Pith. sign in

REVIEW 2 major objections 5 minor 29 references

Singular Dirichlet boundary problems for a class of fully nonlinear parabolic equations in one dimension

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper proves a sharp existence/non-existence classification for one-dimensional fully nonlinear parabolic equations with a singular Dirichlet boundary condition, governed by the growth exponents $\alpha$ of $g$ and, when $\alpha\le…

desk verdict Worth refereeing: the thresholds are plausible and the paper is clearly written, but the proof of Theorem 1.3(a) has a genuine comparison gap that needs a small fix. read the letter →

arxiv 2506.07415 v1 pith:SILZJE4A submitted 2025-06-09 math.AP

Takashi Kagaya This is my paper
classification math.AP MSC 35A0135K6135D40
keywords fullynonlinearparabolicequationssingularDirichletproblemviscositysolutionsexistenceandnon-existencetheorycomparisonprinciplePerron'smethodtravelingwaveblow-upthreshold
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 determines exactly when the initial-value problem for a one-dimensional fully nonlinear parabolic equation $u_t = f(g(u_x)u_{xx})$ on a finite interval, with boundary values forced to diverge to $\infty$, has a viscosity solution. The answer is governed by the growth exponent $\alpha$ of the coefficient $g$ at infinity and, in the regime $\alpha \le 1$, by the growth exponent $\beta$ of $f$. Solutions exist for $\alpha > 2$ when the initial function diverges at the boundary with at most polynomial rate, and for $1 < \alpha \le 2$ even for bounded continuous initial data. For $\alpha < 1$, existence holds precisely when $\beta < 1/(1-\alpha)$; at and above this threshold no viscosity solution exists, because the boundary divergence propagates into the interior instantly. As a direct corollary, the linear heat equation with the singular Dirichlet condition admits no viscosity solution.

What carries the argument

The machinery has three layers. The framework is viscosity solutions, with comparison proved by a doubling-of-variables argument that adds a singularity $1/(T-t)$ in time and rescales one side by a factor $\lambda$; the comparison principles are stated for the relaxed equations $u_t \ge \max\{f((1+\delta)g(u_x)u_{xx}),0\}$ and $u_t \le \min\{f(g(u_x)u_{xx}),0\}$, which handle unbounded sub- and super-solutions. Existence is built by Perron's method inside sandwiches $w_\varepsilon \le u \le v_\varepsilon$ of traveling-wave sub- and super-solutions constructed from functions $h$ with singularities $\psi_\gamma(s)=s^{-\gamma}$ or $-\log s$. Two special families carry the load at the boundary: the functions $\tilde{u}_k$ of Definition 4.1, which are bounded at $t=0$ yet have boundary values tending to $\infty$ as $k\to\infty$ (imported from the earlier singular Neumann paper), and the sub-solutions $v_L$ of Definition 4.7, which equal $1$ at $t=0$ and converge to $\infty$ in the interior as $L\to\infty$ when $\beta \ge 1/(1-\alpha)$. The threshold itself is the sign of $1-\beta(1-\alpha)$: positive sign gives globally solvable super-solution ODEs, non-positive sign makes the non-existence sub-solutions diverge.

What would settle it

Exhibit a viscosity solution, in the sense of Definition 2.1, of $u_t=(1+\varepsilon)u_{xx}$ on $(-b,b)\times(0,\infty)$ with $\lim_{x\to\pm b}u(x,t)=\infty$ for all $t>0$ and any continuous initial datum bounded below; the theorem states this is impossible, so any such solution would refute the classification.

Watch

Extended reading notes

Core claim

On its own terms, the paper proves a complete classification of solvability for (1.1)--(1.3) in the viscosity sense. If $\alpha > 2$, every initial datum satisfying the divergence condition (B2) admits a solution, and the solution is continuous and unique when the initial datum has the precise asymptotic form (B3). If $1 < \alpha \le 2$, existence holds for bounded continuous data and for (B2) data alike, with uniqueness under (B3) for exponents $\gamma_\pm \ge (2-\alpha)/(\alpha-1)$. If $\alpha = 1$, existence holds for bounded or moderately divergent data. If $\alpha < 1$, the problem is solvable exactly when $\beta < 1/(1-\alpha)$, where $\beta$ is the growth rate of $f$; when $\beta \ge 1/(1-\alpha)$, the proof shows that any would-be solution is infinite at every interior point for every positive time, so no solution exists. The heat equation $u_t = (1+\varepsilon)u_{xx}$ is the boundary case, and the theorem puts it in the non-existence regime.

Load-bearing premise

The existence theorem for $1<\alpha\le 2$ with bounded initial data imports the sub-solution family $\tilde{u}_k$ verbatim from Lemma 4.8 of the earlier singular Neumann paper without re-proving it, so that lemma is the load-bearing premise: if it fails, the bounded-data existence claim collapses.

Editorial extensions

If this is right

  • For the signed-power curvature flow (1.6), the theorems remove the earlier smoothness and convexity assumptions on the initial graph, and for $0<\beta_2\le 1$ even allow bounded continuous initial data; uniqueness holds under (B3) with the stated exponents.
  • For the p-Laplace-type heat equation (1.5), global existence or instantaneous blow-up is decided by whether $\beta_1$ is below or at/above $1/(p-1)$; the linear heat equation ($\beta_1=1$, $p=2$) is in the no-solution regime.
  • In the existence regime $\alpha>1$ with (B3) data, the divergence rate $u_0(x)-D_\pm\psi_{\gamma_\pm}(b\mp x)$ is preserved in time, so the solution inherits the exact boundary asymptotics of the initial datum.
  • When $\alpha<1$ and $\beta \ge 1/(1-\alpha)$, the non-existence proof shows any would-be solution would already be infinite at every interior point for every $t>0$, meaning the obstruction is not a boundary-condition failure but instantaneous interior blow-up.

Reading between the lines

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

  • The author does not draw this consequence, but the same sub-solution family $v_L$ could be used to test higher-dimensional analogues: any operator whose diffusion degenerates so fast that $1-\beta(1-\alpha)\le 0$ should exhibit the same instantaneous blow-up under a singular Dirichlet condition.
  • A natural numerical check of the sharpness: for $\alpha<1$ and $\beta$ slightly below $1/(1-\alpha)$, solutions should exist globally while developing boundary layers with the rate $\psi_{(2-\alpha)/(\alpha-1)}$; near the critical curve the layer width should shrink to zero as $\beta$ approaches the threshold.
  • Since the critical curve $\beta_c=1/(1-\alpha)$ resembles classical Fujita exponents, one might expect a large-deviation reading of the threshold in terms of the effective diffusivity $g(s)\sim|s|^{-\alpha}$ and the reaction rate $f(s)\sim|s|^{\beta}$.
  • The comparison machinery with the $(1+\delta)$ modification of the operator suggests a general stability principle: under singular Dirichlet data, the comparison class is stable under relaxing the operator by any positive factor $\delta$, which may extend to equations with drift or second-order terms.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the initial value problem for a class of fully nonlinear parabolic equations in one dimension with the singular Dirichlet boundary condition u(x,t) -> infinity as x approaches the lateral boundary, for all t>0. The main results give existence and non-existence thresholds in terms of the parameter alpha in the growth assumption (A2) on the diffusion coefficient g, the growth exponent beta in (A3) on the nonlinearity f, and the boundedness of the initial data. Theorem 1.1 treats alpha>2 with unbounded initial data satisfying (B2)/(B3); Theorem 1.2 treats 1<alpha<=2, including bounded initial data under (B1); Theorem 1.3 treats alpha<=1, showing non-existence when alpha<1 and beta>=1/(1-alpha), with a formal instantaneous interior blow-up, and existence otherwise. Applications include a p-Laplace-type heat equation and a beta-power curvature flow, with the heat equation case yielding non-existence.

Significance. If the results are correct, the paper gives a sharp classification for a class of fully nonlinear parabolic equations with singular Dirichlet boundary data, going substantially beyond the traveling-wave and singular Neumann results in prior work. The comparison principles for unbounded sub- and super-solutions (Theorems 3.1-3.4) are nontrivial and are developed carefully, and the Perron-method constructions with explicit families of super- and sub-solutions (Definitions 4.1, 4.4, 4.7) are valuable. The corollaries, especially the non-existence for the heat equation with singular Dirichlet boundary condition, are striking and potentially influential. The paper is not machine-checked, and the main proofs are analytic; the correctness of the central claims depends on closing the gaps described below.

major comments (2)
  1. [Section 4.4, proof of Theorem 1.3(a)] The comparison step 'from v_L(·,0)−(M+1) ≤ u0' is invalid. Since v_L(·,0) ≡ 1 (Remark 4.9), the left-hand side equals 1−(M+1) = −M, so the asserted inequality is −M ≤ u0. This does not follow from u0 ≥ M when M < 0; for example, with u0 ≡ −1 one has M = −1 and the left side is 1, while the right side is −1. Thus Theorem 3.3 cannot be applied as written, and the non-existence proof has a genuine gap for initial data with negative infimum. The gap is easily repairable: replace the shift M+1 by C = M−1, so that v_L(·,0)+C = M ≤ u0. Adding a constant preserves the sub-solution property (4.26) and convexity, and letting L→∞ still yields the claimed instantaneous interior blow-up. Because this is the core of the non-existence claim, the correction is load-bearing.
  2. [Section 4.2, Lemma 4.2] The proof of Theorem 1.2 under the bounded-initial-data assumption (B1) relies entirely on Lemma 4.2, imported from [17, Section 4.2, Lemma 4.8], to show that the Perron solution satisfies the singular boundary condition. The lemma is not proved in the manuscript, and it is the only mechanism that produces boundary blow-up for bounded initial data in the range 1 < α ≤ 2. As written, the existence theorem for the (B1) case is not self-contained, and a gap in the cited lemma would invalidate the theorem. The author should either include a proof or a detailed outline of Lemma 4.2, or at least state precisely which properties of the construction are used and why they hold in the present setting.
minor comments (5)
  1. [Title] There is a typographical error in the title on the first page: 'BOUNDAR Y' should be 'BOUNDARY'.
  2. [Section 4.4, proof of Theorem 1.3(a)] In the line after applying the comparison result, the interval '(0,1/c]' should refer to c_L rather than c, and the notation should be consistent with Definition 4.7.
  3. [Section 4.1, Eq. (4.10)] In the definition of w, the second argument is written as 'y' in 'w(x,t):=sup{wε(x,y): ε>0}'; this should be 't'.
  4. [Proof of Theorem 3.4] There is a small typo in the sentence 'Due the the assumption (B3)' — it should read 'Due to the assumption (B3)'.
  5. [Definition 4.1 and Lemma 4.2] The domain of the functions ˜uk is stated as 'Q0' in Lemma 4.2, but Definition 4.1 defines them on [−b,b]×[0,∞); the notation should be clarified for consistency.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the threshold theorems are proved by direct Perron and comparison arguments; the main imported ingredient is a parameter-free sub-solution lemma from the author's earlier work, which is independent support rather than a repackaged conclusion.

full rationale

The central existence and non-existence thresholds in Theorems 1.1-1.3 are derived by explicit constructions and Perron's method rather than by fitting or renaming. The only notable imported ingredient is the family of sub-solutions \tilde u_k in Definition 4.1 and Lemma 4.2, taken from the author's prior paper with Qing Liu [17, Section 4.2, Lemma 4.8]; it is used in the proof of Theorem 1.2 under (B1) to verify the singular boundary condition (1.2). This is a self-citation, and it does make the bounded-initial-data existence proof not fully self-contained, but Lemma 4.2 is a stated, parameter-free result whose assumptions (A1), (A2) with alpha <= 2 do not include the target existence theorem, so by the independence rule it does not constitute circularity. The non-existence proof of Theorem 1.3(a) uses the independently constructed sub-solutions v_L and comparison Theorem 3.3; no fitted parameter or known result is renamed as a prediction. I flag one non-circular correctness gap: in the proof of Theorem 1.3(a), the claim 'from v_L(·,0)-(M+1) <= u0' is not valid when M < 0 because v_L(·,0) = 1 gives the left side equal to -M, and u0 >= M does not imply u0 >= -M; this is repairable by shifting with M-1, but as written the comparison application is a genuine gap. That is a correctness issue, not a circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented physical entities. Its load-bearing input beyond standard PDE machinery is a black-box lemma from the author's earlier work, which is explicitly cited but not reproduced.

assumptions (2)
  • standard math Standard viscosity solution theory: Crandall-Ishii lemma, Perron's method, stability of viscosity solutions (from [12,15]).
    Used throughout Sections 3 and 4 for comparison and existence proofs.
  • domain assumption Lemma 4.2 (imported from Kagaya-Liu [17, Lemma 4.8]): for alpha<=2 the functions ~u_k of Definition 4.1 are continuous sub-solutions to (1.1), uniformly bounded at t=0, with lim_{k->infinity} ~u_k(+-b,t)=infinity for t>0.
    Load-bearing for the proof of Theorem 1.2 under (B1); not re-proven in this manuscript.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Singular Dirichlet boundary problems for a class of fully nonlinear parabolic equations in one dimension." pith.science (2026). https://pith.science/paper/SILZJE4A

@misc{pith2026250607415,
  author       = {Pith},
  title        = {Pith review of: Singular Dirichlet boundary problems for a class of fully nonlinear parabolic equations in one dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SILZJE4A}},
  note         = {Machine review of arXiv:2506.07415}
}
abstract

In this paper, we deal with the initial value problem for a class of fully nonlinear parabolic equations with a singular Dirichlet boundary condition in one space dimension. The interior equation includes, for example, a fully nonlinear $p$-Laplace type heat equation and a $\beta$-power type curvature flow. The singular Dirichlet boundary condition depicts, for example, the asymptoticness of the ends of complete curve to parallel two lines in geometric flow of graphs. We study the dependence of the existence and non-existence of solution to the problem on the interior equation and the boundedness of the initial function.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

29 extracted references · 29 canonical work pages

  1. [17]

    Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension

    Takashi Kagaya and Qing Liu. Singular Neumann boundary problems for a class of fully nonlinear parabolic equations in one dimension. SIAM J. Math. Anal. , Vol. 53, No. 4, pp. 4350–4385, 2021

    work page 2021

  2. [1]

    Altschuler and Lang F

    Steven J. Altschuler and Lang F. Wu. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differential Equations , Vol. 2, No. 1, pp. 101–111, 1994

    work page 1994

  3. [2]

    Altschuler and Lang-Fang Wu

    Steven J. Altschuler and Lang-Fang Wu. Convergence to translating solutions for a class of quasilinear parabolic boundary problems. Math. Ann., Vol. 295, No. 4, pp. 761–765, 1993

    work page 1993

  4. [3]

    Attouchi and G

    A. Attouchi and G. Barles. Global continuation beyond singularities on the boundary for a degenerate diffusive Hamilton-Jacobi equation. J. Math. Pures Appl. (9) , Vol. 104, No. 2, pp. 383–402, 2015

    work page 2015

  5. [4]

    Remarks on the Dirichlet and state-constraint problems for quasi- linear parabolic equations

    Guy Barles and Francesca Da Lio. Remarks on the Dirichlet and state-constraint problems for quasi- linear parabolic equations. Adv. Differential Equations , Vol. 8, No. 8, pp. 897–922, 2003

    work page 2003

  6. [5]

    On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations

    Guy Barles and Francesca Da Lio. On the generalized Dirichlet problem for viscous Hamilton-Jacobi equations. J. Math. Pures Appl. (9) , Vol. 83, No. 1, pp. 53–75, 2004

    work page 2004

  7. [6]

    On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations

    Guy Barles, Alessio Porretta, and Thierry Tabet Tchamba. On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations. J. Math. Pures Appl. (9), Vol. 94, No. 5, pp. 497–519, 2010. 30 TAKASHI KAGAYA

    work page 2010

  8. [7]

    A nonlinear heat equation with singular initial data

    Ha ¨ ım Brezis and Thierry Cazenave. A nonlinear heat equation with singular initial data. J. Anal. Math., Vol. 68, pp. 277–304, 1996

    work page 1996

Show all 29 references
  1. [8]

    Long-time behavior of the mean curvature flow with periodic forcing

    Annalisa Cesaroni and Matteo Novaga. Long-time behavior of the mean curvature flow with periodic forcing. Comm. Partial Differential Equations , Vol. 38, No. 5, pp. 780–801, 2013

    2013

  2. [9]

    Convergence of curve shortening flow to translating soliton

    Beomjun Choi, Kyeongsu Choi, and Panagiota Daskalopoulos. Convergence of curve shortening flow to translating soliton. Amer. J. Math. , Vol. 143, No. 4, pp. 1043–1077, 2021

    2021

  3. [10]

    The Qk flow on complete non-compact graphs

    Kyeongsu Choi and Panagiota Daskalopoulos. The Qk flow on complete non-compact graphs. Calc. Var. Partial Differential Equations , Vol. 61, No. 2, pp. Paper No. 73, 19, 2022

    2022

  4. [11]

    The evolution of complete non-compact graphs by powers of Gauss curvature

    Kyeongsu Choi, Panagiota Daskalopoulos, Lami Kim, and Ki-Ahm Lee. The evolution of complete non-compact graphs by powers of Gauss curvature. J. Reine Angew. Math. , Vol. 757, pp. 131–158, 2019

    2019

  5. [12]

    Crandall, Hitoshi Ishii, and Pierre-Louis Lions

    Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions. User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) , Vol. 27, No. 1, pp. 1–67, 1992

    1992

  6. [13]

    Local solvability and dilation- critical singularities of supercritical fractional heat equations

    Yohei Fujishima, Kotaro Hisa, Kazuhiro Ishige, and Robert Laister. Local solvability and dilation- critical singularities of supercritical fractional heat equations. J. Math. Pures Appl. (9) , Vol. 186, pp. 150–175, 2024

    2024

  7. [14]

    Existence and nonexistence of solutions for the heat equation with a superlinear source term

    Yohei Fujishima and Norisuke Ioku. Existence and nonexistence of solutions for the heat equation with a superlinear source term. J. Math. Pures Appl. (9) , Vol. 118, pp. 128–158, 2018

    2018

  8. [15]

    Surface evolution equations, Vol

    Yoshikazu Giga. Surface evolution equations, Vol. 99 of Monographs in Mathematics. Birkh¨ auser Ver- lag, Basel, 2006. A level set approach

    2006

  9. [16]

    Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in R2

    Norisuke Ioku, Bernhard Ruf, and Elide Terraneo. Existence, non-existence, and uniqueness for a heat equation with exponential nonlinearity in R2. Math. Phys. Anal. Geom. , Vol. 18, No. 1, pp. Art. 29, 19, 2015

    2015

  10. [18]

    Laister, J

    R. Laister, J. C. Robinson, M. Sier˙ z¸ ega, and A. Vidal-L´ opez. A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces. Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire, Vol. 33, No. 6, pp. 1519–1538, 2016

    2016

  11. [19]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions. Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints. I. The model problem. Math. Ann. , Vol. 283, No. 4, pp. 583–630, 1989

    1989

  12. [20]

    On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source

    Songzhe Lian, Hongjun Yuan, Chunling Cao, Wenjie Gao, and Xiaojing Xu. On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source. J. Differential Equations , Vol. 235, No. 2, pp. 544–585, 2007

    2007

  13. [21]

    The Cauchy problem for a parabolic p-Laplacian equation with com- bined nonlinearities

    Heqian Lu and Zhengce Zhang. The Cauchy problem for a parabolic p-Laplacian equation with com- bined nonlinearities. J. Math. Anal. Appl. , Vol. 514, No. 2, pp. Paper No. 126329, 40, 2022

    2022

  14. [22]

    On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source and measures as initial data

    Haifeng Shang and Fengquan Li. On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source and measures as initial data. Nonlinear Anal., Vol. 72, No. 7-8, pp. 3396–3411, 2010

    2010

  15. [23]

    Local existence and nonexistence for fractional semilinear parabolic systems with singular initial data

    Masamitsu Suzuki. Local existence and nonexistence for fractional semilinear parabolic systems with singular initial data. Evol. Equ. Control Theory , Vol. 14, No. 4, pp. 594–609, 2025

    2025

  16. [24]

    Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian

    Thierry Tabet Tchamba. Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian. Asymptot. Anal., Vol. 66, No. 3-4, pp. 161–186, 2010

    2010

  17. [25]

    Solvability of a semilinear heat equation on Riemannian man- ifolds

    Jin Takahashi and Hikaru Yamamoto. Solvability of a semilinear heat equation on Riemannian man- ifolds. J. Evol. Equ. , Vol. 23, No. 2, pp. Paper No. 33, 55, 2023

    2023

  18. [26]

    Berikbol T. Torebek. Critical exponents for the p-Laplace heat equations with combined nonlinearities. J. Evol. Equ. , Vol. 23, No. 4, pp. Paper No. 71, 15, 2023

    2023

  19. [27]

    On the stability of stationary line and grim reaper in planar curvature flow

    Xiaoliu Wang and Weifeng Wo. On the stability of stationary line and grim reaper in planar curvature flow. Bull. Aust. Math. Soc. , Vol. 83, No. 2, pp. 177–188, 2011

    2011

  20. [28]

    Solvability of a semilinear parabolic equation on Riemannian manifolds

    Xinran Wei and Mengmeng Zhang. Solvability of a semilinear parabolic equation on Riemannian manifolds. Partial Differ. Equ. Appl. , Vol. 5, No. 6, pp. Paper No. 32, 22, 2024

    2024

  21. [29]

    Weissler

    Fred B. Weissler. Local existence and nonexistence for semilinear parabolic equations in Lp. Indiana Univ. Math. J. , Vol. 29, No. 1, pp. 79–102, 1980. Takashi Kagaya, Muroran Institute of Technology, Japan Email address : kagaya@muroran-it.ac.jp

    1980

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.