{"id":"b5336cc0-d41f-49a6-867e-fd8e0a009893","arxiv_id":"2507.00959","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For a doubly nonlinear parabolic equation with a mixed local-nonlocal operator and convection, the paper establishes weak-mild well-posedness and conditions for stabilization, finite-time extinction, and blow-up.","lead":"Loïc Constantin and Carlota M. Cuesta prove existence, uniqueness, and long-time behavior for a doubly degenerate parabolic equation that mixes a p-Laplacian, a fractional q-Laplacian, a convection term, and a porous-medium nonlinearity. The paper gives a rigorous weak-mild solution framework for such mixed local-nonlocal diffusion models with external forcing, which appear in nonlocal materials science and population dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The ε-approximate solution definition (A.4) uses the wrong operator, so the proof that the time-discretized solutions are mild—hence Theorems 1.4 and 1.6—is formally invalid as written.","rationale":"The paper's central contribution is existence, uniqueness, and qualitative behavior of weak-mild solutions. The adjective 'mild' is defined through ε-approximate solutions in Definition A.4. A wrong operator in that definition directly severs the proof of the mild property: the discretized objects solve an equation with A, while the definition asks for an entirely different operator. The gap is not a matter of consensus; it is an internal inconsistency in the written argument. It is repairable by replacing the display with the correct resolvent equation for A, which is why the verdict remains CONDITIONAL rather than REJECT. The stabilization theorem's comparison-principle proof (Lemma 4.1) has a separate, genuinely flawed strict-positivity step; the asserted pointwise bound cannot hold for pairs in the interior of the set {u>v}, since γε differences vanish there. This affects Theorem 1.8 and also requires a corrected argument, but it is less central than the definitional failure of the mild-solution concept. I therefore partially agree with the reader's assessment: the reader listed Definition A.4 among the repairable issues but selected Lemma 4.1 as the weakest assumption; I would put the definitional error first. No change in verdict: conditional acceptance with requested revisions.","tokens_in":23773,"tokens_out":15911,"duration_ms":182101,"concrete_test":"Substitute U_i=β(u_i) for the sequence (u_i) constructed in Step 1 of Theorem 1.4. From display (3.1), (U_i-U_{i-1})/Δt + A(U_i)=g_i. Compare term-by-term with the equation in Definition A.4: (U_i-U_{i-1})/Δt + (-Δ)^s_p(|U_i|^{m-1}U_i)=f_i. The mismatch is explicit; replacing A by the fractional p-Laplacian porous-medium operator is required for the equality to hold. This settles that the appendix definition is erroneous and must be corrected for the mild-solution step to follow.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition A.4, which fixes the meaning of the ε-approximate solutions used in Definition 1.2, requires (U_i-U_{i-1})/(t_i-t_{i-1}) + (-Δ)^s_p(|U_i|^{m-1}U_i) = f_i. This is the fractional porous-medium operator from [10], not the operator A of (1.2) associated with problem (P0). In Step 4 of the proof of Theorem 1.4 the authors state that β(u_Δt) is an ε-approximate solution, citing Definition A.4. However, the discrete equations actually solved in Step 1 are (β(u_n)-β(u_{n-1}))/Δt + A_μ u_n = g_n + div f(u_n); writing U_n=β(u_n) converts them to (U_n-U_{n-1})/Δt + A(U_n) = g_n. The two equations match only if the operator in Definition A.4 is replaced by A. As written, the implication from time discretization to mild solution is unsupported, and the central 'weak-mild' notion in Theorems 1.4–1.6 is not rigorously established. This is a load-bearing gap in the main existence/uniqueness claim, not merely a stylistic issue. The reader's Lemma 4.1 concern is secondary but real: the pointwise lower bound on the double-integrand is false when x,y both lie in {u>v} for small ε; that gap affects Theorem 1.8 only and is repairable by testing with (u-v)^+.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the doubly nonlinear parabolic problem (P) with the mixed local-nonlocal operator A_μ = -Δ_p + μ(-Δ)_q^s, a convection term, and a source term. The authors introduce a notion of weak-mild solution, prove local and global existence under growth conditions on g, prove uniqueness and an L^1-contraction estimate under a Lipschitz-type condition, and then analyze stabilization to steady state, finite-time extinction, and finite-time blow-up. The proofs combine time discretization, accretive operator theory, comparison principles, and energy methods.","tokens_in":24008,"tokens_out":9972,"duration_ms":115107,"significance":"If the stated results are fully justified, the paper gives a useful extension of the porous-medium/fractional-p-Laplacian theory in [10] to a mixed local-nonlocal operator with convection, with explicit conditions for existence, uniqueness, stabilization, extinction, and blow-up. The work avoids fitted parameters and is built on standard monotone-operator and Sobolev tools. However, two load-bearing proof gaps at present prevent the central existence and stabilization claims from being considered established: the ε-approximate solution definition in the appendix uses the wrong operator, and the elliptic comparison principle in Lemma 4.1 relies on an unjustified pointwise lower bound. Both appear repairable, but they require genuine corrections before the main theorems can be accepted.","major_comments":[{"comment":"The definition of an ε-approximate solution uses the wrong operator. Definition A.4 requires (U_i - U_{i-1})/(t_i - t_{i-1}) + (-Δ)^s_p(|U_i|^{m-1} U_i) = f_i, which is the fractional porous-medium operator from [10], not the operator A(v) = A_μ(β^{-1}(v)) - div \\vec f(β^{-1}(v)) defined in (1.2) for problem (P0). The time-discretized equations in Step 1 of Theorem 1.4, rewritten with U_n = β(u_n), are (U_n - U_{n-1})/Δt + A(U_n) = g_n. Therefore the assertion in Step 4 that β(u_Δt) is an ε-approximate solution under Definition A.4 is not supported, and the implication from discretization to mild solution — and hence the weak-mild existence claim in Theorems 1.4–1.6 — is formally invalid as written. The natural correction is to replace the operator in Definition A.4 by A, after which the surrounding argument appears to go through, but the definition must be corrected before the proof is valid.","section":"Appendix A, Definition A.4; Step 4 of Theorem 1.4"},{"comment":"The proof of Lemma 4.1 claims that if u > v on a set K of positive measure, then the double-integrand is at least C > 0 on K. This pointwise lower bound is not justified: for x, y ∈ K, the factor γ_ε(u(x)-v(x)) - γ_ε(u(y)-v(y)) may vanish when both values lie in the same saturated regime of γ_ε, or may be arbitrarily small when the two values are close inside (0, ε). No uniform positive lower bound follows, so the displayed strict inequality ⟨A_μu - A_μv, γ_ε(u-v)⟩ ≥ C > 0 is not established. Since Lemma 4.1 underlies Corollary 4.2 and Theorem 1.8, the stabilization result is unsupported as written. The gap appears repairable by testing with (u-v)^+ and deriving positivity of the integral from the strict monotonicity of the operator rather than from a pointwise bound on the integrand.","section":"Lemma 4.1"},{"comment":"The coercivity estimate for J_w bounds the term ∫ h u by ∥h∥_{L^∞(Ω)} ∥u∥_{L^1(Ω)}, but h is assumed to belong to W*, not necessarily to L^∞. This estimate is not valid as written. Since Theorem 2.3 is invoked for h ∈ W* in Corollary 2.4 and Lemma 2.5, the abstract resolvent and density part of the accretive-operator framework has a technical gap. A standard repair is to estimate ∫ h u by ∥h∥_{W*} ∥u∥_{W} and apply Young's inequality; with that change the coercivity proof goes through.","section":"Theorem 2.3"}],"minor_comments":[{"comment":"In the displayed integrand, the denominator is written as |x-y|^{N+sp}, but for the fractional q-Laplacian the kernel should be |x-y|^{d+sq}; this appears to be a typo and should be corrected.","section":"Lemma 4.1"},{"comment":"The set on which the approximating functions g_n are required to vanish is written as '(-∞,0] ∩ [k+1/n, ∞)', which is empty; it should be the union '(-∞,0] ∪ [k+1/n, ∞)'.","section":"Theorem 1.5, Step 2"},{"comment":"Remark 3.3 invokes Barbu's theorem [4, Th. 4.1] for uniqueness of the mild solution, but the m-accretivity and density hypotheses are only verified in Corollary 2.4 under extra conditions (p > d, or μ > 0 and qs > d). Since Theorem 1.6 proves uniqueness directly via (1.4) and Gronwall's lemma, either the remark should be restricted to those extra assumptions or removed.","section":"Remark 3.3"},{"comment":"The notation f_i in Definition A.4 conflicts with the convection vector \\vec f = (f_1, ..., f_d) used throughout the paper; using a different symbol (for example h_i) for the approximate right-hand side would avoid confusion.","section":"Definition A.4"},{"comment":"There is a typo in 'discretization in tiome scheme'; it should read 'time scheme'.","section":"Remark 4.3"}],"recommendation":"major_revision","confidential_remarks":"The two main gaps identified in the report are, in my assessment, repairable within the scope of the manuscript: the ε-approximate solution definition appears to be a simple operator/subscript error, and the comparison principle likely admits a standard repair using (u-v)^+ as a test function. The paper is not circular and I see no evidence of fitted parameters or invented entities. I would encourage the editor to request a revision rather than reject, but the current version should not be accepted as is because the formal definition of the central solution concept is inconsistent with the discretization actually used."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does the obvious next step after Constantin–Giacomoni–Warnault [10]: same doubly nonlinear porous-medium framework, but with the mixed operator A_mu = -Delta_p + mu(-Delta)_q^s and a convection term, plus stabilization, extinction, and blow-up. The combination is new and the qualitative results are useful. The existence route is standard—time discretization, accretivity in L1, Barbu's mild-solution theory—and for the most part it is executed cleanly. The energy method for blow-up with convection in bounded domains is a genuinely nice addition.\n\nI would not hang the verdict on the reader's Lemma 4.1 concern alone. That proof is sloppy; the claim that the double integrand is pointwise bounded below by C>0 on a set of positive measure is not justified as written. But it looks repairable by testing with (u-v)^+ and using strict accretivity. The more serious issue is in the appendix. Definition A.4 defines an epsilon-approximate solution using (-Delta)_p^s(|U_i|^{m-1}U_i), the operator from [10], not the operator A of (1.2) associated with problem (P0). The discrete scheme actually solved in Step 1 of Theorem 1.4 becomes (U_n-U_{n-1})/Delta t + A(U_n) = g_n once U_n=beta(u_n), so the two match only with A in the definition. As written, Step 4's claim that beta(u_Delta t) is an epsilon-approximate solution is not true by the stated definition, and the weak-mild link in Theorems 1.4–1.6 is formally unsupported. This is a substantive gap, but it is almost certainly a copy-paste error from [10] rather than a hidden flaw in the argument; the intended fix is clear.\n\nMinor: Theorem 2.3's functional has a W* norm issue for h in W*, and the proof of Corollary 4.2 for mu=0 is sketched too quickly. The OCR of the arXiv version garbles some exponents, so my reading confidence is limited.\n\nVerdict: worth refereeing. The central claims are probably true, the errors are repairable, and the paper extends an active subfield. I would send it to a referee but flag the appendix definition and the comparison-principle gap clearly. Who it's for: people working on nonlocal porous-medium equations, mixed operators, or extinction/blow-up. I'd cite it after the fix; the blow-up energy method may be worth citing regardless.\n\nRecommendation: engage with it; a serious editor should not desk-reject.","headline":"A solid and useful generalization of the fractional porous-medium work [10] to a mixed local-nonlocal operator with convection, but the appendix's epsilon-approximate definition uses the wrong operator and the weak-mild link is formally unsupported as written.","tokens_in":24638,"tokens_out":2393,"would_cite":true,"duration_ms":26493,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B30","35B44","35D30","35K61","35Q86"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a doubly degenerate equation with mixed local-nonlocal diffusion and convection, the paper proves existence, uniqueness, stabilization, extinction, and blow-up.","keywords":["doubly nonlinear parabolic equation","mixed local-nonlocal operator","fractional q-Laplacian","p-Laplacian","convection term","weak-mild solution","finite-time extinction","finite-time blow-up"],"falsifier":"Construct two admissible functions $u,v$ such that $u>v$ on a set $K$ of positive measure but $u-v$ is constant on $K$, then compute the double-integral integrand in Lemma 4.1; on $K\\times K$ the integrand is identically zero, so the asserted uniform lower bound $C>0$ fails, and one would need a different argument to establish the comparison principle underpinning the stabilization claim.","tokens_in":23520,"feed_emoji":"🔥","tokens_out":11675,"duration_ms":121379,"temperature":0.7,"pith_summary":"The paper aims to give a near-complete theory for a doubly degenerate parabolic equation whose diffusion is the sum of the p-Laplacian and a fractional q-Laplacian, in the presence of a convection term and a source term. It introduces a combined solution concept, weak-mild, and proves local existence under a general growth condition, global existence under a different structural growth condition, and uniqueness plus an L1-contraction estimate under a Lipschitz-type condition on the source. For a time-independent nonnegative source and a positive nonlocal weight, it claims convergence of solutions starting below the stationary state to that stationary state in every L^γ. For power-type sources it claims finite-time extinction for small initial data and finite-time blow-up for large initial energy, with an explicit upper bound on the blow-up time. The reader should care because this covers a model combining local and long-range diffusion with convection, a setting relevant to multi-phase and fracturing materials, and the results indicate that initial size alone can decide whether a solution extinguishes, stabilizes, or explodes.","feed_headline":"Small data die out, large data blow up in one parabolic model","feed_subtitle":"For a p-Laplacian plus fractional q-Laplacian with convection, small data extinguish, large data blow up, and solutions stabilize","key_machinery":"The central object is the nonlinear operator $A v = \\mathcal{A}_\\mu(\\beta^{-1}(v)) - \\operatorname{div}(\\vec{f}(\\beta^{-1}(v)))$ on $L^1(\\Omega)$, with a domain defined through the space $W = W^{1,p}_0(\\Omega)\\cap W^{s,q}_0(\\Omega)$ when $\\mu>0$ and $W=W^{1,p}_0(\\Omega)$ when $\\mu=0$. The proof shows that $A$ is accretive in $L^1$, that its domain is dense, and that the associated elliptic resolvent problems are solvable by minimization and a fixed-point argument. The weak-mild solution concept bridges this semigroup-style operator to the variational weak formulation, and the time-discretization scheme converts parabolic existence into a sequence of elliptic problems. The proof of stabilization uses an elliptic comparison principle for $\\mathcal{A}_\\mu u - \\operatorname{div}(\\vec{f}(u))$ together with monotone convergence, while extinction and blow-up are driven by energy estimates involving the functional $E(u)=\\frac{\\mu}{q}\\|u\\|^q_{W^{s,q}_0} + \\frac1p\\|\\nabla u\\|_p^p - \\frac1{r+1}\\|u\\|_{r+1}^{r+1}$.","core_discovery":"The paper establishes a well-posedness and long-time behavior theory for the doubly degenerate problem\n\n$$\n\\partial_t \\$\\beta$(u) + \\mathcal{A}_\\mu u = \\operatorname{div}(\\vec{f}(u)) + g(t,x,u) \\quad \\text{in } Q_T, \\qquad u=0 \\text{ outside }\\$\\Omega$, \\quad u(0)=u_0,\n$$\n\nwhere $\\mathcal{A}_\\mu u = -\\Delta_p u + \\mu(-\\Delta)_q^s u$ is the sum of the $p$-Laplacian and the fractional $q$-Laplacian. The solution concept is the weak-mild solution: a weak solution in the variational sense whose image $v=\\beta(u)$ is a mild solution of the associated semigroup problem. Existence is obtained by a time-discretization scheme in which each step solves an elliptic problem for the mixed operator with convection, after which compactness arguments produce a limit; accretivity of the operator and density of its domain make the limit simultaneously mild. For odd $\\beta$ satisfying the stated growth and monotonicity conditions, local-in-time existence holds under a general growth bound on $g$, global existence under a structural growth condition, and uniqueness together with an $L^1$-contraction estimate under a Lipschitz-type condition. For $\\mu>0$ and a time-independent nonnegative source $h$, if $0\\le u_0\\le u_{\\mathrm{stat}}$, then the solution converges to the unique stationary solution in every $L^\\gamma$, $\\gamma<\\infty$. For the power model $\\beta(s)=|s|^{1/m-1}s$ and $g(u)=|u|^{r-1}u$, small initial norms give finite-time extinction when $q<r+1<1/m+1$, while sufficiently negative initial energy gives finite-time blow-up with time bound $T_* := \\tilde{c}\\,\\|u_0\\|_{1/m+1}^{1/m-r}$.","pith_inferences":["The blow-up argument is said to be new even in the local case $\\mu=0$, $\\beta=\\mathrm{Id}$, so a natural extension, not claimed by the paper, would be to use the same energy functional to derive a blow-up rate or to treat time-dependent convection $\\vec{f}(t,u)$ under the same growth hypotheses.","If the elliptic comparison principle in Lemma 4.1 can be repaired by a different argument, the stabilization theorem would likely extend to broader initial data and nonlinear sources, as the paper sketches in a remark; at present the stabilization claim depends on that comparison principle.","The small-data extinction and large-data blow-up results suggest a critical-norm dichotomy for the power-law model, though the paper does not prove that the two regimes meet at a sharp threshold; a testable question is whether a critical value of the initial norm separates extinction from blow-up.","The weak-mild framework is developed for constant $p,q\\in(2,\\infty)$, but the same accretivity-plus-discretization architecture should transfer to variable-exponent or other doubly nonlinear versions of the mixed operator whenever the elliptic problem is solvable and the operator is $L^1$-accretive."],"forward_implications":["Under the hypotheses on $\\beta$ and the growth condition $(g1)$, a weak-mild solution exists locally in time; if $g$ satisfies the structural growth condition $(g2)$ instead, the solution is global in time.","When the source satisfies the Lipschitz-type condition $(g3)$, the weak-mild solution is unique, and any two solutions obey the $L^1$-contraction estimate in terms of their initial data and source terms.","For $\\mu>0$ and a time-independent nonnegative $h$, every solution starting between $0$ and the stationary solution converges to that stationary solution in $L^\\gamma$ for every finite $\\gamma$.","For the power model $\\beta(s)=|s|^{1/m-1}s$ and $g(u)=|u|^{r-1}u$, sufficiently small $\\|u_0\\|_{1/m+1}$ gives finite-time extinction when $q<r+1<1/m+1$; under the growth condition on the convection, sufficiently negative initial energy gives finite-time blow-up with the explicit time bound $T_*$.","For a Lipschitz source with $g(0)=0$ and initial data confined to $[0,k]$, there is a global bounded weak-mild solution taking values in $[0,k]$, provided the convection and source are supported appropriately on that interval."],"supporting_citations":[{"why":"Supplies the definition of mild and ε-approximate solutions and the accretive-operator existence results on which the weak-mild framework relies.","marker":"[4]"},{"why":"Provides the integral inequality used to pass from the L1-contraction estimate to uniqueness.","marker":"[8]"},{"why":"Supplies the discretization-in-time scheme and the weak-mild notion for the fractional porous-medium problem that this paper generalizes to the mixed operator with convection.","marker":"[10]"},{"why":"Provides the fractional Sobolev embeddings and compactness used in the functional framework for $W^{s,q}_0(\\Omega)$.","marker":"[12]"},{"why":"Supplies the two-sided monotonicity estimates for the p-Laplacian used throughout the a priori estimates and convergence arguments.","marker":"[29]"},{"why":"Supplies the compactness lemma used to extract a limit from the time-discretized solutions.","marker":"[30]"},{"why":"Contains the chain-rule result used for Lipschitz compositions with Sobolev functions in the convection and test-function arguments.","marker":"[34]"}],"fun_headline_variants":["Small data extinguish, large data blow up in mixed local-nonlocal PDE","Weak-mild solutions: existence, uniqueness, and blow-up or extinction","Mixed operator PDE: solutions stabilize, extinguish, or blow up by size","Data size decides extinction vs blow-up for a doubly degenerate flow","Doubly nonlinear parabolic equation: small data die, large data burst"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The stabilization theorem rests on the elliptic comparison principle in Lemma 4.1; its proof assumes that whenever one solution exceeds another on a set of positive measure a certain double integral is bounded below by a positive constant, and that pointwise lower bound is not guaranteed, so the comparison principle and Theorem 1.8 stand or fall with that estimate.","fun_headline_variants_meta":{"raw":{"variants":["Small data extinguish, large data blow up in mixed local-nonlocal PDE","Weak-mild solutions: existence, uniqueness, and blow-up or extinction","Mixed operator PDE: solutions stabilize, extinguish, or blow up by size","Data size decides extinction vs blow-up for a doubly degenerate flow","Doubly nonlinear parabolic equation: small data die, large data burst"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001151,"raw_usage":{"total_tokens":4926,"prompt_tokens":1251,"completion_tokens":3675,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":867,"completion_tokens_details":{"reasoning_tokens":3579}},"tokens_in":867,"tokens_out":3675,"duration_ms":33576,"temperature":1.0,"reasoning_tokens":3579,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:04:49.259173+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct two admissible functions $u,v$ such that $u>v$ on a set $K$ of positive measure but $u-v$ is constant on $K$, then compute the double-integral integrand in Lemma 4.1; on $K\\times K$ the integrand is identically zero, so the asserted uniform lower bound $C>0$ fails, and one would need a different argument to establish the comparison principle underpinning the stabilization claim.","supporting_citations":[{"cited_title":"Nonlinear differential equations of monotone types in Banach spaces","cited_arxiv_id":null,"evidence_quote":"Supplies the definition of mild and ε-approximate solutions and the accretive-operator existence results on which the weak-mild framework relies."},{"cited_title":"An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications","cited_arxiv_id":null,"evidence_quote":"Provides the integral inequality used to pass from the L1-contraction estimate to uniqueness."},{"cited_title":"Existence and global behaviour of solutions of a parabolic problem involving the fractionalp-Laplacian in porous medium","cited_arxiv_id":null,"evidence_quote":"Supplies the discretization-in-time scheme and the weak-mild notion for the fractional porous-medium problem that this paper generalizes to the mixed operator with convection."},{"cited_title":"Régularité de la solution d’une équation non linéaire dansRN","cited_arxiv_id":null,"evidence_quote":"Supplies the two-sided monotonicity estimates for the p-Laplacian used throughout the a priori estimates and convergence arguments."},{"cited_title":"Compact sets in the spaceLp(0,T ;B)","cited_arxiv_id":null,"evidence_quote":"Supplies the compactness lemma used to extract a limit from the time-discretized solutions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contains the chain-rule result used for Lipschitz compositions with Sobolev functions in the convection and test-function arguments."}],"review_version":1}