Existence of new self-similar solutions of the fast diffusion equation

Kin Ming Hui

REVIEW 3 major objections 3 minor 15 references

Existence of new self-similar solutions of the fast diffusion equation

T0 review · 3 major / 3 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 For n≥3 and 0<m<(n-2)/n, radial solutions of the stationary fast-diffusion equation exist and are unique under prescribed value or decay.

desk verdict Genuine parameter-range extension, but the global proof has an unhandled degeneracy at f=0; worth refereeing, needs repair. read the letter →

arxiv 2505.24131 v1 pith:MCUGNUZY submitted 2025-05-30 math.AP

Kin Ming Hui This is my paper

classification math.AP MSC 35J1535J7035K65

keywords fastdiffusionequationself-similarsolutionbackwardradiallysymmetricfinite-timeextinctiondecayratecontractionmappingdegenerateelliptic

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 existence and uniqueness of two families of radially symmetric positive solutions to the stationary elliptic equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$ for $n\ge 3$ and $0

What carries the argument

The proof works with the radial form of the equation, written as an integral equation for $(f,h)=(f,f_r)$: $r^{n-1} f^{m-1} f_r = -\int_0^r \rho^{n-1}(\alpha f+\beta\rho f_r)\,d\rho$. Local existence is obtained by a contraction mapping on a closed ball in a Banach space of continuous functions, with the map defined by integrating $h$ to get $f$ and using the integral identity to define $h$ from $f$. Sign arguments using $\alpha>0$ show that the solution stays strictly decreasing, and bounds on $f_r$ exclude blow-up and vanishing, allowing the local solution to extend to all of $\mathbb{R}^n$. For the exterior problem, the change of variables $g(r)=r^{-(n-2)/m}f(r^{-1})$ converts the decay condition into the initial condition $g(0)=\eta$, so the same local-existence technique applies to a transformed ODE.

What would settle it

Evaluate the Lipschitz constant of the fixed-point map $\Phi$ on the ball $D_{\epsilon,\eta_0}$ for some allowed parameters near the upper endpoint $\beta=m\rho_1/(n-2-nm)-\delta$; if no $\epsilon>0$ makes the constant less than 1, Lemma 2.1 and hence Theorem 1.1 collapse.

Watch

Extended reading notes

Core claim

The central claim is that the radial elliptic equation has a unique global solution in each of two natural normalizations. For the interior problem, the unique solution has $f(0)=\eta_0$, $f_r(0)=0$, and is strictly decreasing. For the exterior problem, the unique solution satisfies the sharp decay $\lim_{r\to\infty} r^{(n-2)/m}f(r)=\eta$, and it is either strictly decreasing or has exactly one interior critical point. These statements are the paper's Theorems 1.1 and 1.2, and the profiles they produce are exactly the backward self-similar solutions of the fast diffusion equation when $\rho_1=1$.

Load-bearing premise

The global construction rests on Lemma 2.1, a local contraction-mapping existence result whose contraction estimate is not verified in the paper but referred to a cited lemma; if that estimate fails anywhere in the stated parameter range, the global solutions are not established.

Editorial extensions

If this is right

For $\rho_1=1$, the solutions $f_1$ and $f_2$ produce backward self-similar solutions $V_i(x,t)=(T-t)^\alpha f_i((T-t)^\beta x)$ of $u_t=\Delta(u^m/m)$ on $\mathbb{R}^n\times(-\infty,T)$ and $(\mathbb{R}^n\setminus\{0\})\times(-\infty,T)$, respectively.
The interior solutions are strictly radially decreasing, so $f_r(r)<0$ for all $r>0$.
The exterior solutions have the sharp decay $r^{(n-2)/m}f(r)\to\eta$, and the derivative obeys $\lim_{r\to\infty} r^{(n-2)/m+1} f_r(r)=-(n-2)\eta/m$.
Each exterior solution is either globally decreasing or decreases on an outer interval and increases near the origin with exactly one critical point.
For $\beta>0$ in the exterior case, the weighted quantity $\alpha f+\beta r f_r$ stays positive, matching the sign condition used to rule out local minima.

Reading between the lines

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

A scaling check shows $\rho_1$ is removable: replacing $x$ by $x/\sqrt{\rho_1}$ and $\beta$ by $\beta/\rho_1$ turns the general-$\rho_1$ equation into the $\rho_1=1$ case, so the essential new parameter is the ratio $\beta/\rho_1$.
Because uniqueness is proved only within the radial class, a natural next question is whether these profiles are also unique among all solutions of the elliptic equation with the same asymptotics; the paper does not address non-radial uniqueness.
The positivity bounds (3.28) and (3.31) are strong enough to be used as comparison barriers if one wants to prove convergence of general fast-diffusion solutions to these backward profiles.

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.

Desk Editor's Note

read the letter

This paper proves existence and uniqueness of radially symmetric solutions to Δ(f^m/m)+αf+βx·∇f=0 in the subcritical fast diffusion range 0

The soft spot is real and load-bearing. In the proof of Theorem 1.1, to rule out f(ri)→0 at a finite maximal existence time r0, the author extends f with f(r0)=fr(r0)=0 and then invokes "uniqueness of ODE (2.1)" to conclude f≡0. But (2.1) is degenerate at f=0: for m<1 the coefficient f^{m-1} blows up, and the standard Picard-Lindelöf theorem does not apply. Worse, in the natural variable w=f^m/m, fr(r0)=0 does not force w_r(r0)=0, since fr=(mw)^{1/m-1}w_r and 1/m-1>0. So the terminal data do not even pin down the transformed initial condition, and the conclusion needs a separate proof that no nontrivial solution touches zero in finite radius. The paper gives none, and the same move appears in Theorem 3.4 around (3.25). Because this step is what forces the maximal interval to be [0,∞), both the existence and uniqueness claims of Theorem 1.1 rest on it.

The local existence lemma (Lemma 2.1) is sketched by reference to [HKs]; that is probably fine but should be verified for the stated parameter range. Everything else—the derivative bounds, the contradiction at the first zero of f_r, the inversion back to the f variable—checks out. This is a repairable gap rather than a demonstrated false result. The right fix likely involves working with the transformed variable or a comparison principle that does not need degenerate uniqueness.

Who this is for: specialists in nonlinear diffusion and extinction profiles. It deserves a serious referee. I would send it out, but with the expectation of a substantive revision addressing the f=0 uniqueness step.

Referee Report

3 major / 3 minor

Summary. The paper studies radially symmetric solutions of the elliptic equation Δ(f^m/m)+α f+β x·∇f=0 for n≥3, 0<m<(n-2)/n, with α=(2β+ρ1)/(1-m), ρ1>0. Theorem 1.1 claims existence and uniqueness of a global regular solution with f(0)=η0, f_r(0)=0 when -ρ1/2<β<mρ1/(n-2-nm). Theorem 1.2 claims existence and uniqueness of a solution on R^n\{0} with r^{(n-2)/m}f(r)→η when β<mρ1/(n-2-nm). The proofs proceed by a local contraction-mapping lemma, a continuation argument that rules out finite-radius blow-up, and an inversion g(r)=r^{-(n-2)/m}f(1/r) that reduces the second problem to a global problem for g. The resulting solutions are used to construct backward self-similar solutions of the fast diffusion equation u_t=Δ(u^m/m).

Significance. If the results are correct, they extend earlier work of Hsu and of Peletier-Zhang to a broader two-parameter family with prescribed value at the origin or prescribed decay rate, which is a meaningful contribution to the understanding of backward self-similar solutions of the fast diffusion equation. The paper explicitly identifies the parameter range, re-proves a useful extension theorem (Theorem 3.4), and makes the inversion between f and g precise. However, the central existence and uniqueness claims rest on a continuation argument whose key step is not justified, and the local existence lemma is only sketched by reference to [HKs]. These are load-bearing gaps rather than cosmetic issues.

major comments (3)

[§2, proof of Theorem 1.1, exclusion of (2.10)] The step after (2.13) in which the solution is extended to a C^1 function at r0 with f(r0)=0 is not valid: for m<1, f^{1-m} tends to +∞ as f tends to 0, while the integral ∫_0^{r0} ρ^{n-1} f(ρ) dρ is positive, so the right-hand side of (2.13) diverges at r0 rather than defining a finite value. The subsequent assertion that the zero function is the unique solution of (2.1), (2.14) is also not justified by standard ODE uniqueness, because the coefficient f^{m-1} is singular at f=0; in the variable w=f^m/m, the terminal condition f_r(r0)=0 does not imply w_r(r0)=0. Since this is the step that forces r0=∞, the global existence and uniqueness of Theorem 1.1 are not established by the written proof.
[§3, Theorem 3.4 at (3.25)] The same unsupported uniqueness assertion appears in the continuation proof for g. Equation (3.2) has the same degeneracy in g^{m-1} at g=0, so terminal data g(r0)=g_r(r0)=0 do not uniquely determine the zero solution by standard Picard-Lindelöf theory. Moreover, the claim that gr(r0) can be set equal to the right-hand side of (3.24) is problematic because that expression contains g(r0)^{1-m}, which blows up when g(r0)=0. Since Lemma 3.6 and hence Theorem 1.2 rely on Theorem 3.4, the global result for the singular solution inherits the same vulnerability.
[§2, Lemma 2.1] The local existence lemma is only sketched: the proof states that the map Φ is contractive on D_{ε,η0} 'similar to Lemma 2.2 of [HKs]' after choosing ε small, but the verification is not reproduced. In particular, the Lipschitz constant of Φ2 depends on the full range of β in (1.12), including negative β, and on the bound f≥η0/2; the omission is load-bearing because Lemma 2.1 is the sole source of the local solution from which Theorem 1.1 is extended. The details of the contraction estimate, or a precise statement of which hypotheses of Lemma 2.2 of [HKs] apply to the present parameter range, need to be supplied.

minor comments (3)

[Abstract] The expression 'fi(T−t)β x)' is missing a parenthesis; it should read 'fi((T−t)^β x)'.
[§3, proof of Theorem 3.4, last paragraph] In the sentence 'When n−2/(n+1)≤m<(n−2)/n, by a similar argument as above using Lemma 3.2 implies ... which satisfies (3.5) and (3.7)', the final condition should be (3.8), not (3.7), because the local solution in this range is not C^1 at 0 and Lemma 3.2 gives the asymptotic (3.8).
[§2, Lemma 2.1] The definition of Φ2(f,h)(r) is only given for 0<r≤ε; the value at r=0 is not defined by the formula. It should be set to 0, with the limiting argument made explicit, since this is needed for Φ to map D_{ε,η0} into the stated Banach space.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the main existence theorems are not reductions of the paper's inputs; the self-citations to [HKs] supply local lemmas, while the global continuation and new parameter range are argued in the text.

full rationale

The central claims are not circular. The reduction (1.3)-(1.14) is the standard definition of backward self-similar solutions, not a fitted result, and the existence of f is proved by a local fixed-point lemma plus a continuation argument. The paper does rely repeatedly on [HKs], a paper co-authored by the present author, for local existence and regularity facts (Lemmas 2.1, 3.1, 3.2, 3.5, and the 'modification of the proof of Lemma 3.3 of [HKs]' in Lemma 3.7), but those cited lemmas are prior results whose assumptions do not include the target theorems, and the new global parameter range is established in the text through Lemma 2.2 and the continuation/contradiction argument in Theorem 1.1 and Theorem 3.4. Lemma 2.1 is only sketched, and the uniqueness assertion at (2.14)/(3.25) for the degenerate ODE is a genuine mathematical-risk point, but that is a potential gap or error, not circularity: nothing in the cited lemmas is defined in terms of the target conclusions. Therefore the paper has at most minor self-citation, not load-bearing circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No data fitting is involved; ρ1, η0, η, and β are arbitrary parameters in the theorem statements. The proofs rely on standard ODE tools plus a few domain assumptions and one unproved uniqueness assertion for a degenerate equation.

assumptions (5)

standard math Contraction mapping theorem and standard local ODE existence
Used in Lemma 2.1 to produce a unique local solution and in the local lemmas of Section 3.
standard math Maximal-interval blow-up dichotomy for ODEs
Used in Proof of Theorem 1.1 and Proof of Theorem 3.4 to assert that a finite maximal radius implies |f_r|→∞, f→0, or f→∞.
domain assumption Uniqueness of the degenerate ODE with zero data f(r0)=f_r(r0)=0
Invoked in Proof of Theorem 1.1 to rule out f→0 at a finite endpoint; not justified by standard Picard-Lindelöf because the equation degenerates at f=0.
domain assumption Inversion equivalence g(r)=r^{-(n-2)/m} f(r^{-1})
Established by direct computation in Section 3 and is the bridge between decay at infinity for f and the boundary value g(0)=η.
domain assumption Parameter regime n≥3, 0<m<(n-2)/n with β satisfying (1.10) or (1.12)
These inequalities ensure α>0 and alpha_tilde>0, which the monotonicity and sign arguments depend on.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Existence of new self-similar solutions of the fast diffusion equation." pith.science (2026). https://pith.science/paper/MCUGNUZY

@misc{pith2026250524131,
  author       = {Pith},
  title        = {Pith review of: Existence of new self-similar solutions of the fast diffusion equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MCUGNUZY}},
  note         = {Machine review of arXiv:2505.24131}
}

abstract

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\eta>0$, $\eta_0>0$, $\rho_1>0$, $-\frac{\rho_1}{2}<\beta<\frac{m\rho_1}{n-2-nm}$ and $\alpha=\frac{2\beta+\rho_1}{1-m}$. We will prove the existence of radially symmetric solution of the equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n$, which satisfies $f(0)=\eta_0$, $f_r(0)=0$. When $\beta<\frac{m\rho_1}{n-2-nm}$ holds instead, we will also prove the existence of radially symmetric solution of the equation $\Delta(f^m/m)+\alpha f+\beta x\cdot\nabla f=0$, $f>0$, in $\mathbb{R}^n\setminus\{0\}$, which satisfies $\lim_{x\to\infty}|x|^{\frac{n-2}{m}}f(x)=\eta$. As a consequence if $f_1$, $f_2$, are the solutions of the above two problems with $\rho_1=1$, then the function $V_i(x,t)=(T-t)^{\alpha}f_i(T-t)^{\beta} x)$, $i=1,2$, are backward similar solutions of the fast diffusion equation $u_t=\Delta (u^m/m)$ in $\mathbb{R}^n\times (-\infty,T)$ and $(\mathbb{R}^n\setminus\{0\})\times (-\infty,T)$ respectively.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

D.G. Aronson. The porous medium equation in Problems in Nonlinear Diffusion, Lectures given at the 2nd 1985 Session of the Centro Internazionale Matermatico Estivo (CIME) held at Montecatini Terme, Italy, June 10--June 18, 1985 , Springer Lecture Notes Math. vol. 1224, pp. 1--46, 2006

work page 1985
[2]

Daskalopoulos and Carlos E

P. Daskalopoulos and Carlos E. Kenig, Degenerate diffusion-initial value problems and local regularity theory Tracts in Mathematics 1, European Math. Soc., 2007

work page 2007
[3]

Daskalopoulos, J

P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Communications in Analysis and Geometry 27 (2019), no 8, 1757--1798

work page 2019
[4]

Daskalopoulos and N

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion. J. Reine Angew. Math. 622 (2008), 95--119

work page 2008
[5]

Daskalopoulos and N

P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Advances in Math. 240 (2013), 346--369

work page 2013
[6]

Galaktionov and L.A

V.A. Galaktionov and L.A. Peletier, Asymptotic behaviour near finite-time extinction for the fast diffusion equation, Arch. Rational Mech. Anal. 139 (1997), 83--98

work page 1997
[7]

Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal

S.Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal. TMA 75 (2012), no. 7, 3443--3455

work page 2012
[8]

Hsu, Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow, Proc

S.Y. Hsu, Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4239--4249

work page 2014

Show all 15 references

[9]

Hui and Soojung Kim, Asymptotic large time behavior of singular solutions of the fast diffusion equation, Discrete and Continuous Dynamical Systems Series A 37 (2017), no

K.M. Hui and Soojung Kim, Asymptotic large time behavior of singular solutions of the fast diffusion equation, Discrete and Continuous Dynamical Systems Series A 37 (2017), no. 11, 5943-5977

2017
[10]

King, Extremely high concentration dopant diffusion in silicon, IMA Journal of Applied Mathematics 40 (1988), Issue 3, 163--181

J.R. King, Extremely high concentration dopant diffusion in silicon, IMA Journal of Applied Mathematics 40 (1988), Issue 3, 163--181

1988
[11]

King, Self-similar behavior for the equation of fast diffusion equation, Phil

J.R. King, Self-similar behavior for the equation of fast diffusion equation, Phil. Trans. R. Soc. London (A) 343 (1993), 337-375

1993
[12]

Luo and H

T. Luo and H. Zeng, Global existence and smooth solutions and convergence to Barenblatt solutions for the physical vacuum free boundary problem of compressible Euler equations with damping, Commun. on Pure and Applied Math. 69 (2016), no. 7, 1354--1396

2016
[13]

Peletier and H

M.A. Peletier and H. Zhang, Self-similar solutions of a fast diffusion equation that do not conserve mass, D Differential and Integral Eqns 8 (1995), no. 8, 2045--2064

1995
[14]

del Pino and M

M. del Pino and M. Saez, On the extinction profile for solutions of u_t= u^ (N-2)/(N+2) , Indiana University Math. J. 50 ( 2001), no. 1, 611--628

2001
[15]

V \'a zquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and Its Applications vol

J.L. V \'a zquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type, Oxford Lecture Series in Mathematics and Its Applications vol. 33, Oxford University Press Inc., 2006

2006

Pith tools

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

[1] [1]

D.G. Aronson. The porous medium equation in Problems in Nonlinear Diffusion, Lectures given at the 2nd 1985 Session of the Centro Internazionale Matermatico Estivo (CIME) held at Montecatini Terme, Italy, June 10--June 18, 1985 , Springer Lecture Notes Math. vol. 1224, pp. 1--46, 2006

work page 1985

[2] [2]

Daskalopoulos and Carlos E

P. Daskalopoulos and Carlos E. Kenig, Degenerate diffusion-initial value problems and local regularity theory Tracts in Mathematics 1, European Math. Soc., 2007

work page 2007

[3] [3]

Daskalopoulos, J

P. Daskalopoulos, J. King and N. Sesum, Extinction profile of complete non-compact solutions to the Yamabe flow, Communications in Analysis and Geometry 27 (2019), no 8, 1757--1798

work page 2019

[4] [4]

Daskalopoulos and N

P. Daskalopoulos and N. Sesum, On the extinction profile of solutions to fast diffusion. J. Reine Angew. Math. 622 (2008), 95--119

work page 2008

[5] [5]

Daskalopoulos and N

P. Daskalopoulos and N. Sesum, The classification of locally conformally flat Yamabe solitons, Advances in Math. 240 (2013), 346--369

work page 2013

[6] [6]

Galaktionov and L.A

V.A. Galaktionov and L.A. Peletier, Asymptotic behaviour near finite-time extinction for the fast diffusion equation, Arch. Rational Mech. Anal. 139 (1997), 83--98

work page 1997

[7] [7]

Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal

S.Y. Hsu, Singular limit and exact decay rate of a nonlinear elliptic equation, Nonlinear Anal. TMA 75 (2012), no. 7, 3443--3455

work page 2012

[8] [8]

Hsu, Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow, Proc

S.Y. Hsu, Exact decay rate of a nonlinear elliptic equation related to the Yamabe flow, Proc. Amer. Math. Soc. 142 (2014), no. 12, 4239--4249

work page 2014