REVIEW 3 major objections 5 minor 1 cited by
Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term
T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read For a doubly degenerate equation with mixed local-nonlocal diffusion and convection, the paper proves existence, uniqueness, stabilization, extinction, and blow-up.
desk verdict A solid and useful generalization of the fractional porous-medium work [10] to a mixed local-nonlocal operator with convection, but the appendix's epsilon-approximate definition uses the wrong operator and the weak-mild link is formally unsupported as written. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the nonlinear operator $A v = \mathcal{A}_\mu(\beta^{-1}(v)) - \operatorname{div}(\vec{f}(\beta^{-1}(v)))$ on $L^1(\Omega)$, with a domain defined through the space $W = W^{1,p}_0(\Omega)\cap W^{s,q}_0(\Omega)$ when $\mu>0$ and $W=W^{1,p}_0(\Omega)$ when $\mu=0$. The proof shows that $A$ is accretive in $L^1$, that its domain is dense, and that the associated elliptic resolvent problems are solvable by minimization and a fixed-point argument. The weak-mild solution concept bridges this semigroup-style operator to the variational weak formulation, and the time-discretization scheme converts parabolic existence into a sequence of elliptic problems. The proof of stabilization uses an elliptic comparison principle for $\mathcal{A}_\mu u - \operatorname{div}(\vec{f}(u))$ together with monotone convergence, while extinction and blow-up are driven by energy estimates involving the functional $E(u)=\frac{\mu}{q}\|u\|^q_{W^{s,q}_0} + \frac1p\|\nabla u\|_p^p - \frac1{r+1}\|u\|_{r+1}^{r+1}$.
What would settle it
Construct two admissible functions $u,v$ such that $u>v$ on a set $K$ of positive measure but $u-v$ is constant on $K$, then compute the double-integral integrand in Lemma 4.1; on $K\times K$ the integrand is identically zero, so the asserted uniform lower bound $C>0$ fails, and one would need a different argument to establish the comparison principle underpinning the stabilization claim.
Extended reading notes
Core claim
The paper establishes a well-posedness and long-time behavior theory for the doubly degenerate problem $$ \partial_t \$\beta$(u) + \mathcal{A}_\mu u = \operatorname{div}(\vec{f}(u)) + g(t,x,u) \quad \text{in } Q_T, \qquad u=0 \text{ outside }\$\Omega$, \quad u(0)=u_0, $$ where $\mathcal{A}_\mu u = -\Delta_p u + \mu(-\Delta)_q^s u$ is the sum of the $p$-Laplacian and the fractional $q$-Laplacian. The solution concept is the weak-mild solution: a weak solution in the variational sense whose image $v=\beta(u)$ is a mild solution of the associated semigroup problem. Existence is obtained by a time-discretization scheme in which each step solves an elliptic problem for the mixed operator with convection, after which compactness arguments produce a limit; accretivity of the operator and density of its domain make the limit simultaneously mild. For odd $\beta$ satisfying the stated growth and monotonicity conditions, local-in-time existence holds under a general growth bound on $g$, global existence under a structural growth condition, and uniqueness together with an $L^1$-contraction estimate under a Lipschitz-type condition. For $\mu>0$ and a time-independent nonnegative source $h$, if $0\le u_0\le u_{\mathrm{stat}}$, then the solution converges to the unique stationary solution in every $L^\gamma$, $\gamma<\infty$. For the power model $\beta(s)=|s|^{1/m-1}s$ and $g(u)=|u|^{r-1}u$, small initial norms give finite-time extinction when $q<r+1<1/m+1$, while sufficiently negative initial energy gives finite-time blow-up with time bound $T_* := \tilde{c}\,\|u_0\|_{1/m+1}^{1/m-r}$.
Load-bearing premise
The stabilization theorem rests on the elliptic comparison principle in Lemma 4.1; its proof assumes that whenever one solution exceeds another on a set of positive measure a certain double integral is bounded below by a positive constant, and that pointwise lower bound is not guaranteed, so the comparison principle and Theorem 1.8 stand or fall with that estimate.
Editorial extensions
If this is right
- Under the hypotheses on $\beta$ and the growth condition $(g1)$, a weak-mild solution exists locally in time; if $g$ satisfies the structural growth condition $(g2)$ instead, the solution is global in time.
- When the source satisfies the Lipschitz-type condition $(g3)$, the weak-mild solution is unique, and any two solutions obey the $L^1$-contraction estimate in terms of their initial data and source terms.
- For $\mu>0$ and a time-independent nonnegative $h$, every solution starting between $0$ and the stationary solution converges to that stationary solution in $L^\gamma$ for every finite $\gamma$.
- For the power model $\beta(s)=|s|^{1/m-1}s$ and $g(u)=|u|^{r-1}u$, sufficiently small $\|u_0\|_{1/m+1}$ gives finite-time extinction when $q<r+1<1/m+1$; under the growth condition on the convection, sufficiently negative initial energy gives finite-time blow-up with the explicit time bound $T_*$.
- For a Lipschitz source with $g(0)=0$ and initial data confined to $[0,k]$, there is a global bounded weak-mild solution taking values in $[0,k]$, provided the convection and source are supported appropriately on that interval.
Reading between the lines
- The blow-up argument is said to be new even in the local case $\mu=0$, $\beta=\mathrm{Id}$, so a natural extension, not claimed by the paper, would be to use the same energy functional to derive a blow-up rate or to treat time-dependent convection $\vec{f}(t,u)$ under the same growth hypotheses.
- If the elliptic comparison principle in Lemma 4.1 can be repaired by a different argument, the stabilization theorem would likely extend to broader initial data and nonlinear sources, as the paper sketches in a remark; at present the stabilization claim depends on that comparison principle.
- The small-data extinction and large-data blow-up results suggest a critical-norm dichotomy for the power-law model, though the paper does not prove that the two regimes meet at a sharp threshold; a testable question is whether a critical value of the initial norm separates extinction from blow-up.
- The weak-mild framework is developed for constant $p,q\in(2,\infty)$, but the same accretivity-plus-discretization architecture should transfer to variable-exponent or other doubly nonlinear versions of the mixed operator whenever the elliptic problem is solvable and the operator is $L^1$-accretive.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the doubly nonlinear parabolic problem (P) with the mixed local-nonlocal operator A_μ = -Δ_p + μ(-Δ)_q^s, a convection term, and a source term. The authors introduce a notion of weak-mild solution, prove local and global existence under growth conditions on g, prove uniqueness and an L^1-contraction estimate under a Lipschitz-type condition, and then analyze stabilization to steady state, finite-time extinction, and finite-time blow-up. The proofs combine time discretization, accretive operator theory, comparison principles, and energy methods.
Significance. If the stated results are fully justified, the paper gives a useful extension of the porous-medium/fractional-p-Laplacian theory in [10] to a mixed local-nonlocal operator with convection, with explicit conditions for existence, uniqueness, stabilization, extinction, and blow-up. The work avoids fitted parameters and is built on standard monotone-operator and Sobolev tools. However, two load-bearing proof gaps at present prevent the central existence and stabilization claims from being considered established: the ε-approximate solution definition in the appendix uses the wrong operator, and the elliptic comparison principle in Lemma 4.1 relies on an unjustified pointwise lower bound. Both appear repairable, but they require genuine corrections before the main theorems can be accepted.
major comments (3)
- [Appendix A, Definition A.4; Step 4 of Theorem 1.4] The definition of an ε-approximate solution uses the wrong operator. Definition A.4 requires (U_i - U_{i-1})/(t_i - t_{i-1}) + (-Δ)^s_p(|U_i|^{m-1} U_i) = f_i, which is the fractional porous-medium operator from [10], not the operator A(v) = A_μ(β^{-1}(v)) - div \vec f(β^{-1}(v)) defined in (1.2) for problem (P0). The time-discretized equations in Step 1 of Theorem 1.4, rewritten with U_n = β(u_n), are (U_n - U_{n-1})/Δt + A(U_n) = g_n. Therefore the assertion in Step 4 that β(u_Δt) is an ε-approximate solution under Definition A.4 is not supported, and the implication from discretization to mild solution — and hence the weak-mild existence claim in Theorems 1.4–1.6 — is formally invalid as written. The natural correction is to replace the operator in Definition A.4 by A, after which the surrounding argument appears to go through, but the definition must be corrected before the proof is valid.
- [Lemma 4.1] The proof of Lemma 4.1 claims that if u > v on a set K of positive measure, then the double-integrand is at least C > 0 on K. This pointwise lower bound is not justified: for x, y ∈ K, the factor γ_ε(u(x)-v(x)) - γ_ε(u(y)-v(y)) may vanish when both values lie in the same saturated regime of γ_ε, or may be arbitrarily small when the two values are close inside (0, ε). No uniform positive lower bound follows, so the displayed strict inequality ⟨A_μu - A_μv, γ_ε(u-v)⟩ ≥ C > 0 is not established. Since Lemma 4.1 underlies Corollary 4.2 and Theorem 1.8, the stabilization result is unsupported as written. The gap appears repairable by testing with (u-v)^+ and deriving positivity of the integral from the strict monotonicity of the operator rather than from a pointwise bound on the integrand.
- [Theorem 2.3] The coercivity estimate for J_w bounds the term ∫ h u by ∥h∥_{L^∞(Ω)} ∥u∥_{L^1(Ω)}, but h is assumed to belong to W*, not necessarily to L^∞. This estimate is not valid as written. Since Theorem 2.3 is invoked for h ∈ W* in Corollary 2.4 and Lemma 2.5, the abstract resolvent and density part of the accretive-operator framework has a technical gap. A standard repair is to estimate ∫ h u by ∥h∥_{W*} ∥u∥_{W} and apply Young's inequality; with that change the coercivity proof goes through.
minor comments (5)
- [Lemma 4.1] In the displayed integrand, the denominator is written as |x-y|^{N+sp}, but for the fractional q-Laplacian the kernel should be |x-y|^{d+sq}; this appears to be a typo and should be corrected.
- [Theorem 1.5, Step 2] The set on which the approximating functions g_n are required to vanish is written as '(-∞,0] ∩ [k+1/n, ∞)', which is empty; it should be the union '(-∞,0] ∪ [k+1/n, ∞)'.
- [Remark 3.3] Remark 3.3 invokes Barbu's theorem [4, Th. 4.1] for uniqueness of the mild solution, but the m-accretivity and density hypotheses are only verified in Corollary 2.4 under extra conditions (p > d, or μ > 0 and qs > d). Since Theorem 1.6 proves uniqueness directly via (1.4) and Gronwall's lemma, either the remark should be restricted to those extra assumptions or removed.
- [Definition A.4] The notation f_i in Definition A.4 conflicts with the convection vector \vec f = (f_1, ..., f_d) used throughout the paper; using a different symbol (for example h_i) for the approximate right-hand side would avoid confusion.
- [Remark 4.3] There is a typo in 'discretization in tiome scheme'; it should read 'time scheme'.
Circularity Check
No significant circularity: the existence/uniqueness chain builds on Barbu's accretive-operator theory and independent estimates; the Definition A.4 operator mismatch is a formal proof gap, not a circular reduction.
full rationale
I walked the claimed derivation chain. The central existence claims are not derived from their own conclusions: Theorem 2.1 proves L1-accretivity of A(v)=A_mu(beta^{-1}(v))-div f(beta^{-1}(v)) by combining [4, Th. 3.5] with a direct integration-by-parts argument for the convection part; Lemma 2.5 proves density of D(A); Theorem 2.3 gives elliptic solvability via minimization plus Schaefer's fixed point theorem. These ingredients feed Barbu's theory [4] externally, and the time-discretization estimates in the proof of Theorem 1.4 are independent a priori bounds (boundedness, dt-estimate, compactness). Theorems 1.5, 1.6, 1.9 and 1.10 likewise follow from the same discretization or from energy methods, not from restating assumptions. The only self-citations are [10] as methodological precedent ('As in [10] we will use the accretive operator theory combined with a discretization in time scheme') and Remark 4.3 ('the arguments employed in [10]) can be adapted'); neither carries the proof: accretivity, maximality and uniqueness come from [4] and from in-paper computations. Under the review rule I explicitly flag two non-circular gaps. (i) Definition A.4 defines an epsilon-approximate solution of (P0) with the operator (-Delta)^s_p(|U|^{m-1}U), i.e. the fractional porous-medium operator, whereas the discretized equations actually solved in Step 1 of Theorem 1.4 are (beta(u_n)-beta(u_{n-1}))/Delta t + A_mu u_n = g_n + div f(u_n); setting U=beta(u) gives (U_n-U_{n-1})/Delta t + A(U_n)=g_n with A as in (1.2). Hence Step 4's assertion that beta(u_Delta t) is an epsilon-approximate solution 'see Definition A.4' is unsupported as written: the identification requires a corrected Definition A.4 using A. This is a missing-support/correctness issue, not circularity. (ii) In Lemma 4.1, the claim that the double-integral integrand is >= C>0 on a set K with u>v is not justified (for x,y both in K the gamma_epsilon-difference vanishes); this affects Theorem 1.8 and is again a proof gap, not a circular reduction. Neither issue equates a prediction to an input by definition, and neither depends on a self-citation chain. The paper is self-contained against external benchmark results ([4], [30], [34], standard Sobolev embeddings), and no fitted parameter is renamed as a prediction. The minor self-citations are not load-bearing, so the circularity score is at the low end (2).
Assumptions & free parameters
assumptions (7)
- standard math Sobolev embedding and compactness theorems for W^{1,p}_0(Ω) and W^{s,q}_0(Ω)
- standard math Barbu's theory of nonlinear accretive operators in Banach spaces
- standard math Schaefer fixed point theorem and direct method of calculus of variations
- standard math Aubin-Simon compactness lemma
- standard math Lipschitz chain rule for f_i composed with W^{1,p} functions (Theorem A.3)
- domain assumption Structural growth conditions (β1), (β2), (g1)-(g3), (f1)
- domain assumption Elliptic comparison principle for A_μ u - div f(u) (Lemma 4.1)
Cite this review
Pith. "Pith review of Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term." pith.science (2026). https://pith.science/paper/5URP535Q
@misc{pith2026250700959,
author = {Pith},
title = {Pith review of: Doubly nonlinear parabolic equation involving a mixed local-nonlocal operator and a convection term},
year = {2026},
howpublished = {\url{https://pith.science/paper/5URP535Q}},
note = {Machine review of arXiv:2507.00959}
}
abstract
In this paper we study a doubly degenerate parabolic equation involving a convection term and the operator $\mathcal{A}_\mu u:=-\Delta_p u +\mu (-\Delta)^s_q u$ which is a linear combination of the $p$-Laplacian and the fractional $q$-Laplacian, and results in a mixed local-nonlocal nonlinear operator. The problem we study is the following, \begin{equation*} \begin{cases} \partial_t \beta(u)+ \mathcal{A}_\mu u= div (\overset{\to}{f}(u))+g(t,x,u) \quad \text{in} \;Q_T:=(0,T)\times \Omega, u=0 \quad \text{in} \; (0,T)\times (\mathbb{R}^d \backslash \Omega), u(0)=u_0 \text{ in } \Omega. \end{cases}\ \end{equation*} We discuss existence, uniqueness and qualitative behavior of, what we call {\it weak-mild} solutions, that is weak solutions of this problem that when interpreted as $v=\beta(u)$ they are a mild solutions. In particular, we investigate stabilization to steady state, extinction and blow up in finite time and show how the occurrence of such behaviors depend on specific conditions on the nonlinearities $\beta$ (typically of porous media type), $\overset{\to}{f}$ and the source term $g$, and on their relation, in terms of certain regularity and growth conditions.
Forward citations
Cited by 1 Pith paper
-
Local Existence, Uniqueness, Regularity, and Global Behavior of Evolution Equations Involving Mixed Local and Nonlocal Operators
Weak energy solutions of a doubly nonlinear mixed local–nonlocal parabolic problem exist locally, are unique via a new Díaz–Saa inequality, extend globally by barriers, and converge to a nontrivial stationary state.
Reference graph
Works this paper leans on
-
[10]
Loïc Constantin, Jacques Giacomoni, and Guillaume Warnault. Existence and global behaviour of solutions of a parabolic problem involving the fractionalp-Laplacian in porous medium. Nonlinear Anal. Real World Appl., 87:Paper No. 104416, 2026
work page 2026
-
[1]
B. Abdellaoui, A. Attar, R. Bentifour, and I. Peral. On fractional p-Laplacian parabolic problem with general data.Ann. Mat. Pura Appl. (4), 197(2):329–356, 2018
work page 2018
-
[2]
Nathaël Alibaud, Jørgen Endal, Espen R Jakobsen, and Ola Mæhlen. Nonlocal degenerate parabolic-hyperbolic equations on bounded domains.Annales de l’Institut Henri Poincaré C, 2025
work page 2025
-
[3]
Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type
Carlo Alberto Antonini and Matteo Cozzi. Global gradient regularity and a Hopf lemma for quasilinear operators of mixed local-nonlocal type. J. Differential Equations, 425:342–382, 2025
work page 2025
-
[4]
Nonlinear differential equations of monotone types in Banach spaces
Viorel Barbu. Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York, 2010. 27
work page 2010
-
[5]
A Brezis-Oswald approach for mixed local and nonlocal operators.Commun
Stefano Biagi, Dimitri Mugnai, and Eugenio Vecchi. A Brezis-Oswald approach for mixed local and nonlocal operators.Commun. Contemp. Math., 26(2):Paper No. 2250057, 28, 2024
work page 2024
-
[6]
Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators
Stefano Biagi, Fabio Punzo, and Eugenio Vecchi. Global solutions to semilinear parabolic equations driven by mixed local-nonlocal operators. Bull. Lond. Math. Soc., 57(1):265–284, 2025
work page 2025
-
[7]
Functional analysis, Sobolev spaces and partial differential equations
Haim Brezis. Functional analysis, Sobolev spaces and partial differential equations . Universitext. Springer, New York, 2011
work page 2011
Show all 35 references
-
[8]
An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications
Thierry Cazenave and Alain Haraux. An introduction to semilinear evolution equations, volume 13 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998
1998
-
[9]
A doubly nonlinear evolution problem involving the fractional p-laplacian
Timthy Collier and Daniel Hauer. A doubly nonlinear evolution problem involving the fractional p-laplacian. arXiv preprint arXiv:2110.13401, 2021
2021 arXiv
-
[11]
Gradient regularity in mixed local and nonlocal problems
Cristiana De Filippis and Giuseppe Mingione. Gradient regularity in mixed local and nonlocal problems. Math. Ann., 388(1):261–328, 2024
2024
-
[12]
Hitchhiker’s guide to the fractional Sobolev spaces.Bull
Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci. Hitchhiker’s guide to the fractional Sobolev spaces.Bull. Sci. Math., 136(5):521–573, 2012
2012
-
[13]
Drapaca and S
C.S. Drapaca and S. Sivalogonathan. A fractional model of continuum mechanics.Journal of Elasticity, 107:105–123, 2012
2012
-
[14]
Faraci, D
F. Faraci, D. Motreanu, and D. Puglisi. Positive solutions of quasi-linear elliptic equations with dependence on the gradient. Calc. Var. Partial Differential Equations, 54(1):525–538, 2015
2015
-
[15]
Luiz F. O. Faria, Olímpio H. Miyagaki, and Dumitru Motreanu. Comparison and positive solutions for problems with the(p,q )-Laplacian and a convection term.Proc. Edinb. Math. Soc. (2), 57(3):687–698, 2014
2014
-
[16]
Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations
Prashanta Garain and Erik Lindgren. Higher Hölder regularity for mixed local and nonlocal degenerate elliptic equations. Calc. Var. Partial Differential Equations, 62(2):Paper No. 67, 36, 2023
2023
-
[17]
Existence and global behavior of weak solutions to a doubly nonlinear evolution fractionalp-Laplacian equation
Jacques Giacomoni, Abdelhamid Gouasmia, and Abdelhafid Mokrane. Existence and global behavior of weak solutions to a doubly nonlinear evolution fractionalp-Laplacian equation. Electron. J. Differential Equations, pages Paper No. 9, 37, 2021
2021
-
[18]
Evolutionaryp-Laplacian with convection and reaction under dynamic boundary condition.Bound
Shanming Ji, Jingxue Yin, and Rui Huang. Evolutionaryp-Laplacian with convection and reaction under dynamic boundary condition.Bound. Value Probl., pages 2015:194, 15, 2015
2015
-
[19]
Propagation profile of support for evolution p-Laplacian with convection in half space.J
Chunhua Jin, Jingxue Yin, and Sining Zheng. Propagation profile of support for evolution p-Laplacian with convection in half space.J. Math. Anal. Appl., 416(2):710–723, 2014. 28
2014
-
[20]
Existence for doubly nonlinear fractionalp-Laplacian equations
Nobuyuki Kato, Masashi Misawa, Kenta Nakamura, and Yoshihiko Yamaura. Existence for doubly nonlinear fractionalp-Laplacian equations. Ann. Mat. Pura Appl. (4), 203(6):2481– 2527, 2024
2024
-
[21]
G. I. Laptev. Weak solutions of second-order quasilinear parabolic equations with double nonlinearity. Mat. Sb., 188(9):83–112, 1997
1997
-
[22]
Mazón, Julio D
José M. Mazón, Julio D. Rossi, and Julián Toledo. Fractionalp-Laplacian evolution equations. J. Math. Pures Appl. (9), 105(6):810–844, 2016
2016
-
[23]
Radulescu, and Raffaella Servadei.Variational methods for nonlocal fractional problems, volume 162 ofEncyclopedia of Mathematics and its Applications
Giovanni Molica Bisci, Vicentiu D. Radulescu, and Raffaella Servadei.Variational methods for nonlocal fractional problems, volume 162 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2016. With a foreword by Jean Mawhin
2016
-
[24]
Nonhomogeneous degenerate quasilinear problems with convection.Nonlinear Anal
Dumitru Motreanu and Elisabetta Tornatore. Nonhomogeneous degenerate quasilinear problems with convection.Nonlinear Anal. Real World Appl., 71:Paper No. 103800, 14, 2023
2023
-
[25]
Global existence and gradient estimates for the quasilinear parabolic equations of m-Laplacian type with a nonlinear convection term
Mitsuhiro Nakao and Caisheng Chen. Global existence and gradient estimates for the quasilinear parabolic equations of m-Laplacian type with a nonlinear convection term. J. Differential Equations, 162(1):224–250, 2000
2000
-
[26]
Nonlinear partial differential equations with applications, volume 153 of International Series of Numerical Mathematics
Tomáš Roubíček. Nonlinear partial differential equations with applications, volume 153 of International Series of Numerical Mathematics. Birkhäuser/Springer Basel AG, Basel, second edition, 2013
2013
-
[27]
Cauchy problem for doubly degenerate parabolic equation with gradient source.Nonlinear Anal., 113:323–338, 2015
Haifeng Shang and Junxiang Cheng. Cauchy problem for doubly degenerate parabolic equation with gradient source.Nonlinear Anal., 113:323–338, 2015
2015
-
[28]
Reformulationofelasticitytheoryfordiscontinuitiesandlong-rangeforces,
J.S.Silling. Reformulationofelasticitytheoryfordiscontinuitiesandlong-rangeforces,. Journal of the Mechanics and Physics of Solids, 48:175–209, 2000
2000
-
[29]
Régularité de la solution d’une équation non linéaire dansRN
Jacques Simon. Régularité de la solution d’une équation non linéaire dansRN. In Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977), volume 665 ofLecture Notes in Math., pages 205–227. Springer, Berlin, 1978
1977
-
[30]
Compact sets in the spaceLp(0,T ;B)
Jacques Simon. Compact sets in the spaceLp(0,T ;B). Ann. Mat. Pura Appl. (4), 146:65–96, 1987
1987
-
[31]
On parabolic problems involving fractional p-Laplacian via topological degree
Ihya Talibi, Abdellah Taqbibt, Brahim El Boukari, Jalila El Ghordaf, and M’hamed El Omari. On parabolic problems involving fractional p-Laplacian via topological degree. Filomat, 38(20):7173–7181, 2024
2024
-
[32]
Solvability of doubly nonlinear parabolic equation withp-Laplacian
Shun Uchida. Solvability of doubly nonlinear parabolic equation withp-Laplacian. Evol. Equ. Control Theory, 11(3):975–1000, 2022
2022
-
[33]
The Dirichlet problem for the fractionalp-Laplacian evolution equation
Juan Luis Vázquez. The Dirichlet problem for the fractionalp-Laplacian evolution equation. J. Differential Equations, 260(7):6038–6056, 2016
2016
-
[34]
William P. Ziemer. Weakly differentiable functions, volume 120 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1989
1989
-
[35]
Zimmermann.A Continuum Theory with Long-Range Forces for Solids
M. Zimmermann.A Continuum Theory with Long-Range Forces for Solids. PhD thesis, MIT, 2005. 29
2005
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