparison principle (Proposition 2.2 with g = β) gives v1 ≤ 0; equality is impossible because the right-hand side is -Δt ≠ 0. Hence v1 is negative on a set of positive measure, so the approximate solution is already negative after one time step, and no nonnegative weak-mild solution can exist. The homogeneous case h(θ) = ⌈θ⌉^{q/m} used in Theorem 1.12 is nonnegative on the nonnegative cone, so Theorem 1.12 may survive, but Theorem 1.9 as stated is false. The fix is to add a hypothesis such as h(t,x,θ) ≥ 0 for θ ≥ 0 (or h(t,x,0) ≥ 0 with monotonicity).","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the doubly nonlinear parabolic problem with fractional p-Laplacian and porous-medium structure, ∂_t β(v)+(-Δ)_p^s v = h(t,x,v), where β(v)=⌈v⌉^{1/m}. It establishes local and global existence of weak-mild solutions, an L1 contraction property, uniqueness under a local Lipschitz condition, and several qualitative properties for the power-type source h(v)=⌈v⌉^{q/m}: global existence and stabilization to a nontrivial stationary solution in the sub-homogeneous regime p>q/m+1, and finite-time extinction or blow-up in the opposite regime. The proofs combine time discretization, accretive operator theory, comparison principles, and energy estimates.","tokens_in":26112,"tokens_out":21207,"duration_ms":178176,"significance":"If the results are correct, this is a substantial contribution to the theory of fractional porous-medium-type equations with sources. The paper provides a fairly complete existence-uniqueness framework, including L1 contraction and energy inequalities, and gives new stabilization and blow-up/extinction results for this class. The proofs are detailed and largely self-contained, with explicit comparison arguments and accretivity proofs. The reliance on existing boundary regularity results is clearly indicated, and the qualitative conclusions (stationary state with two-sided distance-power behavior, convergence in all L^r) are natural and genuinely new for the fractional p-Laplacian porous-medium setting. However, the false statement of Theorem 1.9 and the incomplete verification of the hypotheses in Proposition 3.4 are load-bearing issues that require correction.","major_comments":[{"comment":"Theorem 1.9 is false as stated because it does not impose any sign condition on h. The proof begins 'By Theorem 3.1, for R>0 and T>0, there exists a nonnegative weak solution v of (Ph,R),' but Theorem 3.1 only gives existence of a weak solution, and Remark 3.2 explicitly states that nonnegativity requires h and v0 to be nonnegative. This is not a mere proof gap: take h≡-1 and v0≡0. The first step of the discretization in Theorem 3.1 is β(v1)+Δt(-Δ)_p^s v1 = -Δt. Applying Proposition 2.2 with g=β and comparing to the zero solution gives v1≤0, and equality is impossible because the right-hand side is nonzero, so v1 is negative on a set of positive measure. Thus no nonnegative approximate solution exists for this data, so no nonnegative weak-mild solution exists. The theorem needs an additional hypothesis such as h(t,x,θ)≥0 for θ≥0 (or h(t,x,0)≥0 together with monotonicity of h in θ). The homogeneous power case h(θ)=⌈θ⌉^{q/m} is nonnegative on the nonnegative cone, so Theorem 1.12 may survive, but Theorem 1.9 as stated is false.","section":"§3.4, Theorem 1.9"},{"comment":"The proof of uniqueness of the weak-mild solution invokes [2, Th. 4.1] after establishing only accretivity of A (Theorem A.2) and density of D(A) (Corollary 1.7). The cited theorem typically requires a range condition, e.g., R(I+λA)=L1(Ω) (or at least the appropriate full-range condition for the Crandall-Liggett generation theorem). The manuscript verifies the resolvent equation only for f∈L∞ via Theorem 2.3, so R(I+λA) contains a dense subset of L1, not necessarily all of L1. Since uniqueness of mild solutions actually follows from accretivity alone, the conclusion is likely correct, but the proof as written relies on an unverified hypothesis. This affects Theorem 1.8(iii) and the semigroup identities used in the proof of Theorem 1.12 (Step 3). Please either verify the full range condition (e.g., by an approximation argument from L∞ data) or replace the citation by a direct uniqueness argument based on the L1 contraction inequality.","section":"§3.2, Proposition 3.4"}],"minor_comments":[{"comment":"The test function φ=⌈β(v_n)⌉^{r-1} is claimed to be admissible for 'some r≥m'. For r∈[m,m+1), the map t↦sign(t)|t|^{(r-1)/m} is not Lipschitz, and φ need not belong to W^{s,p}_0(Ω). Since the argument allows any r≥1, the proof should take r≥m+1 (where the map is Lipschitz) or use a truncation/approximation argument.","section":"§3.1, Step 2 of Theorem 3.1"},{"comment":"The notation v0 is used simultaneously for the initial datum and for the overlined/underlined sub-supersolutions of (Qstat). Please use ̲{v}_0 and ̄{v}_0 (or another convention) to avoid ambiguity.","section":"§4.1.2, Theorem 1.12 (Step 1)"},{"comment":"In the displayed inequality (3.8), the left-hand side reads ‖β(u)-β(u)‖_{L1(Ω)}; this should be ‖β(u)-β(v)‖_{L1(Ω)}.","section":"§3.2, Proposition 3.6"},{"comment":"The abstract contains a repeated word: 'We also study further the the homogeneous case' should read 'We also study further the homogeneous case'.","section":"§1, Abstract"},{"comment":"The displayed inequality 'M′_r(t) ≤ M′_r(t) + c/2 M^α_r(t) ≤ C M^γ_r(t) − c/2 M^α_r(t)' contains a redundant term; it should be 'M′_r(t) + c/2 M^α_r(t) ≤ C M^γ_r(t) − c/2 M^α_r(t)'.","section":"§4.2.1, equation (4.3)"},{"comment":"In the final computation, the notation switches between Z(t) and Z(T) inconsistently; the argument should use a single symbol for the L∞(Q_T) norm and track the dependence on T carefully.","section":"§4.2.2, Case 3 of Theorem 1.14"},{"comment":"The two-sided boundary estimate c d(·,∂Ω)^s ≤ w ≤ C d(·,∂Ω)^s is imported verbatim from [10, Th. 1.5] and [13, Th. 2.7]. Please confirm that the hypotheses of those theorems (regularity of Ω, sign of the right-hand side, and the precise notion of solution) are satisfied in the present setting, since this estimate is load-bearing for the ordering in Theorem 1.12 and the construction in Theorem 1.9.","section":"§3.4 and §4.1.1"}],"recommendation":"major_revision","confidential_remarks":"The skeptic's counterexample is valid and directly contradicts Theorem 1.9; this must be fixed before publication. The range-condition gap in Proposition 3.4 is also genuine, though likely repairable. The remaining results appear sound and the paper makes a solid contribution to the field, so I encourage a major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the bad news: Theorem 1.9 is false as stated. The stress-test note is right. The proof says \"By Theorem 3.1, for R > 0 and T > 0, there exists a nonnegative weak solution v of (Ph,R),\" but Theorem 3.1 only gives existence, and Remark 3.2 explicitly says nonnegativity requires h and v0 nonnegative. With h(t,x,θ) ≡ -1 and v0 ≡ 0, the first discrete step solves (1/Δt)β(v1) + (-Δ)_p^s v1 = -1, while zero solves the same equation with right-hand side 0. Comparison gives v1 ≤ 0, and equality is impossible, so v1 is negative on a set of positive measure. No nonnegative ∞-weak-mild solution exists. The missing hypothesis is a sign condition like h(t,x,θ) ≥ 0 for θ ≥ 0.\n\nNow the good news. The paper does something new for m > 1 with non-Lipschitz source h: existence of weak-mild solutions via L1-accretivity of A: u ↦ (-Δ)_p^s ⌈u⌉^m, plus density and comparison; uniqueness via a Gronwall argument under the local Lipschitz condition (H); and a stabilization/extinction/blow-up trichotomy for the power case. The time-discretization proofs are detailed and mostly convincing. The L1 contraction property (3.8) and the energy inequality (3.10) look correct. The boundary-decay estimates are imported from [10] and [13] rather than reproved; that is acceptable if the authors state them as standing assumptions, but it should be explicit.\n\nOther soft spots: Proposition 3.4 invokes [2, Th. 4.1] after showing only accretivity and density, without verifying the range condition. The reader says Theorem 3.9 gives an independent uniqueness route, so this is minor. In the stabilization proof, the lower bound on the stationary solution v∞ ≥ c d(·,∂Ω)^s is load-bearing for the sub-supersolution ordering; if that fails, the convergence argument breaks. Again, that is imported from the literature.\n\nWho is this for? Specialists in nonlinear nonlocal parabolic equations. The methods are worth knowing, but the paper needs a fix before it can be used. I would send it to peer review, but only after the authors add the sign hypothesis and clarify the boundary-regularity imports. The power-case theorems likely survive, so the paper is salvageable with a revision.","headline":"Theorem 1.9 is false as stated (missing sign condition on h), but the paper's core methods are solid and the power-case results are likely salvageable with a small fix.","tokens_in":26694,"tokens_out":4436,"would_cite":false,"duration_ms":34679,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R11","35K55","35B40","47H06","35A01"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves existence, uniqueness, and long-time stabilization for a porous-medium fractional p-Laplacian parabolic problem, with convergence to a unique nontrivial stationary solution in the sub-homogeneous regime.","keywords":["fractional p-Laplacian","porous medium equation","weak-mild solution","global existence","stabilization","finite-time extinction","blow-up","comparison principle"],"falsifier":"Compute the unique positive solution of $(-\\Delta)_p^s v = v^{q/m}$ on a bounded smooth domain with zero exterior values, for parameters $p > q/m+1$, and measure the ratio $v(x)/d(x,\\partial\\Omega)^s$ near the boundary; if the ratio tends to $0$ or $\\infty$, the assumed two-sided boundary decay is false and the stabilization theorem's ordering argument is void.","tokens_in":25590,"feed_emoji":"🌊","tokens_out":11153,"duration_ms":103103,"temperature":0.7,"pith_summary":"The paper studies a nonlinear parabolic equation whose diffusion term is the fractional $p$-Laplacian applied to a porous-medium nonlinearity, $\\partial_t u + (-\\Delta)_p^s(|u|^{m-1}u) = h$. Rewriting it in terms of $v = |u|^{m-1}u$, the authors prove that a weak-mild solution exists for bounded initial data, is unique under a local Lipschitz condition on $h$, and is global in time when the source grows at most linearly. For the power source $h(v) = |v|^{q/m}$, global bounded solutions exist in the sub-homogeneous range $p > q/m + 1$, and when the initial datum is trapped between constant multiples of $d(\\cdot,\\partial\\Omega)^s$, every such solution converges in every $L^r$ to the unique nontrivial stationary solution. In the complementary range $p < q/m + 1$, small initial data extinguish in finite time for $q \\le 1$, while negative-energy data blow up or grow without bound depending on $q$. These results matter because they extend existence, uniqueness, and qualitative long-time behavior from the fractional Laplacian and nonlocal porous-medium cases to the doubly nonlinear fractional $p$-Laplacian with sources.","feed_headline":"Fractional porous flows stabilize to a unique steady state","feed_subtitle":"When the source is sub-homogeneous, fractional porous flows converge to a single steady state","key_machinery":"The engine is the operator $A: u \\mapsto (-\\Delta)_p^s(\\lceil u \\rceil^m)$, shown to be accretive in $L^1(\\Omega)$ with dense domain, which places the problem inside the abstract theory of mild solutions. Around this, the paper constructs a time-discretization scheme in which each step solves an elliptic problem $\\beta(v_n) + \\Delta t\\,(-\\Delta)_p^s v_n = \\text{data}$ with an $L^\\infty$ right-hand side, whose solution exists, is H\\\"older continuous, and obeys a comparison principle. The quantitative input is two-sided boundary decay: elliptic solutions with constant or sub-homogeneous right-hand sides, and the stationary solution $v_\\infty$, satisfy $c\\,d(x,\\partial\\Omega)^s \\le v \\le C\\,d(x,\\partial\\Omega)^s$, which lets a sub-supersolution method trap the evolution between monotone barriers that converge to $v_\\infty$; semigroup arguments then identify the limits. A pointwise energy identity and an energy inequality govern the super-homogeneous regime, producing finite-time extinction or blow-up. A weak-mild solution is a function that is a weak solution in the variational sense and whose $\\beta(v)$ is a mild solution in the semigroup sense.","core_discovery":"On its own terms, the paper establishes that the auxiliary problem $\\partial_t \\beta(v) + (-\\Delta)_p^s v = h(t,x,v)$ admits a $T$-weak-mild solution for any $L^\\infty \\cap W^{s,p}_0$ initial datum when $h$ satisfies a polynomial growth bound, that this solution is unique when $h$ is locally Lipschitz with respect to $\\beta(v)$, and that for $q \\le 1$ the solution exists for all time. For the model source $h(v) = \\lceil v \\rceil^{q/m}$, it proves a dichotomy governed by the sign of $p - (q/m+1)$: when $p > q/m+1$ there is a global nonnegative solution that stabilizes to the unique nontrivial stationary solution $v_\\infty$, with convergence in every $L^r$; when $p < q/m+1$, small initial data lead to finite-time extinction for $q \\le 1$, and initial data with nonpositive energy lead to finite-time blow-up for $q > 1$ or to unbounded growth as $t \\to \\infty$ for $q \\le 1$.","pith_inferences":["The boundary condition on the initial datum in the stabilization result, equivalence to $d(\\cdot,\\partial\\Omega)^s$, is likely an artifact of the available boundary-regularity tools; a finer boundary theory would probably widen the admissible class. The paper does not claim this.","The critical case $p = q/m+1$ is left open; by analogy with classical porous-medium results one would expect a borderline dichotomy sensitive to $m$ and the integrability of the data. This is an editorial expectation, not a paper claim.","The same monotone-barrier construction should extend to non-autonomous sources that are asymptotically sub-homogeneous in $v$, yielding convergence to a possibly time-dependent profile; the paper proves only the autonomous case.","A numerical check of the predicted stabilization is feasible: solve the evolution problem with $v_0 = \\lambda v_\\infty$ for $\\lambda > 1$ and monitor $\\|v(t) - v_\\infty\\|_{L^r}$; the paper proves convergence but gives no rate, so any observed rate would be new information."],"forward_implications":["If the theorems hold, the doubly nonlinear fractional porous-medium equation with a sub-homogeneous source has a global weak-mild solution for any bounded initial datum lying between multiples of $d(\\cdot,\\partial\\Omega)^s$, and that solution forgets its initial data, converging in every $L^r$ to the unique stationary state.","Uniqueness for sources satisfying the local Lipschitz condition means the time-discrete scheme used in the proof converges to a well-defined solution rather than to a spurious limit.","For $q \\le 1$, global existence holds for all bounded initial data, so no finite-time blow-up can occur in that range regardless of data size.","In the super-homogeneous range, the energy threshold $E(v_0) \\le 0$ is sufficient to force finite-time blow-up for $q > 1$ and unbounded growth for $q \\le 1$; a small $L^{r+1/m}$ norm forces finite-time extinction when $q \\le 1$.","The comparison principle gives monotone dependence on initial data for global solutions, so solutions inherit the ordering of their data for all time."],"supporting_citations":[{"why":"Supplies the abstract accretive-operator and mild-solution theory that defines the weak-mild solution and proves $L^1$-uniqueness.","marker":"[2]"},{"why":"Provides the Gronwall lemma and the time-differentiation identity used in the uniqueness and energy arguments.","marker":"[6]"},{"why":"Supplies the Hopf-type lemma and strong maximum principle giving the lower bound $v \\ge c\\,d(x,\\partial\\Omega)^s$ for fractional $p$-Laplacian solutions.","marker":"[10]"},{"why":"Gives the doubly nonlinear model and the comparison and boundedness arguments adapted here to $m>1$ with power sources.","marker":"[12]"},{"why":"Supplies the fine boundary regularity and upper H\\\"older bounds used to control solutions and the stationary state near $\\partial\\Omega$.","marker":"[13]"},{"why":"Provides the algebraic inequalities in Property B.4 used in the energy and extinction estimates.","marker":"[19]"}],"fun_headline_variants":["Fractional porous flows: unique stationary states and dichotomy","Porous p-fractional flows: global solutions or finite-time blow-up","Stabilization and blow-up dichotomy in fractional porous media","Unique steady states and finite-time blow-up in porous flows","Dichotomy in fractional porous media flows: extinction vs blow-up"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stabilization result rests on the imported boundary-regularity fact that both the elliptic building blocks and the stationary state $v_\\infty$ are bounded above and below by constant multiples of $d(x,\\partial\\Omega)^s$ near the boundary; if that two-sided decay fails, the monotone barrier construction in the stabilization theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Fractional porous flows: unique stationary states and dichotomy","Porous p-fractional flows: global solutions or finite-time blow-up","Stabilization and blow-up dichotomy in fractional porous media","Unique steady states and finite-time blow-up in porous flows","Dichotomy in fractional porous media flows: extinction vs blow-up"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000753,"raw_usage":{"total_tokens":3359,"prompt_tokens":961,"completion_tokens":2398,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":577,"completion_tokens_details":{"reasoning_tokens":2312}},"tokens_in":577,"tokens_out":2398,"duration_ms":16110,"temperature":1.0,"reasoning_tokens":2312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:23:59.096836+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the unique positive solution of $(-\\Delta)_p^s v = v^{q/m}$ on a bounded smooth domain with zero exterior values, for parameters $p > q/m+1$, and measure the ratio $v(x)/d(x,\\partial\\Omega)^s$ near the boundary; if the ratio tends to $0$ or $\\infty$, the assumed two-sided boundary decay is false and the stabilization theorem's ordering argument is void.","supporting_citations":[{"cited_title":"Nonlinear differential equations of monotone types in Banach spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the abstract accretive-operator and mild-solution theory that defines the weak-mild solution and proves $L^1$-uniqueness."},{"cited_title":"An introduction to semilinear evolution equations , volume 13","cited_arxiv_id":null,"evidence_quote":"Provides the Gronwall lemma and the time-differentiation identity used in the uniqueness and energy arguments."},{"cited_title":"A hopf's lemma and a strong minimum principle for the fractional p-laplacian","cited_arxiv_id":null,"evidence_quote":"Supplies the Hopf-type lemma and strong maximum principle giving the lower bound $v \\ge c\\,d(x,\\partial\\Omega)^s$ for fractional $p$-Laplacian solutions."},{"cited_title":"Existence and global behavior of weak solutions to a doubly nonlinear evolution fractional p - L aplacian equation","cited_arxiv_id":null,"evidence_quote":"Gives the doubly nonlinear model and the comparison and boundedness arguments adapted here to $m>1$ with power sources."},{"cited_title":"Fine boundary regularity for the singular fractional p-laplacian","cited_arxiv_id":null,"evidence_quote":"Supplies the fine boundary regularity and upper H\\\"older bounds used to control solutions and the stationary state near $\\partial\\Omega$."},{"cited_title":"R \\'e gularit \\'e de la solution d’une \\'e quation non lin \\'e aire dans R ^n","cited_arxiv_id":null,"evidence_quote":"Provides the algebraic inequalities in Property B.4 used in the energy and extinction estimates."}],"review_version":1}