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Global existence for a Leibenson type equation with reaction on Riemannian manifolds

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2505.08304 v1 pith:ZWTV3VDH submitted 2025-05-13 math.AP

Giulia Meglioli , Francescantonio Oliva , Francesco Petitta This is my paper · ORCID
classification math.AP MSC 35K5735B4458J3535K5935K6535R01
keywords LeibensonequationdoublynonlinearparabolicRiemannianmanifoldsglobalexistenceSobolevinequalityPoincarésmoothingestimatesMoseriteration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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].

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $f(u)\le C u^q$.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

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.

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 (2)
  1. [Theorem 2.3, Eq. (2.4), §5] 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.
  2. [Lemma 3.6, Eq. (3.25)] 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.
minor comments (4)
  1. [Lemma 3.7, Eq. (3.36)] 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).
  2. [Proof of Theorem 2.2, Eq. (4.26)] 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.
  3. [Proposition 3.9 and surrounding text] 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.
  4. [Theorem 2.2, Eq. (2.3)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the global-existence and L∞-decay claims are derived from the stated Sobolev/Poincaré inequalities via internally proved estimates; self-citations are methodological only.

full rationale

I walked the derivation chain of Theorems 2.2 and 2.3. The central inputs are the Sobolev inequality (1.3) and, for Theorem 2.3, the Poincaré inequality (1.4), together with smallness of the initial datum in specific Lebesgue norms. The paper proves all auxiliary estimates internally: the Caccioppoli-type bounds (Lemmas 3.3 and 3.4), the Moser iteration (Lemma 3.6), the L^s-L^s decay estimates (Lemmas 3.7 and 3.8), and Proposition 3.9. The constants ε0, ε1, c, and Γ are chosen from these hypotheses; no parameter is fitted to the target estimate. The self-citations [17], [18], and [20] appear in the introduction as related results and as methodological inspiration ('we adapt the methods exploited in [30], [17] and [20]'), but the proofs do not invoke theorems from those papers as inputs; the estimates are re-derived from (1.3) and (1.4). The geometric inequalities are assumptions, not consequences of the conclusion, and the bootstrap S(t)≤1 is internal to the proof. The reviewer's concern about the local estimate (2.4) concerns a possible correctness gap in the R→∞ passage (whether the local L^s norm on B_R controls the limiting global solution), not circularity: the estimate does not reduce by definition to its own assumptions. No step satisfies the required test of exhibiting a specific reduction of a claimed result to its inputs, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on geometric inequalities assumed as hypotheses (Sobolev and Poincaré), on standard PDE regularity theory, and on a small-data condition. No parameters are fitted to data and no new entities are postulated.

assumptions (6)
  • domain assumption Global Sobolev inequality (1.3) on M: ||v||_{L^{p*}} ≤ (1/S_p) ||∇v||_{L^p} for all v in C_c^∞(M).
    Assumed in Theorem 2.2 and used throughout Lemmas 3.4 and 3.6. It holds for Cartan-Hadamard manifolds, so it is a geometric hypothesis rather than a derived fact.
  • domain assumption Poincaré inequality (1.4) on M: ||v||_{L^p} ≤ (1/C_p) ||∇v||_{L^p} for all v in C_c^∞(M).
    Additional assumption in Theorem 2.3, used in Lemma 3.8 and Proposition 3.9. It holds for Cartan-Hadamard manifolds with sectional curvature ≤ -c < 0.
  • domain assumption Parameter range (1.2): 1 < p < N, m > 1, m(p-1) ≥ 1, q > m(p-1).
    Defines the slow diffusion regime on which the smoothing exponents and the L∞ decay formulas are based.
  • domain assumption The initial datum u0 is nonnegative and small in the stated Lebesgue spaces (2.2) or (2.5).
    Smallness is the mechanism that keeps the reaction term controlled through the bootstrap condition S(t) ≤ 1 and is essential for global existence.
  • standard math Standard elliptic regularity and compactness results for the regularized problems on geodesic balls, cited to [28] and [6].
    Used in Remark 3.2 and in the passages to the limit as k→∞, R→∞, and h→∞.
  • standard math Moser iteration lemma from Ladyzhenskaya-Solonnikov-Uraltseva [28, Chapter 2, Lemma 5.6].
    Used in Lemma 3.6 to conclude that the truncated L^r norms converge to zero and to obtain the L∞ bound.

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Pith. "Pith review of Global existence for a Leibenson type equation with reaction on Riemannian manifolds." pith.science (2026). https://pith.science/paper/ZWTV3VDH

@misc{pith2026250508304,
  author       = {Pith},
  title        = {Pith review of: Global existence for a Leibenson type equation with reaction on Riemannian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZWTV3VDH}},
  note         = {Machine review of arXiv:2505.08304}
}
abstract

We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = \Delta_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume $m(p-1)\ge1$, $m>1$ and $q>m(p-1)$. In particular, under the assumptions that $M$ supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided $q>m(p-1)+\frac pN$ and the initial datum is small enough; namely, we establish an explicit bound on the $L^\infty$ norm of the solution at all positive times, in terms of the $L^1$ norm of the data. Under the additional assumption that a Poincar\'e-type inequality also holds in $M$, we can establish the same result in the larger interval, i.e. $q>m(p-1)$. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that $M$ is non-compact and has infinite measure.

Figures

Figures reproduced from arXiv: 2505.08304 by the authors.

Figure 1
Figure 1. The cylinders Ui = BR × (θi , t) Remark 3.2. Let us observe that a solution u to problem (3.2) in the sense of Definition 3.1 is, in particular, a distributional solution of the equation in (3.2). Hence, a standard cut-off and density argument allows us to check that, for any 0 < t1 < t2 < T one has Z BR u(x, t2) φ(x, t2) dµ − Z t2 t1 Z BR u φt dµ dt + Z t2 t1 Z BR |∇u m| p−2 ⟨∇u m, ∇φ⟩ dµ dt + ε Z t2 t1 Z BR ⟨∇u, ∇… view at source ↗
Figure 2
Figure 2. Constants approximating C0 In (3.27) we set τ1 = tn+1 ; τ2 = tn ; a1 = hn ; a2 = hn . (3.29) Then, by using (3.28) and (3.29) and the fact that, for any t ∈ (0, T), S(t) ≤ 1, we can say that there exists C2 > 1 such that C˜ (2γ C1) β ≤ C n 2 t . Due to the latter and (3.29), (3.27) reads sup tn+1<τ<t Z BR [Ghn (u)]s dµ ≤ C n 2 t −β+1  sup tn<τ<t Z BR [Ghn (u)]r dµλ , (3.30) with β and λ as in (3.24), (3.26), respe… view at source ↗

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Existence results for Leibenson's equation on Riemannian manifolds

    math.AP 2026-01 unverdicted novelty 6.0 of 10

    The Cauchy problem for ∂t u = Δp u^q on Riemannian manifolds admits a unique weak solution when p>1, q>0, pq≥1 for any initial data in L1(M) ∩ L∞(M).

  2. Gradient estimates for Leibenson's equation on Riemannian manifolds

    math.AP 2025-06 conditional novelty 6.0 of 10

    Gradient estimates for solutions of ∂_t u = Δ_p u^q are proved on manifolds with Ricci curvature bounded below, in both slow and fast diffusion regimes, under a uniform bound on |∇v|^{p-2} v.

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