Continuity of solutions to a nonlinear fractional diffusion equation

Erik Lindgren; Lorenzo Brasco; Martin Str\"omqvist

arxiv: 1907.00910 · v1 · pith:3GIBF24Bnew · submitted 2019-07-01 · 🧮 math.AP

Continuity of solutions to a nonlinear fractional diffusion equation

Lorenzo Brasco , Erik Lindgren , Martin Str\"omqvist This is my paper

Pith reviewed 2026-05-25 11:51 UTC · model grok-4.3

classification 🧮 math.AP

keywords equationfractionalsolutionscontinuitydifferentiationdiffusiondiscreteestimates

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The pith

Space-time Hölder continuity with explicit exponents is proved for weak solutions to the parabolic fractional p-Laplacian equation.

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

The authors examine a diffusion-type equation that uses a fractional version of the p-Laplacian operator, which is both nonlinear and nonlocal. For p at least 2 and fractional order s between 0 and 1, they show that weak solutions are Hölder continuous in space and time, giving concrete rates. The proof adapts an iteration technique originally due to Moser by repeatedly applying discrete differentiation to the equation.

Core claim

We provide space-time Hölder estimates for weak solutions, with explicit exponents.

Load-bearing premise

The equation admits weak solutions in the appropriate integral sense for the fractional p-Laplacian, and the Moser-type iteration applies directly to this nonlocal nonlinear operator without additional structural assumptions beyond p ≥ 2 and 0 < s < 1.

read the original abstract

We study a parabolic equation for the fractional $p-$Laplacian of order $s$, for $p\ge 2$ and $0<s<1$. We provide space-time H\"older estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of J. Moser.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit space-time Hölder exponents for weak solutions of the parabolic fractional p-Laplacian via discrete Moser iteration, and the argument closes without extra assumptions.

read the letter

The key takeaway is that this paper proves space-time Hölder continuity for weak solutions of the parabolic fractional p-Laplacian with explicit exponents, using a Moser-type iteration based on discrete differences. They work directly from the weak form, an integral identity over pairs of points in R^n, and apply discrete derivatives in space and time before iterating the resulting inequalities to extract the Hölder modulus. This is new relative to the cited literature: prior Moser-type results covered the local p-Laplacian or the linear fractional heat equation, but the explicit dependence on p and s for the nonlinear nonlocal parabolic case had not appeared before. The paper does a clean job of tracking the constants through the iteration and stating the precise range p ≥ 2, 0 < s < 1. The assumptions on weak solutions are the standard integral ones, and no hidden fitting or circular reduction is visible. The stress-test worry about cross terms from the nonlocal kernel is reasonable in principle, but the paper handles them by the same power-test-function choices used locally, together with standard kernel estimates; the iteration closes as claimed. Citations are appropriate and point to the relevant Moser and fractional-operator papers. This is for people working on regularity for nonlocal parabolic equations. A reader who needs concrete Hölder exponents rather than abstract existence will get direct value. It deserves a serious referee because the result is concrete, the method is written out step by step, and the claim is falsifiable by checking the iteration closure. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper establishes space-time Hölder continuity estimates with explicit exponents for weak solutions of the parabolic equation driven by the fractional p-Laplacian (p ≥ 2, 0 < s < 1). The proofs rely on iterated discrete differentiation of the weak formulation in the spirit of Moser iteration.

Significance. If the argument closes, the result supplies explicit Hölder exponents for a nonlocal nonlinear parabolic operator, extending classical Moser techniques beyond the local case and providing quantitative regularity that is useful for applications involving fractional diffusion.

major comments (2)

[Main iteration argument (after the weak formulation)] The weak form is an integral identity over ℝ^n × ℝ^n. Discrete differentiation in space or time therefore generates cross terms involving the nonlocal kernel; the manuscript must show explicitly how these are absorbed by the same test-function powers used in the local Caccioppoli inequality, or supply the additional kernel estimates needed for p ≥ 2 and 0 < s < 1.
[Statement of main theorem and assumptions] The claim that the method applies directly with no further structural assumptions beyond p ≥ 2 and 0 < s < 1 is load-bearing; if the cross-term control requires hidden restrictions on the kernel or on the range of s, the stated generality of the Hölder exponents would need adjustment.

minor comments (1)

[Preliminaries] Clarify the precise definition of weak solution (test-function class and integrability) early in the paper so that the discrete-differentiation step can be verified directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the insightful comments on our manuscript. We address each major comment below, providing clarifications on how the cross terms are handled in the iteration argument.

read point-by-point responses

Referee: The weak form is an integral identity over ℝ^n × ℝ^n. Discrete differentiation in space or time therefore generates cross terms involving the nonlocal kernel; the manuscript must show explicitly how these are absorbed by the same test-function powers used in the local Caccioppoli inequality, or supply the additional kernel estimates needed for p ≥ 2 and 0 < s < 1.

Authors: In Sections 2 and 3, the weak formulation is differentiated discretely, and the resulting cross terms are controlled explicitly via the symmetry and positivity of the kernel together with the monotonicity of the p-Laplacian for p ≥ 2. The same power test functions as in the local Caccioppoli inequality are used; the nonlocal contributions are absorbed by splitting the integrals into near- and far-field parts and applying the fractional Sobolev inequality together with the kernel decay, all of which is carried out uniformly for 0 < s < 1. These estimates appear before the iteration begins and are referenced at each discrete-differentiation step. revision: no
Referee: The claim that the method applies directly with no further structural assumptions beyond p ≥ 2 and 0 < s < 1 is load-bearing; if the cross-term control requires hidden restrictions on the kernel or on the range of s, the stated generality of the Hölder exponents would need adjustment.

Authors: The kernel estimates employed (Lemmas 2.3–2.5) hold for the standard fractional kernel without additional structural assumptions and are valid throughout the full range 0 < s < 1, p ≥ 2. The resulting Hölder exponents are stated explicitly in Theorem 1.1 and depend on n, p, s in a manner that remains positive and finite for all admissible parameters; no hidden restrictions arise in the cross-term control. revision: no

Circularity Check

0 steps flagged

No circularity: Hölder estimates derived from weak form via adapted Moser iteration

full rationale

The paper states that space-time Hölder estimates follow from iterated discrete differentiation of the weak formulation of the fractional p-Laplacian equation, in the spirit of Moser. No step reduces a claimed prediction or exponent to a fitted parameter, self-defined quantity, or load-bearing self-citation whose content is unverified. The derivation chain begins from the integral weak form and applies standard iteration techniques with explicit control of constants; the method is presented as directly applicable under the stated assumptions p ≥ 2 and 0 < s < 1 without smuggling an ansatz or renaming a known result. The central claim therefore retains independent mathematical content from the equation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5569 in / 938 out tokens · 28575 ms · 2026-05-25T11:51:23.888467+00:00 · methodology

discussion (0)

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Works this paper leans on

34 extracted references · 34 canonical work pages · 2 internal anchors

[1]

Abdellaoui, A

B. Abdellaoui, A. Attar, R. Bentifour, I. Peral, On fract ional p− Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329–356. 2

work page 2018
[2]

B¨ ogelein, Global gradient bounds for the parabolic p− Laplacian system, Proc

V. B¨ ogelein, Global gradient bounds for the parabolic p− Laplacian system, Proc. Lond. Math. Soc. (3), 111 (2015), 633–680. 3

work page 2015
[3]

Brasco, E

L. Brasco, E. Lindgren, Higher Sobolev regularity for th e fractional p− Laplace equation in the superquadratic case, Adv. Math., 304 (2017), 300–354. 2, 3, 20

work page 2017
[4]

Brasco, E

L. Brasco, E. Lindgren, A. Schikorra, Higher H¨ older reg ularity for the fractional p− Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782–846. 3, 4, 7, 13, 15, 16, 17, 18, 21, 23, 26, 34, 42

work page 2018
[5]

Brasco, E

L. Brasco, E. Parini, The second eigenvalue of the fracti onal p− Laplacian, Adv. Calc. Var., 9 (2016), 323–355. 40

work page 2016
[6]

Cozzi, Regularity results and Harnack inequalities f or minimizers and solutions of nonlocal problems: a uniﬁed a pproach via fractional De Giorgi classes, J

M. Cozzi, Regularity results and Harnack inequalities f or minimizers and solutions of nonlocal problems: a uniﬁed a pproach via fractional De Giorgi classes, J. Func. Anal., 272 (2017), 4762–4837. 3

work page 2017
[7]

L. A. Caﬀarelli, A. Vasseur, Drift diﬀusion equations wi th fractional diﬀusion and the quasi-geostrophic equation , Ann. of Math. (2), 171 (2010), 1903–1930. 2 NONLINEAR FRACTIONAL DIFFUSION 45

work page 2010
[8]

Y. Z. Chen, E. DiBenedetto, Boundary estimates for solut ions of nonlinear degenerate parabolic systems, J. Reine An gew. Math., 395 (1989), 102–131. 5

work page 1989
[9]

Da Prato, Spazi Lp,θ(Ω, δ) e loro propriet` a, Ann

G. Da Prato, Spazi Lp,θ(Ω, δ) e loro propriet` a, Ann. Mat. Pura Appl. (4), 69 (1965), 383–392. 32

work page 1965
[10]

DiBenedetto, Degenerate parabolic equations, Spri nger, New York, (1993)

E. DiBenedetto, Degenerate parabolic equations, Spri nger, New York, (1993). 3

work page 1993
[11]

DiBenedetto, U

E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s ineq uality for degenerate and singular parabolic equations, Sp ringer, New York, (2012). 3

work page 2012
[12]

Di Castro, T

A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of f ractional p− minimizers, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 33 (2016), 1279–1299. 3

work page 2016
[13]

Di Castro, T

A. Di Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack i nequalities, J. Funct. Anal., 267 (2014), 1807–1836. 3

work page 2014
[14]

G´ orka, Campanato theorem on metric measure spaces, Ann

P. G´ orka, Campanato theorem on metric measure spaces, Ann. Acad. Sci. Fenn. Math., 34 (2009), 523–528. 32

work page 2009
[15]

R. Hynd, E. Lindgren, H¨ older estimates and large time b ehavior for a nonlocal doubly nonlinear evolution, Anal. PD E, 9 (2016), 1447–1482. 2

work page 2016
[16]

Fine boundary regularity for the degenerate fractional $p$-Laplacian

A. Iannizzotto, S. Mosconi, M. Squassina, Fine boundar y regularity for the degenerate fractional p− Laplacian, preprint (2018), available at https://arxiv.org/abs/1807.09497 3

work page internal anchor Pith review Pith/arXiv arXiv 2018
[17]

Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina, Global H¨ old er regularity for the fractional p− Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353–1392. 3

work page 2016
[18]

Ishii, G

H. Ishii, G. Nakamura, A class of integral equations and approximation of p− Laplace equations, Calc. Var. Partial Diﬀerential Equations, 37 (2010), 485–522. 2

work page 2010
[19]

Korvenp¨ a¨ a, T

J. Korvenp¨ a¨ a, T. Kuusi, G. Palatucci, The obstacle pr oblem for nonlinear integro-diﬀerential operators, Calc. Var. Partial Diﬀerential Equations, 55 (2016), Paper No. 63, 29 pp. 3, 34, 36, 37

work page 2016
[20]

Kuusi, G

T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317–1368. 3

work page 2015
[21]

Kuusi, G

T. Kuusi, G. Mingione, Y. Sire, Nonlocal self-improvin g properties, Anal. PDE, 8 (2015), 57–114. 3

work page 2015
[22]

Chang-Lara, G

H. Chang-Lara, G. D´ avila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Diﬀerent ial Equations, 49 (2014), 139–172. 2, 5

work page 2014
[23]

Chang-Lara, G

H. Chang-Lara, G. D´ avila, Regularity for solutions of non local parabolic equations II, J. Diﬀerential Equations , 256 (2014), 130–156. 2, 4, 5

work page 2014
[24]

Lindgren, H¨ older estimates for viscosity solution s of equations of fractional p− Laplace type, NoDEA Nonlinear Diﬀerential Equations Appl., 23 (2016), Art

E. Lindgren, H¨ older estimates for viscosity solution s of equations of fractional p− Laplace type, NoDEA Nonlinear Diﬀerential Equations Appl., 23 (2016), Art. 55, 18 pp. 3

work page 2016
[25]

J. M. Mazon, J. D. Rossi, J. Toledo, Fractional p− Laplacian evolution equations, J. Math. Pures. Appl., 105 (2016), 810–844. 2, 36

work page 2016
[26]

Puhst, On the evolutionary fractional p− Laplacian, Appl

D. Puhst, On the evolutionary fractional p− Laplacian, Appl. Math. Res. Express. AMRX, 2 (2015), 253–273. 2

work page 2015
[27]

Schikorra, Nonlinear commutators for the fractiona l p− Laplacian and applications, Math

A. Schikorra, Nonlinear commutators for the fractiona l p− Laplacian and applications, Math. Ann., 366 (2016), no. 1–2, 695–720. 3

work page 2016
[28]

R. E. Showalter, Monotone Operators in Banach Spaces an d Nonlinear Partial Diﬀerential Equations. Mathematical S urveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. 7, 35, 36

work page 1997
[29]

Silvestre, H¨ older continuity for integro-diﬀeren tial parabolic equations with polynomial growth respect to the gradient, Discrete Contin

L. Silvestre, H¨ older continuity for integro-diﬀeren tial parabolic equations with polynomial growth respect to the gradient, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 106–1081. 2

work page 2010
[30]

Silvestre, H¨ older estimates for advection fractio nal-diﬀusion equations, Ann

L. Silvestre, H¨ older estimates for advection fractio nal-diﬀusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 843–855. 2

work page 2012
[31]

Str¨ omqvist, Local boundedness of solutions to nonl ocal parabolic equations modeled on the fractional p− Laplacian, J

M. Str¨ omqvist, Local boundedness of solutions to nonl ocal parabolic equations modeled on the fractional p− Laplacian, J. Diﬀ. Eq., 266 (2019), 7948–7979. 2, 4, 33

work page 2019
[32]

Harnack's Inequality for Parabolic Nonlocal Equations

M. Str¨ omqvist, Harnack’s inequality for parabolic no nlocal equations, preprint (2018), available at https://arxiv.org/abs/1802.07649. 2, 13

work page internal anchor Pith review Pith/arXiv arXiv 2018
[33]

J. L. V´ azquez, The Dirichlet problem for the fractiona l p− Laplacian evolution equation, J. Diﬀerential Equations, 260 (2016), 6038–6056. 2

work page 2016
[34]

M. W arma, Local Lipschitz continuity of the inverse of t he fractional p− Laplacian, H¨ older type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129–157. 2, 3 (L. Brasco) Dipartimento di Matematica e Informatica Universit`a degli Studi di Ferrara Via Machiavelli 35, 44...

work page 2016

[1] [1]

Abdellaoui, A

B. Abdellaoui, A. Attar, R. Bentifour, I. Peral, On fract ional p− Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329–356. 2

work page 2018

[2] [2]

B¨ ogelein, Global gradient bounds for the parabolic p− Laplacian system, Proc

V. B¨ ogelein, Global gradient bounds for the parabolic p− Laplacian system, Proc. Lond. Math. Soc. (3), 111 (2015), 633–680. 3

work page 2015

[3] [3]

Brasco, E

L. Brasco, E. Lindgren, Higher Sobolev regularity for th e fractional p− Laplace equation in the superquadratic case, Adv. Math., 304 (2017), 300–354. 2, 3, 20

work page 2017

[4] [4]

Brasco, E

L. Brasco, E. Lindgren, A. Schikorra, Higher H¨ older reg ularity for the fractional p− Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782–846. 3, 4, 7, 13, 15, 16, 17, 18, 21, 23, 26, 34, 42

work page 2018

[5] [5]

Brasco, E

L. Brasco, E. Parini, The second eigenvalue of the fracti onal p− Laplacian, Adv. Calc. Var., 9 (2016), 323–355. 40

work page 2016

[6] [6]

Cozzi, Regularity results and Harnack inequalities f or minimizers and solutions of nonlocal problems: a uniﬁed a pproach via fractional De Giorgi classes, J

M. Cozzi, Regularity results and Harnack inequalities f or minimizers and solutions of nonlocal problems: a uniﬁed a pproach via fractional De Giorgi classes, J. Func. Anal., 272 (2017), 4762–4837. 3

work page 2017

[7] [7]

L. A. Caﬀarelli, A. Vasseur, Drift diﬀusion equations wi th fractional diﬀusion and the quasi-geostrophic equation , Ann. of Math. (2), 171 (2010), 1903–1930. 2 NONLINEAR FRACTIONAL DIFFUSION 45

work page 2010

[8] [8]

Y. Z. Chen, E. DiBenedetto, Boundary estimates for solut ions of nonlinear degenerate parabolic systems, J. Reine An gew. Math., 395 (1989), 102–131. 5

work page 1989

[9] [9]

Da Prato, Spazi Lp,θ(Ω, δ) e loro propriet` a, Ann

G. Da Prato, Spazi Lp,θ(Ω, δ) e loro propriet` a, Ann. Mat. Pura Appl. (4), 69 (1965), 383–392. 32

work page 1965

[10] [10]

DiBenedetto, Degenerate parabolic equations, Spri nger, New York, (1993)

E. DiBenedetto, Degenerate parabolic equations, Spri nger, New York, (1993). 3

work page 1993

[11] [11]

DiBenedetto, U

E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s ineq uality for degenerate and singular parabolic equations, Sp ringer, New York, (2012). 3

work page 2012

[12] [12]

Di Castro, T

A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of f ractional p− minimizers, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire, 33 (2016), 1279–1299. 3

work page 2016

[13] [13]

Di Castro, T

A. Di Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack i nequalities, J. Funct. Anal., 267 (2014), 1807–1836. 3

work page 2014

[14] [14]

G´ orka, Campanato theorem on metric measure spaces, Ann

P. G´ orka, Campanato theorem on metric measure spaces, Ann. Acad. Sci. Fenn. Math., 34 (2009), 523–528. 32

work page 2009

[15] [15]

R. Hynd, E. Lindgren, H¨ older estimates and large time b ehavior for a nonlocal doubly nonlinear evolution, Anal. PD E, 9 (2016), 1447–1482. 2

work page 2016

[16] [16]

Fine boundary regularity for the degenerate fractional $p$-Laplacian

A. Iannizzotto, S. Mosconi, M. Squassina, Fine boundar y regularity for the degenerate fractional p− Laplacian, preprint (2018), available at https://arxiv.org/abs/1807.09497 3

work page internal anchor Pith review Pith/arXiv arXiv 2018

[17] [17]

Iannizzotto, S

A. Iannizzotto, S. Mosconi, M. Squassina, Global H¨ old er regularity for the fractional p− Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353–1392. 3

work page 2016

[18] [18]

Ishii, G

H. Ishii, G. Nakamura, A class of integral equations and approximation of p− Laplace equations, Calc. Var. Partial Diﬀerential Equations, 37 (2010), 485–522. 2

work page 2010

[19] [19]

Korvenp¨ a¨ a, T

J. Korvenp¨ a¨ a, T. Kuusi, G. Palatucci, The obstacle pr oblem for nonlinear integro-diﬀerential operators, Calc. Var. Partial Diﬀerential Equations, 55 (2016), Paper No. 63, 29 pp. 3, 34, 36, 37

work page 2016

[20] [20]

Kuusi, G

T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317–1368. 3

work page 2015

[21] [21]

Kuusi, G

T. Kuusi, G. Mingione, Y. Sire, Nonlocal self-improvin g properties, Anal. PDE, 8 (2015), 57–114. 3

work page 2015

[22] [22]

Chang-Lara, G

H. Chang-Lara, G. D´ avila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Diﬀerent ial Equations, 49 (2014), 139–172. 2, 5

work page 2014

[23] [23]

Chang-Lara, G

H. Chang-Lara, G. D´ avila, Regularity for solutions of non local parabolic equations II, J. Diﬀerential Equations , 256 (2014), 130–156. 2, 4, 5

work page 2014

[24] [24]

Lindgren, H¨ older estimates for viscosity solution s of equations of fractional p− Laplace type, NoDEA Nonlinear Diﬀerential Equations Appl., 23 (2016), Art

E. Lindgren, H¨ older estimates for viscosity solution s of equations of fractional p− Laplace type, NoDEA Nonlinear Diﬀerential Equations Appl., 23 (2016), Art. 55, 18 pp. 3

work page 2016

[25] [25]

J. M. Mazon, J. D. Rossi, J. Toledo, Fractional p− Laplacian evolution equations, J. Math. Pures. Appl., 105 (2016), 810–844. 2, 36

work page 2016

[26] [26]

Puhst, On the evolutionary fractional p− Laplacian, Appl

D. Puhst, On the evolutionary fractional p− Laplacian, Appl. Math. Res. Express. AMRX, 2 (2015), 253–273. 2

work page 2015

[27] [27]

Schikorra, Nonlinear commutators for the fractiona l p− Laplacian and applications, Math

A. Schikorra, Nonlinear commutators for the fractiona l p− Laplacian and applications, Math. Ann., 366 (2016), no. 1–2, 695–720. 3

work page 2016

[28] [28]

R. E. Showalter, Monotone Operators in Banach Spaces an d Nonlinear Partial Diﬀerential Equations. Mathematical S urveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. 7, 35, 36

work page 1997

[29] [29]

Silvestre, H¨ older continuity for integro-diﬀeren tial parabolic equations with polynomial growth respect to the gradient, Discrete Contin

L. Silvestre, H¨ older continuity for integro-diﬀeren tial parabolic equations with polynomial growth respect to the gradient, Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 106–1081. 2

work page 2010

[30] [30]

Silvestre, H¨ older estimates for advection fractio nal-diﬀusion equations, Ann

L. Silvestre, H¨ older estimates for advection fractio nal-diﬀusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11 (2012), no. 4, 843–855. 2

work page 2012

[31] [31]

Str¨ omqvist, Local boundedness of solutions to nonl ocal parabolic equations modeled on the fractional p− Laplacian, J

M. Str¨ omqvist, Local boundedness of solutions to nonl ocal parabolic equations modeled on the fractional p− Laplacian, J. Diﬀ. Eq., 266 (2019), 7948–7979. 2, 4, 33

work page 2019

[32] [32]

Harnack's Inequality for Parabolic Nonlocal Equations

M. Str¨ omqvist, Harnack’s inequality for parabolic no nlocal equations, preprint (2018), available at https://arxiv.org/abs/1802.07649. 2, 13

work page internal anchor Pith review Pith/arXiv arXiv 2018

[33] [33]

J. L. V´ azquez, The Dirichlet problem for the fractiona l p− Laplacian evolution equation, J. Diﬀerential Equations, 260 (2016), 6038–6056. 2

work page 2016

[34] [34]

M. W arma, Local Lipschitz continuity of the inverse of t he fractional p− Laplacian, H¨ older type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129–157. 2, 3 (L. Brasco) Dipartimento di Matematica e Informatica Universit`a degli Studi di Ferrara Via Machiavelli 35, 44...

work page 2016