{"id":"178ade46-184d-4da3-b7aa-beb2ffc94f81","arxiv_id":"1908.02489","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"For any initial data, a weakly mixing incompressible flow suppresses finite-time blow-up in the generalized Keller-Segel system with fractional dissipation 0<α<2 and β in [2,d], giving a global H^3 solution.","lead":"This mathematics paper proves that a wisely chosen, strongly mixing fluid flow prevents the density blow-up that normally occurs in a generalized Keller-Segel chemotaxis model, for every fractional diffusion strength between 0 and 2 and attraction exponent from 2 to the dimension. Generalists should care because it sharpens the theoretical boundary between ambient stirring and aggregation singularities in biological and physical transport models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The global L∞ estimate in Prop. 3.4 requires every iterate (ρ(jτ)−ρ̄)/‖ρ(jτ)−ρ̄‖_{L2} to lie in the RAGE compact set K; this is asserted only for j=0 and never proved for j≥1, so the repeated contraction leading to (3.51) is not justified.","rationale":"The reader's weakest assumption is exactly the load-bearing point: the RAGE compactness condition is applied to normalized fluctuations whose L2 denominator is being contracted, and no argument shows that the normalized H^{α/2} norm stays in the fixed set K. This is not a stylistic gap; it is the step that converts the time-averaged dissipation lower bound (3.49) into the repeated L2 contraction (3.51), which in turn produces the small Lp state (3.52) and the global L∞ estimate (3.56). Without membership of every iterate in K, Lemma 3.5 cannot be applied at j=1,2,...,k−1, and the global bound does not follow. The initial membership assertion is also not implied by (3.39), though it is patchable by choosing N larger. The iterated membership problem is more serious but still plausibly repairable, e.g. by choosing N from the stopping level B1 and the a priori H^3 bound on the local interval, or by replacing the repeated RAGE application with a different mechanism once the Lp norm is small. Because the gap is localized to one iteration step and the surrounding estimates are coherent, the appropriate verdict remains CONDITIONAL rather than REJECT or ACCEPT. The paper is also transparent about the unhandled range d<β<d+1, which supports the conditional posture.","tokens_in":23533,"tokens_out":17617,"duration_ms":208285,"concrete_test":"Write out the induction for the block iteration in Prop. 3.4. Assume φ0∈K and the L2 contraction (3.50)-(3.51) holds on [0,τ]; derive the sharpest available bound on ‖φ1‖_{H^{α/2}} from Lemma 3.3, Lemma 3.5, the local H^3 bound, and the equation on [0,τ]. Then test whether λ_N chosen by (3.39), supplemented by an explicit condition such as λ_N^{α/2} ≥ C(H3,B1,α), guarantees φ1∈K. In particular, check the case where the contraction factor brings D1 close to or below B1; if φ1∉K for parameters allowed by (3.39), the RAGE step at j=1 fails and the proof of (3.51) collapses. This is a purely analytical verification of one line of the manuscript, with no numerics needed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.5 (RAGE) is invoked in (3.42) with φ0=(ρ0−ρ̄)/‖ρ0−ρ̄‖_{L2}∈K, but the 'repeat the above process k times' step after (3.51) re-applies the lemma at t=τ,2τ,...,(k−1)τ. For the j-th step one needs φj=(ρ(jτ)−ρ̄)/‖ρ(jτ)−ρ̄‖_{L2}∈K. The sentence 'Let (ρ0−ρ̄)/‖ρ0−ρ̄‖_{L2}∈K' is an assertion, and condition (3.39) does not imply it; enlarging N would repair j=0, but not the later iterates. Interpolation gives ‖φj‖_{H^{α/2}} ≤ C‖ρ(jτ)−ρ̄‖_{H^3}^{α/6} ‖ρ(jτ)−ρ̄‖_{L2}^{−α/6}, and the denominator is being contracted by the L2 estimate (3.49)-(3.51). The proof supplies no bound showing this ratio remains below λ_N^{α/2}; the H^3 control from Prop. 3.1 depends on the very L∞ bound being proved. Since the contraction is iterated to cover all n∈Z+, the L2 deviation can fall below any fixed positive level, so no fixed finite N can keep all φj in K. Proposition 4.4 inherits the gap verbatim ('According to the proof of Proposition 3.4').","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the generalized parabolic-elliptic Keller-Segel system on the torus with fractional dissipation (−Δ)^{α/2}, 0<α<2, β∈[2,d], d≥2, and advection by an incompressible flow. The main result, Theorem 1.1, asserts that for every nonnegative initial datum in H^3∩L^∞ there exists a smooth incompressible flow u such that the unique solution of (1.1) is global and belongs to C(R_+;H^3). The proof strategy is to first establish an L^∞-criterion (Proposition 3.1), then use the RAGE theorem for weakly mixing flows to obtain a local L^2 contraction of the deviation from the mean, iterate this contraction to make the L^2 deviation small, and finally use a nonlinear maximum principle on the torus to convert the resulting L^p control into a global L^∞ bound. The case β=d is treated in Section 3 and the case β∈[2,d) in Section 4.","tokens_in":23723,"tokens_out":22433,"duration_ms":229691,"significance":"If the argument were correct, the result would be a substantial improvement over previous work: it would extend mixing-induced suppression of blow-up for the generalized Keller-Segel system to the full range 0<α<2 and β∈[2,d], removing restrictions such as α>d/2 or the restriction to classical diffusion. The overall strategy, combining the RAGE theorem with the nonlinear maximum principle, is natural and attractive, and the appendix gives a self-contained proof of the nonlinear maximum principle on the torus. The main claim is clearly stated and falsifiable, and the paper does not rely on hidden fitting or ad-hoc numerical assumptions. However, the proof of the central global L^∞ estimate contains a load-bearing gap in the iteration of the mixing argument, so the theorem as stated is not established by the present manuscript.","major_comments":[{"comment":"The iteration of the RAGE-based contraction is not justified. Lemma 3.5 is applied in (3.42) to the initial normalized fluctuation φ0=(ρ0−ρ̄)/‖ρ0−ρ̄‖_{L2}, and the membership φ0∈K is asserted immediately before (3.42). Even for j=0 this membership is not a consequence of the choice of N in (3.39); it is an additional condition that could be met by enlarging N, but the manuscript does not state this. More seriously, the 'repeat the above process k times' step after (3.51) requires φj=(ρA(jτ)−ρ̄)/‖ρA(jτ)−ρ̄‖_{L2}∈K for j=1,...,k−1. No argument for this membership is supplied. After the L2 contraction (3.49)–(3.51), the denominator ‖ρA(jτ)−ρ̄‖_{L2} shrinks, while the numerator is only known through an H^3 bound that itself depends on the L^∞ bound being proved; interpolation gives no uniform control of ‖φj‖_{\\dot H^{α/2}} below λ_N^{α/2}. Since the contraction is iterated until ‖ρA(kτ)−ρ̄‖_{L2}≤B1, no fixed finite N can keep all iterates in the fixed compact set K. This gap is load-bearing: the global L^∞ estimate and hence Theorem 1.1 rest on it.","section":"§3.2, Proposition 3.4, equations (3.39)–(3.51)"},{"comment":"The passage from the single-time estimate (3.52) to the uniform L^p bound (3.53) is not justified as written. Inequality (3.52) gives ‖ρA(kτ)−ρ̄‖_{Lp}≤D only at the one time t=kτ, while the maximum-principle differential inequality (3.54) is integrated over the whole interval 0≤t≤kτ and therefore requires a bound on ‖ρA(t)‖_{Lp} for every t in that interval. The cited 'Theorem 2.6 and (3.52)' does not provide this: Theorem 2.6 is only a local well-posedness statement. On the first block one could obtain (3.53) from (3.41) via interpolation, but the text does not say this; for later blocks, where the same issue recurs in the sentence 'by the same argument with above', the missing uniform L^p or L^∞ control is exactly the quantity being proved. The final assertion that the same argument applies to the solution of (1.1) for all n∈Z+ is therefore unsupported.","section":"§3.2, equations (3.52)–(3.56) and final paragraph of Proposition 3.4"},{"comment":"Proposition 4.4 is proved by the one-sentence statement 'According to the proof of Proposition 3.4.' It therefore inherits verbatim the two gaps identified above. In addition, the equation (4.15) for the maximum contains the constant C0 multiplying the quadratic term, so the local time-scale estimates in Lemma 4.2 and the repeated contraction need to be rechecked with the modified constants; no such verification is given. Consequently, the proof of Theorem 1.1 for β∈[2,d) is not completed.","section":"§4.2, Proposition 4.4"}],"minor_comments":[{"comment":"The displayed condition for choosing N is notationally unclear: expressions such as '2400/23(C∞+ρ̄)' and '(1−B1^2/(B0^2−ρ̄^2))^{1/τ1}' should be written as explicit inequalities for λ_N^{α/2}, and the third condition should be λ_N^{α/2} ≥ (2/τ1)ln((B0^2−ρ̄^2)/B1^2).","section":"§3.2, (3.39)"},{"comment":"The sentence 'Let (ρ0−ρ̄)/‖ρ0−ρ̄‖_{L2} ∈ K' should be stated as an additional condition on N and included in the selection of N; as written it appears as an assertion that is not implied by (3.39).","section":"§3.2, before (3.42)"},{"comment":"In the estimate of the term ρ̄∫(ρ−ρ̄)ΔK∗ρ dx, the displayed bound loses a factor of ‖ρ‖_{L∞}: one expects C0ρ̄‖ρ‖_{L∞}(‖ρ‖_{L∞}+ρ̄) rather than C0ρ̄(‖ρ‖_{L∞}+ρ̄). This affects the source term in (4.18).","section":"§4.2, Lemma 4.2, (4.17)"},{"comment":"The proof is carried out for the rescaled equation (3.37) with velocity Au, while Theorem 1.1 is stated for (1.1) with velocity u. Since the theorem allows choosing u, this can be repaired by declaring the final flow to be A times a fixed weakly mixing flow and noting that the weakly mixing property is preserved under time rescaling, but the passage is not explained.","section":"§3.2, final paragraph of Proposition 3.4"},{"comment":"There are numerous typographical errors, including 'location solution' for 'local solution', 'we definite' for 'we define', 'downward rectification' for 'floor', and 'Combing' for 'Combining'. These should be corrected in revision.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The intended strategy is coherent and the claimed range of parameters would be valuable if established, but the current proof of the central L^∞ estimate is incomplete: the RAGE-based iteration is only justified at the initial time, and the step from a contracted state to a global L^∞ bound is not supplied. A revision would need a new argument that either maintains the normalized fluctuations in a fixed compact set, or replaces the fixed-N iteration with an adaptive mechanism, and that provides the uniform L^p control needed for the nonlinear maximum principle on each interval. Without such an addition, Theorem 1.1 is not supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, this is a genuine extension: for beta in [2,d], d>=2, they handle every alpha in (0,2), where Kiselev-Xu and Hopf-Rodrigo needed alpha=2 or alpha>max{beta-d/2,1}. If the proof is fixed, the result closes the parameter range for mixing suppression on the torus. Second, the proof as written has a load-bearing gap in the iteration in Prop 3.4: the RAGE lemma requires the normalized fluctuation to stay in a fixed compact set K, but the paper only asserts this for the initial datum and never checks it for the later iterates. This gap is real, but patchable.\n\nThe architecture is coherent: local L2 and L∞ estimates, comparison with a passive scalar, RAGE time-averaging, then the nonlinear maximum principle to upgrade Lp control to L∞. The torus version of the nonlinear maximum principle is a nice tool and is used correctly. The paper is also honest about leaving d<beta<d+1 open, and the citations are to the right prior work; there is no hidden fitting or self-citation.\n\nThe main soft spot is the K-membership issue. It can be fixed: choose N first so that the normalized initial datum is in K, then also choose N so that lambda_N^{alpha/2} dominates C_H3^{alpha/6} B1^{-alpha/6}, using the uniform H^3 bound on the contraction interval. That uniform H^3 bound follows from the local L∞ bound and Proposition 3.1, so it is not circular. But the manuscript does none of this; it simply asserts membership. The stress-test note overstates the difficulty: no fixed N can keep all iterates in K if you contract forever, but the proof only contracts down to the positive level B1, so a finite N can work. Still, the check has to be written.\n\nSmaller issues: the expression for tau1 in Lemma 4.2 looks like it can be non-positive for some parameter ranges, probably benign but needs a fix. The Appendix's WLOG M>=1/4 is actually fine if you use translation invariance on the torus, but the proof should say that. And the beta in [2,d) section is largely a one-sentence reduction to Prop 3.4; a referee will want the iteration written out there too.\n\nWho is this for? Specialists in parabolic-elliptic aggregation and mixing-enhanced dissipation. The central claim is likely correct, but the current draft is not rigorous enough as written. I would send it to a serious referee with the expectation of major revision, not desk reject.","headline":"Real parameter-range extension with a repairable but real gap in the heart of the proof; worth refereeing with expectation of major revision.","tokens_in":24570,"tokens_out":9099,"would_cite":true,"duration_ms":95039,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35B45","35R11","35Q92"],"pacs":[],"model":"deepseek-v4-flash","headline":"Mixing by a weakly mixing flow suppresses blow-up in the generalized Keller-Segel equation for every fractional diffusion order $0<\\alpha<2$ in all dimensions $d\\ge2$.","keywords":["Keller-Segel equations","fractional dissipation","mixing","relaxation enhancing flows","RAGE theorem","nonlinear maximum principle","blow-up suppression","global classical solution"],"falsifier":"A concrete check is to test whether (3.39) implies $\\|(\\rho_0-\\bar\\rho)/\\|\\rho_0-\\bar\\rho\\|_{L^2}\\|_{\\dot H^{\\alpha/2}}^2\\le\\lambda_N^{\\alpha/2}$; if some admissible initial datum satisfies (3.39) but not this inequality, then (3.43) fails and the contraction (3.49)--(3.51), on which the global $L^\\infty$ bound rests, is unsupported. A numerical iteration of the substep map would also reveal whether the normalized $\\dot H^{\\alpha/2}$ norm stays below $\\lambda_N^{\\alpha/2}$ at each step.","tokens_in":23106,"feed_emoji":"🌊","tokens_out":11361,"duration_ms":103784,"temperature":0.7,"pith_summary":"The paper claims that finite-time blow-up, known to occur for large-data solutions of the unadvected generalized Keller-Segel equation with fractional dissipation, is suppressed completely when a suitable incompressible flow is added. For every fractional diffusion order $0<\\alpha<2$, every $\\beta\\in[2,d]$, and every dimension $d\\ge2$, the authors construct a smooth, weakly mixing divergence-free flow $u$ such that the advected equation (1.1) has a unique global-in-time classical solution for every non-negative initial datum in $H^3(\\mathbb{T}^d)\\cap L^\\infty(\\mathbb{T}^d)$. This covers parameter ranges inaccessible to earlier suppression results, which required stronger dissipation such as $\\alpha>d/2$ or $\\alpha=2$. The proof couples a nonlinear maximum principle on the torus with the relaxation-enhancing effect of weak mixing, which forces the $L^2$ deviation of the density from its mean to contract over short time substeps.","feed_headline":"Mixing kills blow-up in Keller-Segel for every fractional exponent","feed_subtitle":"A suitable incompressible flow turns large-data finite-time blow-up into a global classical solution.","key_machinery":"Four components carry the argument. (i) The weakly mixing incompressible flow $u$, a divergence-free vector field whose transport group $U^t$ has purely continuous spectrum; this is the mixing agent. (ii) The RAGE theorem (Lemma 3.5), which states that for any compact set $K$ of unit vectors in $L^2$ and any finite-rank projection $P_N$ onto the first $N$ eigenfunctions of $(-\\Delta)^{\\alpha/2}$, the long-time average $T^{-1}\\int_0^T\\|P_N U^t\\varphi\\|_{L^2}^2\\,dt$ is uniformly small over $\\varphi\\in K$; this gives the quantitative enhancement of dissipation by mixing. (iii) The nonlinear maximum principle on the torus (Lemma 5.1), which at a spatial maximum $x_t$ of $\\rho$ yields either $\\rho(t,x_t)\\le C\\|\\rho\\|_{L^p}$ or the pointwise lower bound $(-\\Delta)^{\\alpha/2}\\rho(t,x_t)\\ge C\\rho(t,x_t)^{1+p\\alpha/d}\\|\\rho\\|_{L^p}^{-p\\alpha/d}$, reducing the maximum evolution to a differential inequality. (iv) The $L^\\infty$-criterion (Propositions 3.1 and 4.1), which upgrades a uniform $L^\\infty$ bound to an $H^3$ bound through a Gagliardo-Nirenberg interpolation energy estimate. The Proposition 3.4 iteration chooses a large coupling constant $A$ so that each substep of duration $\\tau=T_c/A$ contracts $\\|\\rho-\\bar\\rho\\|_{L^2}$ by the fixed factor $1-\\lambda_N^{\\alpha/2}\\tau/2$.","core_discovery":"Theorem 1.1 states that for $0<\\alpha<2$, $\\beta\\in[2,d]$, $d\\ge2$, and any non-negative $\\rho_0\\in H^3(\\mathbb{T}^d)\\cap L^\\infty(\\mathbb{T}^d)$, there exists a smooth incompressible flow $u$—weakly mixing in the sense that its transport operator has purely continuous spectrum—such that the unique solution $\\rho$ of (1.1) is global in time and lies in $C(\\mathbb{R}_+;H^3(\\mathbb{T}^d))$. The discovery is that mixing supplies the missing dissipation at every scale: the RAGE time-averaging bound shrinks the $L^2$ fluctuation $\\|\\rho-\\bar\\rho\\|_{L^2}$ over intervals of length $\\tau=T_c/A$, and the nonlinear maximum principle converts the resulting $L^p$ control into a uniform-in-time $L^\\infty$ bound. Once the $L^\\infty$ norm is controlled, the $L^\\infty$-criterion upgrades the solution to a global classical solution. The proof treats separately the singular kernel case $\\beta=d$, where $\\Delta K$ is not integrable and $B(\\rho)=\\nabla(-\\Delta)^{-1}\\rho$, and the case $2\\le\\beta<d$, where $\\Delta K\\in L^1$.","pith_inferences":["The same substep contraction mechanism should be reproducible with other dissipative operators—for instance hyperdissipation or nonlinear diffusion—provided a nonlinear maximum principle and an $L^\\infty$-criterion are available; the paper's structure suggests mixing suppresses blow-up whenever those two ingredients exist.","A numerical implementation of the substep map with a simple weakly mixing flow (e.g., a time-periodic shear) could measure the actual contraction rate of $\\|\\rho-\\bar\\rho\\|_{L^2}$ and check whether it matches the predicted factor $1-\\lambda_N^{\\alpha/2}\\tau/2$, which would make the RAGE time $T_c$ the operative bottleneck.","The authors note that the range $d<\\beta<d+1$ remains open because $\\Delta K$ is not integrable there; one natural extension is to split the kernel into a singular part controlled by the $H^3$ energy and a regular part handled by the same mixing iteration."],"forward_implications":["Large initial data that would blow up in finite time for the unadvected equation become globally well-posed once advected by a suitably chosen weakly mixing flow, for every $0<\\alpha<2$, $\\beta\\in[2,d]$, $d\\ge2$.","The global solution belongs to $C(\\mathbb{R}_+;H^3(\\mathbb{T}^d))$, and the same argument upgrades to $C(\\mathbb{R}_+;H^k(\\mathbb{T}^d))$ whenever the initial datum lies in $H^k$, $k\\ge2$.","The result includes the supercritical fractional regime $\\alpha<d/2$, where the unadvected equation is known to blow up and where no prior mixing-suppression theorem applied.","Both the non-integrable kernel case $\\beta=d$ and the integrable-kernel case $2\\le\\beta<d$ are covered by the same strategy, with the energy estimates adapted to the structure of $\\Delta K$.","The mixing flow can be chosen in advance from the class of weakly mixing flows; no smallness condition on the initial data is imposed."],"supporting_citations":[{"why":"Defines relaxation-enhancing flows and supplies the RAGE-type time-averaging lemma that gives the quantitative mixing effect.","marker":"[11]"},{"why":"Previous suppression result for the same equation under the restrictions $\\alpha>\\max\\{\\beta-d/2,1\\}$ or $\\alpha=2$, and the $L^p$-obstacle (1.6) that this paper bypasses.","marker":"[25]"},{"why":"Earlier blow-up suppression for the classical Keller-Segel system ($\\alpha=2$, $\\beta=d$) whose $L^2$-estimate strategy is adapted here.","marker":"[28]"},{"why":"Establishes finite-time blow-up for large data in the unadvected generalized Keller-Segel equation, the phenomenon prevented here.","marker":"[4]"},{"why":"Source of the positivity lemma (Lemma 2.1) used to extract dissipation from the fractional Laplacian in the $L^2$ energy estimates.","marker":"[13]"},{"why":"Provides the proof of the nonlinear maximum principle on the torus used in Lemma 5.1.","marker":"[23]"},{"why":"Reference for the RAGE theorem underlying Lemma 3.5.","marker":"[16]"}],"fun_headline_variants":["Mixing prevents blow-up in fractional Keller-Segel","Large-data Keller-Segel tamed by mixing flow","Global solutions via mixing in Keller-Segel","Mixing as a cure for Keller-Segel blow-up","Fractional Keller-Segel: mixing stops blow-up"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of Proposition 3.4 requires that at the start of every substep $t_j=j\\tau$ the normalized fluctuation $(\\rho(t_j)-\\bar\\rho)/\\|\\rho(t_j)-\\bar\\rho\\|_{L^2}$ lies in the fixed compact set $K$ defined by $\\|\\varphi\\|_{\\dot H^{\\alpha/2}}^2\\le\\lambda_N^{\\alpha/2}$, but the paper only asserts this membership for the initial datum and does not show that the spectral cutoff $N$ chosen in (3.39) forces it for later iterates, whose $L^2$ deviations shrink so their normalized $\\dot H^{\\alpha/2}$ norms can grow.","fun_headline_variants_meta":{"raw":{"variants":["Mixing prevents blow-up in fractional Keller-Segel","Large-data Keller-Segel tamed by mixing flow","Global solutions via mixing in Keller-Segel","Mixing as a cure for Keller-Segel blow-up","Fractional Keller-Segel: mixing stops blow-up"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000197,"raw_usage":{"total_tokens":1335,"prompt_tokens":884,"completion_tokens":451,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":500,"completion_tokens_details":{"reasoning_tokens":371}},"tokens_in":500,"tokens_out":451,"duration_ms":4945,"temperature":1.0,"reasoning_tokens":371,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:49:03.973368+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to test whether (3.39) implies $\\|(\\rho_0-\\bar\\rho)/\\|\\rho_0-\\bar\\rho\\|_{L^2}\\|_{\\dot H^{\\alpha/2}}^2\\le\\lambda_N^{\\alpha/2}$; if some admissible initial datum satisfies (3.39) but not this inequality, then (3.43) fails and the contraction (3.49)--(3.51), on which the global $L^\\infty$ bound rests, is unsupported. A numerical iteration of the substep map would also reveal whether the normalized $\\dot H^{\\alpha/2}$ norm stays below $\\lambda_N^{\\alpha/2}$ at each step.","supporting_citations":[{"cited_title":"Constantin, A","cited_arxiv_id":null,"evidence_quote":"Defines relaxation-enhancing flows and supplies the RAGE-type time-averaging lemma that gives the quantitative mixing effect."},{"cited_title":"Hopf and J","cited_arxiv_id":null,"evidence_quote":"Previous suppression result for the same equation under the restrictions $\\alpha>\\max\\{\\beta-d/2,1\\}$ or $\\alpha=2$, and the $L^p$-obstacle (1.6) that this paper bypasses."},{"cited_title":"Kiselev and X","cited_arxiv_id":null,"evidence_quote":"Earlier blow-up suppression for the classical Keller-Segel system ($\\alpha=2$, $\\beta=d$) whose $L^2$-estimate strategy is adapted here."},{"cited_title":"Biler and G","cited_arxiv_id":null,"evidence_quote":"Establishes finite-time blow-up for large data in the unadvected generalized Keller-Segel equation, the phenomenon prevented here."},{"cited_title":"C´ ordoba and D","cited_arxiv_id":null,"evidence_quote":"Source of the positivity lemma (Lemma 2.1) used to extract dissipation from the fractional Laplacian in the $L^2$ energy estimates."},{"cited_title":"Granero-Belinch´ on","cited_arxiv_id":null,"evidence_quote":"Provides the proof of the nonlinear maximum principle on the torus used in Lemma 5.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reference for the RAGE theorem underlying Lemma 3.5."}],"review_version":1}