{"id":"360999b5-a747-462e-a0a7-1f17ac3123c6","arxiv_id":"2505.08304","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Global weak solutions with explicit L∞ decay are shown to exist for u_t = Δ_p u^m + u^q on complete non-compact manifolds with Sobolev (and possibly Poincaré) inequalities, for small initial data and q above a critical exponent.","lead":"This paper proves that a doubly nonlinear porous medium equation with a power reaction term admits global-in-time solutions on certain infinite-volume curved spaces, provided the initial data are small and the reaction exponent lies above a critical threshold. The result extends Fujita-type critical exponent theory from Euclidean space and from special diffusion cases to general Riemannian manifolds satisfying Sobolev or Poincaré inequalities.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.3's local L∞ estimate (2.4) does not follow from the R→∞ limit and is false for data supported outside B_R: finite-speed propagation makes the solution positive inside B_R at large times while the right-hand side is zero.","rationale":"I read the paper as aiming to prove global existence and explicit L∞ decay for a doubly nonlinear reaction-diffusion equation under Sobolev and, optionally, Poincaré inequalities. The energy estimates and Moser iteration in Sections 3 and 4 appear broadly coherent, and I do not see an obvious fatal error in Theorem 2.2 or in the existence part of Theorem 2.3. The most load-bearing problem is the local L∞ estimate (2.4) in Theorem 2.3. The proof passes to the limit R→∞ in Dirichlet problems on balls; for a fixed ball B_R, the limiting solution is obtained from solutions on larger balls that see initial data outside B_R, so the right-hand side is controlled by the global L^s norm, not by ∥u0∥_{L^s(B_R)}. Moreover, because m(p−1)>1 in Theorem 2.3, finite propagation speed implies that data initially outside B_R will eventually influence B_R, so the local estimate with a zero local initial norm cannot hold for all t. The reader's weakest_assumption identified the Sobolev/Poincaré inequalities as the fragile premise; I disagree, because the proof's approximation argument fails in a concrete, checkable way even when those inequalities hold. The existence result may be salvageable by replacing the local norm in (2.4) with the global L^s norm, so I recommend conditional acceptance only if that statement is corrected; if the authors insist on the estimate as written, the paper should not be accepted in its present form.","tokens_in":31690,"tokens_out":18787,"duration_ms":194944,"concrete_test":"Check the claim on H^N: set p=2, m=2, q=3, N≥3, choose R=1 and let u0=εφ where φ∈C_c^∞(M) is supported in B_3\\B_1, with ε small enough that ∥u0∥_{L^s} and ∥u0∥_{L^{qN/p}} are below the threshold ε1 of Theorem 2.3. By the finite propagation speed result of [14] for m(p−1)>1, the support of the weak solution expands at finite speed; take t larger than the crossing time, so there exists x∈B_1 with u(x,t)>0. Then (2.4) with R=1 and r>s gives 0<∥u(t)∥_{L∞(B_1)} ≤ Γ t^{-β}·0=0, a contradiction. Independently, re-examine the last paragraph of the proof of Theorem 2.3 and write the R'→∞ and then R→∞ limits separately; the right-hand side after the limit is ∥u0∥_{L^s(M)}, not ∥u0∥_{L^s(B_R)}.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 5 proves Theorem 2.3 by applying Proposition 5.3 to Dirichlet approximants u_R^{h,k} on B_R and then passing R→∞. Proposition 5.3 is a bound for the Dirichlet solution on B_R in terms of the initial norm over the same ball B_R, which is natural because the approximating problem only sees the data on B_R. However, the limiting global solution at a point x∈B_R is the increasing limit of u_{R'}^{h,k}(x,t) for R'>R, and u_{R'} is driven by data on B_{R'}, including mass outside B_R. The estimate for u_{R'} therefore controls the limit only through the global norm ∥u0∥_{L^s(M)}, not through the local norm ∥u0∥_{L^s(B_R)}. This is not a harmless strengthening. For example, take M=H^N, p=2, m=2, q=3 and a small datum supported in an annulus outside B_R. Since m(p−1)=2>1, the equation has finite propagation speed (see [14]), so for sufficiently large t the support reaches B_R and ∥u(t)∥_{L∞(B_R)}>0, while the right-hand side of (2.4) is identically zero because u0≡0 on B_R. Thus (2.4) is false as stated. The proof of Theorem 2.3 at best yields an estimate with the global L^s norm on the right; the local version would require an additional argument, for instance a proper cut-off with time-dependent radii, which is absent. The existence part may be repairable, but the theorem as stated is not supported.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies global-in-time existence for the doubly nonlinear reaction-diffusion equation u_t = Delta_p u^m + u^q on complete non-compact Riemannian manifolds of infinite volume. Under a Sobolev inequality on M and a smallness condition on the initial datum in L^s ∩ L^1, Theorem 2.2 claims global existence for q > m(p-1) + p/N together with an explicit L^1-to-L^∞ decay estimate. Under the additional Poincaré inequality, Theorem 2.3 claims global existence for the wider range q > m(p-1) and a local L^∞ estimate on geodesic balls. The proofs use Caccioppoli estimates, Moser-type iteration, bootstrap arguments involving S(t) = sup τ ||u(τ)||_{L∞}^{q-1}, and passage to the limit in Dirichlet approximations.","tokens_in":1361,"tokens_out":1513,"duration_ms":109886,"significance":"If the main results were valid, the paper would give a useful extension of known Fujita-type global existence results from the porous medium and p-Laplace cases to the doubly nonlinear Leibenson equation with reaction, with explicit decay rates. The proof strategy is standard and the estimates are explicit; the authors also address the sharpness of the exponent in Theorem 2.2 via a cited non-existence result. However, the local estimate stated in Theorem 2.3 is false as written, and this materially weakens the paper's contribution. The existence part may be salvageable with a global-norm estimate, but the advertised local statement is not supported by the proof.","major_comments":[{"comment":"The local L∞ estimate (2.4) does not follow from the Dirichlet-approximation argument and is, in fact, false in the slow-diffusion regime. In the proof, the approximating solution u^{R'}_{h,k} on B_{R'} is bounded in terms of the initial norm over B_{R'}; after R' → ∞ this controls a global L^s norm, not the local norm ||u0||_{L^s(B_R)}. More concretely, take M = H^N with N > 2, p = 2, m = 2, q = 3, so m(p-1) = 2 > 1, and choose a small nonnegative initial datum supported outside B_R. The equation then has finite propagation speed [14], so for sufficiently large t the solution is positive on B_R, while the right-hand side of (2.4) is zero because u0 vanishes on B_R. Thus (2.4) is false as stated. The theorem can be repaired by replacing the local L^s norm with the global L^s(M) norm, or by giving a genuinely local argument with moving cut-offs; as written, the assertion is not supported.","section":"Theorem 2.3, Eq. (2.4), §5"},{"comment":"The passage to the limit i → ∞ in inequality (3.25) discards the term ε^i J_i, but no uniform bound for J_i is supplied. Since J_i involves gradients of G_{k_i}(u) over time intervals that expand as i grows, the reader needs an explicit argument (for example, using boundedness of u and the choice of ε small enough) to justify that ε^i J_i → 0. The final estimate (3.27) depends on this step, so it should be proved rather than assumed.","section":"Lemma 3.6, Eq. (3.25)"}],"minor_comments":[{"comment":"The displayed identity in (3.36) omits the time derivative and the inequality sign; as written it is not a valid evolution identity. The intended differential inequality should be stated explicitly, and the same issue affects the derivation of (3.39).","section":"Lemma 3.7, Eq. (3.36)"},{"comment":"Proposition 4.2 is proved for the regularized problem (3.2), which contains the term εΔu, but it is then applied to the approximating problem (4.26), which does not contain that term. Since ε enters only through nonnegative terms in the estimates, the results likely extend to ε = 0, but this should be stated explicitly rather than left implicit.","section":"Proof of Theorem 2.2, Eq. (4.26)"},{"comment":"There are several typographical/indexing errors: the symbol ¯m appears undefined in 'for any 1 ≤ n ≤ ¯m', a spurious ρ appears in exponents such as ||u(·,t_n)||_{L^{s_n}ρ(B_R)}^{s_n}, and some constants such as C_2^n are typeset ambiguously. These should be corrected throughout Section 3.2.","section":"Proposition 3.9 and surrounding text"},{"comment":"The estimate (2.3) is stated for all t > 0, but the right-hand side is singular as t → 0; this is standard for smoothing estimates, but the authors should specify that the constant c is independent of t and that the bound is meaningful for t bounded away from zero.","section":"Theorem 2.2, Eq. (2.3)"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the false local estimate in Theorem 2.3. The existence part of the theorem may well be repairable by replacing the local norm with a global norm, but the current statement is incorrect and needs a substantive revision. I do not see other issues that would require rejection if the authors are willing to correct the theorem and fill the small technical gaps indicated above."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it closes a real parameter gap: global existence for the doubly nonlinear reaction-diffusion equation u_t = Δ_p u^m + u^q on non-compact manifolds, covering the combined range m>1, p>1, which had been open for the Sobolev-only and Sobolev+Poincaré settings. Second, the local smoothing estimate in Theorem 2.3, display (2.4), is not supported by the proof and, as written, is probably false.\n\nWhat the paper does well: the strategy is standard for this area—Caccioppoli estimates, Moser iteration, bootstrap, approximation on geodesic balls—and it is executed carefully for the main existence result. Theorem 2.2, which gives global existence and the decay estimate (2.3) under the Sobolev inequality alone, looks sound. The smallness conditions are existential but that is normal here. The literature is cited sensibly, and the authors correctly note that the Poincaré case has no Euclidean analogue.\n\nThe soft spot is Theorem 2.3. Proposition 5.3 is proved for the Dirichlet problem on a fixed ball B_R, with the initial norm over that same ball. The proof of Theorem 2.3 then passes R→∞ and h→∞. But the limit object is the global solution, and the global solution on B_R is not controlled by the Dirichlet solution on B_R: when m(p−1)>1 the equation has finite propagation speed, so mass initially outside B_R can enter B_R at large times. A datum with u0≡0 on B_R but nonzero outside would give a solution that is positive on B_R for large t while the right-hand side of (2.4) is zero. The estimate is therefore false as stated. The existence part of Theorem 2.3 can probably be repaired by replacing the local L^s(B_R) norm with the global L^s(M) norm in (2.4), but that is not what the theorem claims. There are also smaller typos in displayed equations, for example in Lemma 3.7's energy inequality, but those are cosmetic.\n\nWho is this for: researchers working on Fujita-type phenomena for quasilinear parabolic equations on manifolds. The main existence result is worth knowing, but the overclaimed local estimate means the paper needs a serious revision before it can be cited. A referee should ask the authors to either prove a genuine local estimate with time-dependent cut-offs or change Theorem 2.3 to a global-norm statement. I would send it to review, but not accept it in this form.","headline":"The global-existence results for the doubly nonlinear equation are a natural and likely correct extension of the literature, but Theorem 2.3's local L∞ estimate (2.4) is overclaimed and appears false as stated because of finite-speed propagation from data outside B_R.","tokens_in":32637,"tokens_out":4977,"would_cite":false,"duration_ms":55285,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K57","35B44","58J35","35K59","35K65","35R01"],"pacs":[],"model":"deepseek-v4-flash","headline":"Global-in-time weak solutions exist for $u_t=\\Delta_p u^m+u^q$ on noncompact Riemannian manifolds under a Sobolev inequality when $q>m(p-1)+p/N$, and for all $q>m(p-1)$ when a Poincaré inequality is added, provided the initial datum is…","keywords":["Leibenson equation","doubly nonlinear parabolic equation","Riemannian manifolds","global existence","Sobolev inequality","Poincaré inequality","smoothing estimates","Moser iteration"],"falsifier":"Take a complete noncompact infinite-volume manifold satisfying (1.3) and choose $q=m(p-1)+p/N$ with arbitrarily small nonzero compactly supported initial data: if any such datum blows up in finite time, the asserted sharpness is false. Alternatively, on a manifold satisfying both (1.3) and (1.4), solve (1.1) with $q>m(p-1)$ and a datum meeting the smallness condition, and check whether the local decay bound (2.4) holds for all $t>0$; a violation would refute Theorem 2.3.","tokens_in":31478,"feed_emoji":"🌐","tokens_out":9491,"duration_ms":88722,"temperature":0.7,"pith_summary":"This paper proves global-in-time existence for the doubly nonlinear reaction-diffusion equation $u_t=\\Delta_p u^m+u^q$ posed on a complete, noncompact Riemannian manifold of infinite volume. The aim is to show that small nonnegative initial data do not lead to finite-time blow-up, despite the superlinear reaction term $u^q$. Under only a global Sobolev inequality on the manifold, the authors establish existence for every $T>0$ when $q>m(p-1)+p/N$, together with an explicit decay of the $L^\\infty$ norm in time. If the manifold additionally satisfies a Poincaré inequality, the same conclusion holds for the whole range $q>m(p-1)$, a situation with no Euclidean analogue because the manifold is noncompact and has infinite measure.","feed_headline":"Small data prevent blow-up on curved manifolds","feed_subtitle":"Doubly nonlinear diffusion with reaction stays global when the manifold has the right inequalities.","key_machinery":"The argument is carried by Caccioppoli-type energy estimates followed by a Moser iteration on truncated level sets, with truncation $G_k(u)=u-T_k(u)$ where $T_k$ is the standard cut-off at height $k$. The Sobolev inequality (1.3) is used at each iteration step to convert an $L^r$ control of the truncated solution into a gradient bound and then a higher-power $L^s$ bound, yielding a local smoothing estimate on cylinders (Lemma 3.6). A bootstrap on the auxiliary quantities $S(t)=\\sup_{0<\\tau<t}\\tau\\|u(\\tau)\\|_{L^\\infty}^{q-1}$, $F(t)$, and $M(t)$ shows that smallness of the initial datum keeps $S(t)\\le 1$ for all times, which turns the local estimate into a global bound and allows passage to the limit in approximating problems. In the Poincaré-inequality case, an extra $L^{s_0}\\to L^s$ decay estimate (Proposition 3.9) is what removes the $p/N$ shift from the critical exponent.","core_discovery":"The central discovery is that the critical growth exponent for global existence is lowered by the geometry of the manifold. On a complete noncompact infinite-volume manifold supporting the Sobolev inequality, problem (1.1) with $1<p<N$, $m(p-1)\\ge 1$, $m>1$ and $q>m(p-1)+p/N$ admits a weak solution for any $T>0$ whenever the initial datum lies in $L^s\\cap L^{1+m}\\cap L^1$ and is sufficiently small in $L^s$ and $L^1$; the solution satisfies the uniform bound $\\|u(t)\\|_{L^\\infty}\\le c\\,(\\|u_0\\|_{L^1}^p/t^N)^{1/[N(m(p-1)-1)+p]}$ for all $t>0$. When a Poincaré inequality is added, the same type of result holds for every $q>m(p-1)>1$, with the local decay estimate (2.4). The authors also state that the Sobolev-only exponent is sharp: below it, nonexistence holds for every initial datum, as shown by the cited result [48].","pith_inferences":["Inference beyond the paper: if the same geometric mechanism is what lowers the critical exponent, then on manifolds with a spectral gap the Fujita-type threshold should be governed by the bottom of the $L^2$ spectrum, and not by the Euclidean dimension alone.","Inference beyond the paper: a concrete test would be to solve (1.1) numerically on hyperbolic space with $q>m(p-1)$ and small data and check whether the decay in (2.4) is already visible at moderate times; the theorems predict no blow-up for this whole range.","Inference beyond the paper: the method suggests that the existence result should extend to reaction terms $f(u)$ growing at most like $u^q$ for the same $q$, provided the same smallness assumptions are imposed; the proofs here use only the power bound $f(u)\\le C u^q$."],"forward_implications":["For any manifold satisfying (1.3), small data in $L^s\\cap L^1$ with $s>[q-m(p-1)]N/p$ produce a weak solution defined for all $T>0$, so the reaction term $u^q$ is kept under control for the full time interval.","The explicit bound (2.3) gives a quantitative long-time decay rate: the supremum norm of the solution vanishes at least like $t^{-N/[N(m(p-1)-1)+p]}$, with the $L^1$ norm of the data as the only datum-dependent factor.","Adding the Poincaré inequality removes the $p/N$ shift, so the threshold for global existence becomes $q>m(p-1)$; any superlinear power $u^q$ is then admissible for small data.","The Sobolev-only threshold is sharp in the sense that $q=m(p-1)+p/N$ is the critical value; the paper cites nonexistence for all data below it, so the theorem sits exactly on the boundary."],"supporting_citations":[{"why":"Establishes nonexistence of global solutions for q below the critical exponent, which is the sharpness reference for Theorem 2.2.","marker":"[48]"},{"why":"Proves the p=2 porous-medium-with-reaction case on manifolds using the Sobolev inequality, the base case this paper extends.","marker":"[17]"},{"why":"Proves the m=1 p-Laplacian case and supplies the L^1-to-L^infty smoothing scheme used here.","marker":"[20]"},{"why":"Supplies the Caccioppoli-type estimates and Moser iteration for degenerate parabolic equations with source on which Lemmas 3.3 and 3.4 are built.","marker":"[30]"},{"why":"Provides the parabolic iteration lemma and regularization tools invoked in the Moser argument and in Remark 3.2.","marker":"[28]"},{"why":"Supports the validity of the Poincaré inequality on Cartan-Hadamard manifolds with sectional curvature bounded away from zero, the geometric assumption behind Theorem 2.3.","marker":"[12]"}],"fun_headline_variants":["Geometry cuts blow-up threshold on manifolds","Small data, big geometry: global solutions on manifolds","Manifold inequalities rewrite blow-up conditions","Noncompact infinite-volume spaces prevent blow-up","Critical exponent for existence shrinks on manifolds"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument depends on the global Sobolev inequality (1.3) being valid on the noncompact manifold; if it fails, every Caccioppoli estimate and the Moser iteration collapse, and in the wider-exponent theorem the global Poincaré inequality (1.4) is equally load-bearing.","fun_headline_variants_meta":{"raw":{"variants":["Geometry cuts blow-up threshold on manifolds","Small data, big geometry: global solutions on manifolds","Manifold inequalities rewrite blow-up conditions","Noncompact infinite-volume spaces prevent blow-up","Critical exponent for existence shrinks on manifolds"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001253,"raw_usage":{"total_tokens":5163,"prompt_tokens":1002,"completion_tokens":4161,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":4091}},"tokens_in":618,"tokens_out":4161,"duration_ms":29645,"temperature":1.0,"reasoning_tokens":4091,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:58:08.612629+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a complete noncompact infinite-volume manifold satisfying (1.3) and choose $q=m(p-1)+p/N$ with arbitrarily small nonzero compactly supported initial data: if any such datum blows up in finite time, the asserted sharpness is false. Alternatively, on a manifold satisfying both (1.3) and (1.4), solve (1.1) with $q>m(p-1)$ and a datum meeting the smallness condition, and check whether the local decay bound (2.4) holds for all $t>0$; a violation would refute Theorem 2.3.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes nonexistence of global solutions for q below the critical exponent, which is the sharpness reference for Theorem 2.2."},{"cited_title":"Meglioli, F","cited_arxiv_id":null,"evidence_quote":"Proves the p=2 porous-medium-with-reaction case on manifolds using the Sobolev inequality, the base case this paper extends."},{"cited_title":"Meglioli, F","cited_arxiv_id":null,"evidence_quote":"Proves the m=1 p-Laplacian case and supplies the L^1-to-L^infty smoothing scheme used here."},{"cited_title":"Martynenko, A","cited_arxiv_id":null,"evidence_quote":"Supplies the Caccioppoli-type estimates and Moser iteration for degenerate parabolic equations with source on which Lemmas 3.3 and 3.4 are built."},{"cited_title":"Ladyzhenskaya, V.A","cited_arxiv_id":null,"evidence_quote":"Provides the parabolic iteration lemma and regularization tools invoked in the Moser argument and in Remark 3.2."},{"cited_title":"Grigor’yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds , Bull","cited_arxiv_id":null,"evidence_quote":"Supports the validity of the Poincaré inequality on Cartan-Hadamard manifolds with sectional curvature bounded away from zero, the geometric assumption behind Theorem 2.3."}],"review_version":1}