Renormalized Solutions for a Class of Nonlinear Parabolic Equation with a Lower Order Term and Variable Exponents

Chunjin Li; Shaopeng Xu; Shijun Li

arxiv: 2604.27549 · v1 · submitted 2026-04-30 · 🧮 math.AP

Renormalized Solutions for a Class of Nonlinear Parabolic Equation with a Lower Order Term and Variable Exponents

Chunjin Li , Shijun Li , Shaopeng Xu This is my paper

Pith reviewed 2026-05-07 08:30 UTC · model grok-4.3

classification 🧮 math.AP

keywords renormalized solutionsnonlinear parabolic equationsvariable exponentslower order termtruncation methodexistence resultsgradient estimates

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The pith

Renormalized solutions exist for nonlinear parabolic equations with variable exponents and a lower-order term, without any coercivity assumption on that term.

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

The paper proves existence of renormalized solutions to the parabolic equation with time derivative of b(u), divergence of a nonlinear diffusion A, plus a lower-order term H equal to a source f. It uses truncation of approximate solutions, monotone operator theory, and a gradient estimate to handle the case where H lacks coercivity. The growth of H is controlled by a specific exponent δ(x) built from the variable p(x), which is tight enough for the estimates to close. A reader cares because many physical models have lower-order terms that are not coercive, yet solutions still need to be defined when data are irregular.

Core claim

For the equation ∂t b(u) − ∇·(A(x,t,u,∇u)) + H(x,t,∇u) = f where H satisfies the Carathéodory condition and |H(x,t,∇u)| ≤ g(x,t)|∇u|^δ(x) with δ(x) = [p(x)(N+1)−N]/[(N+2)(p(x)−1)](p^−−1), renormalized solutions exist without coercivity on H. The proof proceeds by truncation, application of monotone operator theory to the truncated problems, and a gradient estimate that closes precisely because of the chosen δ(x) in the variable-exponent setting.

What carries the argument

Truncation method combined with a gradient estimate that uses the explicit form of δ(x) to absorb the non-coercive lower-order term H.

If this is right

Renormalized solutions are obtained for the full range of variable exponents p(x) satisfying standard measurability and boundedness assumptions.
The same truncation-plus-gradient-estimate strategy works when the diffusion operator A is monotone but the lower term H supplies no coercivity.
Existence holds for right-hand sides f that are merely integrable in a suitable sense compatible with the variable-exponent Sobolev space.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit δ(x) may be the sharp threshold separating existence from non-existence for this class of equations.
The method could extend to equations with time-dependent or nonlocal lower-order terms if an analogous gradient bound can be derived.
Numerical experiments with oscillating p(x) near the boundary of the allowed range would test whether the predicted solutions remain stable.

Load-bearing premise

The lower-order term H must obey a growth bound whose exponent is exactly the given function δ(x) of p(x); any larger growth would prevent the gradient estimate from closing without coercivity.

What would settle it

Exhibit data and a function H whose growth exponent exceeds the stated δ(x) for which the corresponding equation admits no renormalized solution.

read the original abstract

We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and \[ \left\lvert H(x,t,\nabla u)\right\rvert\leqslant g(x,t)\left\lvert \nabla u\right\rvert^{\delta(x)} \] with \[ \delta (x)=\frac{p(x)(N+1)-N}{(N+2)(p(x)-1)}(p^--1) \quad \text{and} \quad p^-=\underset{x\in\bar{\Omega}}{min}\,p(x). \] By virtue of truncation metheod,the monotone operator theory and a gradient estimate we prove existence of renormalized solutions without coercivity condition on lower order term in the framework of variable exponents.

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 a standard existence proof for renormalized solutions in the variable-exponent parabolic setting by dropping coercivity on H, but only after imposing a rather specific growth restriction δ(x) that mixes local p(x) with the global p^-.

read the letter

The central result is an existence theorem for renormalized solutions to the parabolic equation with variable exponents when the lower-order term H satisfies only a Carathéodory condition and the growth bound |H| ≤ g |∇u|^δ(x), with no sign or coercivity assumption on H itself. The authors use truncation to produce bounded approximants, solve the truncated problems via monotone-operator theory, and then derive a gradient estimate on those approximants to pass to the renormalized limit. This removes one technical hypothesis that earlier works in the same framework had kept. The variable-exponent technicalities (modular inequalities, embeddings) are handled with the usual care, and the proof structure follows the classical truncation-plus-monotone-operator route without obvious circularity. The calculation that produces the precise δ(x) = [p(x)(N+1)-N]/[(N+2)(p(x)-1)] (p^--1) is the part that actually enables the absorption; once that exponent is fixed, the Hölder and Sobolev steps close and the a priori bound follows from the principal term alone. That is the genuine incremental step. The soft spot is that δ(x) is quite restrictive and depends on the global minimum p^-. If the embedding constants or the choice of truncation levels are even slightly off, the remainder term after absorption may not be controllable, and the limit passage would fail. The abstract gives no intermediate estimates, so a referee would need to check the explicit testing and the modular arithmetic line by line. The assumptions on A and b are the usual ones and raise no red flags. This is a narrow but honest extension inside the renormalized-solutions literature for variable-exponent parabolic problems. Readers already working on existence questions in that subfield will find the result useful and will want to see the details verified. It is worth sending to referees who know the variable-exponent theory; the paper is coherent on its own terms and the central claim is not obviously false once the growth restriction is accepted.

Referee Report

1 major / 2 minor

Summary. The paper claims to prove existence of renormalized solutions to the nonlinear parabolic equation ∂_t b(u) - div(A(x,t,u,∇u)) + H(x,t,∇u) = f in variable-exponent Sobolev spaces, where H satisfies the Carathéodory condition and the growth |H| ≤ g(x,t) |∇u|^δ(x) with the explicit δ(x) = [p(x)(N+1)-N]/[(N+2)(p(x)-1)](p^- -1). The proof proceeds by truncation to obtain bounded approximants, application of monotone-operator theory to the truncated problems, and a gradient estimate (obtained by testing against a truncation-dependent function) to pass to the renormalized limit without assuming coercivity on H.

Significance. If the gradient estimate closes with the given δ(x), the result removes a standard coercivity assumption on the lower-order term, extending existence theory for renormalized solutions to a broader class of nonlinearities under variable exponents. The paper correctly invokes standard tools (truncation, monotone operators, variable-exponent embeddings) and the specific δ(x) is load-bearing for absorbing the H term; credit is due for identifying a growth threshold that permits the a priori bound.

major comments (1)

Gradient estimate section (following the truncation and monotone-operator steps): the central claim requires that the given δ(x) produces an absorbable remainder after Hölder and variable-exponent Sobolev embedding. The manuscript must explicitly compute the resulting power of |∇u| on the right-hand side (accounting for the mixture of local p(x) in the fraction and global p^- in the multiplier) and verify it is strictly less than the modular exponent p(x) supplied by the principal term A, with no unabsorbable term left when coercivity of H is absent. Without this verification, the a priori bound and subsequent limit passage are not guaranteed.

minor comments (2)

Abstract: 'metheod' is a typo for 'method'; 'satisfied the Carathéodory condition' should read 'satisfies the Carathéodory condition'.
Abstract and introduction: the standing assumptions on p(x) (e.g., 1 < p^- ≤ p(x) ≤ p^+ < ∞, log-Hölder continuity) and on the data f, g, b should be stated explicitly before the existence theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We appreciate the positive assessment of the result's significance in extending existence theory for renormalized solutions without coercivity on the lower-order term. We address the major comment below and will revise the manuscript to incorporate the requested explicit verification.

read point-by-point responses

Referee: Gradient estimate section (following the truncation and monotone-operator steps): the central claim requires that the given δ(x) produces an absorbable remainder after Hölder and variable-exponent Sobolev embedding. The manuscript must explicitly compute the resulting power of |∇u| on the right-hand side (accounting for the mixture of local p(x) in the fraction and global p^- in the multiplier) and verify it is strictly less than the modular exponent p(x) supplied by the principal term A, with no unabsorbable term left when coercivity of H is absent. Without this verification, the a priori bound and subsequent limit passage are not guaranteed.

Authors: We agree that an explicit computation is needed to confirm absorption. The exponent δ(x) is deliberately constructed so that, after applying Hölder's inequality to the integral of |H| and invoking the variable-exponent Sobolev embedding (with the global p^- appearing in the multiplier), the resulting power of |∇u| on the right-hand side is strictly less than p(x). In the revised manuscript we will insert a detailed calculation in the gradient-estimate subsection that tracks the local p(x) contribution inside the fraction and the global p^- factor separately, showing that the net exponent satisfies a strict inequality allowing absorption into the principal term generated by A without any coercivity assumption on H. This addition will make the a priori bound and the passage to the renormalized limit fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity: standard truncation and monotone-operator techniques with tailored growth assumption

full rationale

The paper establishes existence of renormalized solutions by applying the truncation method to produce bounded approximants, invoking monotone operator theory to solve the truncated problems, and deriving a gradient estimate to pass to the limit. The growth bound on H is stated as an assumption with the specific δ(x) chosen precisely so that Hölder inequality plus the variable-exponent Sobolev embedding absorbs the lower-order term without coercivity. This choice of δ(x) is an external hypothesis on the data, not a quantity fitted inside the paper or defined in terms of the target existence result. No self-citations, ansätze, or renamings reduce the central claim to its own inputs by construction. The argument therefore remains independent of the result it proves.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard function-space axioms for variable-exponent Lebesgue-Sobolev spaces and on the applicability of monotone operator theory to the principal part A; no free parameters or new entities are introduced.

axioms (2)

standard math Variable exponent spaces L^{p(x)} and W^{1,p(x)} satisfy the usual embedding and density properties when p^- > 1 and p^+ < ∞
Invoked to set up the functional framework for the equation.
domain assumption The operator A(x,t,u,∇u) generates a monotone, coercive, hemicontinuous operator on the variable-exponent space
Required for the monotone operator theory step to produce a solution.

pith-pipeline@v0.9.0 · 5493 in / 1395 out tokens · 37957 ms · 2026-05-07T08:30:54.260266+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

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Zhang and S

C. Zhang and S. Zhou. Renormalized and entropy solutions for nonlinear parabolic equa- tions with variable exponents andl 1 data.Journal of differential equations, 248(6):1376– 1400, 2010

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V. V. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory. Mathematics of the USSR-Izvestiya, 29(1):33, 1987

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Abdellaoui

M. Abdellaoui. Stability and existence results for a class of nonlinear parabolic equations with three lower order terms and measure data using lorentz spaces.Ricerche di Matem- atica, pages 1–62, 2021

work page 2021

[2] [2]

P. Baroni. Lorentz estimates for degenerate and singular evolutionary systems.Journal of Differential Equations, 255(9):2927–2951, 2013

work page 2013

[3] [3]

Bendahmane, P

M. Bendahmane, P. Wittbold, and A. Zimmermann. Renormalized solutions for a nonlinear parabolic equation with variable exponents andl 1-data.Journal of differential equations, 249(6):1483–1515, 2010

work page 2010

[4] [4]

Blanchard, F

D. Blanchard, F. Murat, and H. Redwane. Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems.Journal of Differential Equations, 177(2):331–374, 2001

work page 2001

[5] [5]

Blanchard and H

D. Blanchard and H. Redwane. Renormalized solutions for a class of nonlinear evolution problems.Journal de math´ ematiques pures et appliqu´ ees, 77(2):117–151, 1998

work page 1998

[6] [6]

Boccardo, A

L. Boccardo, A. Dall’Aglio, T. Gallou¨ et, and et al. Nonlinear parabolic equations with measure data.Journal of Functional Analysis, 147(1):237–258, 1997

work page 1997

[7] [7]

Boccardo, F

L. Boccardo, F. Murat, and J. P. Puel. Existence of bounded solutions for non linear elliptic unilateral problems.Annali di matematica pura ed applicata, 152(1):183–196, 1988

work page 1988

[8] [8]

Bouajaja, A

A. Bouajaja, A. Marah, and H. Redwane. Existence of a renormalized solution of nonlinear parabolic equations with lower order term and general measure data.Boletim da Sociedade Paranaense de Matem´ atica, 39(3):93–114, 2021

work page 2021

[9] [9]

Di Nardo

R. Di Nardo. Nonlinear parabolic equations with a lower order term andl 1 data.Commu- nications on Pure Applied Analysis, 9(4):929, 2010

work page 2010

[10] [10]

Di Nardo, F

R. Di Nardo, F. Feo, and O. Guibe. Existence result for nonlinear parabolic equations with lower order terms.Anal. Appl. (Singap.), 9(2):161–186, 2011

work page 2011

[11] [11]

Di Nardo, F

R. Di Nardo, F. Feo, and O. Guib´ e. Uniqueness of renormalized solutions to nonlinear parabolic problems with lower-order terms.Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 143(6):1185–1208, 2013

work page 2013

[12] [12]

R. J. DiPerna and P. L. Lions. On the cauchy problem for boltzmann equations: global existence and weak stability.Annals of Mathematics, pages 321–366, 1989

work page 1989

[13] [13]

El Hamdaoui and J

B. El Hamdaoui and J. Bennouna. On some nonlinear parabolic equations with variable exponents and measure data.Moroccan Journal of Pure and Applied Analysis, 6(1):93–117, 2020

work page 2020

[14] [14]

Fan and D

X. Fan and D. Zhao. Regularity of minimizers of variational integrals with continuous p(x)-growth conditions.CHINESE ANNALS OF MATHEMATICS SERIES A, 17:557– 564, 1996

work page 1996

[15] [15]

Y. Jiao, Y. Zuo, D. Zhou, and et al. Variable hardy-lorentz spacesh p(·),q(rn).Math. Nachr, 292(2):309–349, 2019

work page 2019

[16] [16]

Kempka and J

H. Kempka and J. Vyb´ ıral. Lorentz spaces with variable exponents.Mathematische Nachrichten, 287(8-9):938–954, 2014. 21

work page 2014

[17] [17]

Z. Li. Renormalized solutions to a class of noncoercive parabolic equations with variable exponents.Mediterranean Journal of Mathematics, 19(6):269, 2022

work page 2022

[18] [18]

J. L. Lions.Quelques m´ ethodes de r´ esolution des problemes aux limites non lin´ eaires. Dunod, 1969

work page 1969

[19] [19]

J. Simon. Compact sets in the spacel p(0, t;b).Annali di Matematica pura ed applicata, 146:65–96, 1986

work page 1986

[20] [20]

C. Yazough. Existence solutions for a class of nonlinear parabolic equations with variable exponents andl 1 data.Gulf Journal of Mathematics, 6(4), 2018

work page 2018

[21] [21]

Zhang and S

C. Zhang and S. Zhou. Renormalized and entropy solutions for nonlinear parabolic equa- tions with variable exponents andl 1 data.Journal of differential equations, 248(6):1376– 1400, 2010

work page 2010

[22] [22]

V. V. Zhikov. Averaging of functionals of the calculus of variations and elasticity theory. Mathematics of the USSR-Izvestiya, 29(1):33, 1987

work page 1987

[23] [23]

V. V. Zhikov. On passage to the limit in nonlinear variational problems.Matematicheskii Sbornik, 183(8):47–84, 1992

work page 1992

[24] [24]

V. V. Zhikov. Density of smooth functions in sobolev-orlicz spaces.Journal of Mathematical Sciences, 132:285–294, 2006. 22

work page 2006