The Diffusive Exchange Driven Growth Model with unbounded kernels

Ram Gopal Jaiswal; Saumyajit Das

REVIEW 3 major objections 5 minor 35 references

The Diffusive Exchange Driven Growth Model with unbounded kernels

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Pith's one-line read This paper proves global-in-time existence of non-negative weak solutions for the diffusive exchange-driven growth system in arbitrary spatial dimension, for separable kernels with linearly growing donor rates and sublinear receiver rates.

desk verdict First diffusive EDG existence in arbitrary dimension, but the final limit drops the diffusion coefficient and the theorem as stated is not proved. read the letter →

arxiv 2608.09229 v1 pith:BA7RRT3T submitted 2026-08-10 math.AP

Saumyajit Das , Ram Gopal Jaiswal This is my paper · ORCID

classification math.AP MSC 35A0135B4535D3035K5135K5535K5735Q9282C22

keywords diffusiveexchange-drivengrowthinfinitereaction–diffusionsystemsseparableexchangekernelsglobal-in-timeweaksolutionsentropy–entropydissipationFisherinformationrenormalized

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

The paper targets the diffusive exchange-driven growth system, an infinite set of reaction–diffusion equations in which clusters of every size exchange single monomers, and asks whether the whole infinite system has a solution for all times. It answers yes, in any spatial dimension, for separable kernels $K_{i,j}=b_i a_j$ whose donor rates $b_i$ grow at most linearly and whose receiver rates $a_j$ grow sublinearly. The proof works by truncating to finitely many cluster sizes, damping the quadratic source terms, and then passing to the limit twice, with a uniform Fisher-information bound as the engine of compactness. If the theorem is right, the diffusive EDG model is globally well-posed for this kernel class, and the finite-time gelation phenomena seen in the spatially homogeneous model are excluded here.

What carries the argument

The load-bearing object is the relative entropy with kernel-adapted weights, $E_N(f^N)=\sum_{i=0}^N\int_\Omega f^N_i\log(f^N_i/Q_i)$, whose time derivative splits into a non-positive diffusion contribution and an exchange contribution $D_N$ that is non-negative thanks to the separable structure of the kernel. This produces the entropy–entropy dissipation identity $E_N(t)+\sum_i d_i\int_0^t\int_\Omega |\nabla f^N_i|^2/f^N_i+\int_0^t D_N=E_N(0)$, hence the uniform Fisher-information bound (2.9). The second ingredient is the renormalized-solution formalism: a truncation-to-identity of the densities is used to obtain equations that survive the low regularity of the $L^1((0,T);W^{1,1}(\Omega))$ solutions, with measure-valued error terms that vanish as the truncation level goes to infinity. Together these two tools carry the passage through damping, truncation, and finally $N\to\infty$.

What would settle it

One concrete check: for a fixed smooth initial datum and the kernel $K_{i,j}=(i+1)(j+1)^\alpha$ with $\alpha=1$, compute the truncated-system Fisher-information sum $\sum_{i=0}^N\int_0^T\int_\Omega|\nabla f^N_i|^2/f^N_i$ and the weighted entropy with $Q_j$ as defined in the paper; if either fails to stay bounded as $N\to\infty$ and the damping is removed, the sublinearity assumption $\alpha<1$ is exactly the boundary of the method, while boundedness at $\alpha=1$ would show the threshold is not sharp.

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Extended reading notes

Core claim

The central discovery is Theorem 1.2: under the growth conditions (1.6) on the separable exchange kernel, the lower-and-upper bounds (1.8) on the diffusion coefficients, and the moment and relative-entropy conditions (1.9)–(1.10) on the initial data, the full infinite diffusive EDG system (1.1) admits a global-in-time non-negative weak solution in the sense of Definition 1.1, in any space dimension. The solution is constructed as the limit of solutions to the $N$-species truncated systems (1.11), and the limit is shown to exist because the entropy–entropy dissipation identity (2.7) yields a Fisher-information bound (2.9) that is uniform in both the damping parameter and the truncation size. The proof identifies the weights $Q_0=1$, $Q_j=a_{j-1}Q_{j-1}/b_j$ as the correct reference measure for the relative entropy, and the strict sublinearity $\alpha<1$ as what makes these weights summable and the entropy bounded below.

Load-bearing premise

The entire argument hinges on the initial data having finite entropy with respect to the kernel-generated weights $Q_j$, together with the strictly sublinear growth $\alpha<1$ of the receiver rates; if the entropy is infinite or $\alpha=1$, the Fisher-information bound and the compactness it provides give no control.

Editorial extensions

If this is right

For kernels satisfying (1.6)–(1.8), the infinite diffusive EDG system has a well-defined time evolution for all $t\ge0$, so the model can be used without worrying about finite-time blow-up or instantaneous gelation.
The uniform moment bounds (2.10) transfer to the limiting solution, so the total number of clusters and the total mass remain controlled for all times.
The proof shows that the entropy and Fisher-information estimates survive both limiting steps, giving quantitative control on the full infinite system, not only on finite truncations.
The strict sublinearity $\alpha<1$ gives a precise, checkable condition under which the renormalized-limit strategy succeeds, marking where new ideas would be needed.

Reading between the lines

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

The weights $Q_j$ are the natural equilibrium profile of the no-diffusion model; the entropy construction suggests that the long-time limit of the diffusive system should converge to this profile, extending equilibrium results known for the spatially homogeneous EDG equation.
A direct numerical test of the theorem is available: with $\alpha<1$ the Fisher-information sum $\sum_i\int_0^T\int_\Omega |\nabla f_i|^2/f_i$ should stay bounded as the truncation size grows, while at $\alpha=1$ the weighted entropy should fail to be controlled; such a simulation would probe whether the sublinearity threshold is sharp in practice.
The same two-level approximation—damping plus truncation-to-identity—looks transferable to collision-induced breakage equations with monomer production, where the monomer equation shares the non-sign-definite structure that blocks monotone methods.

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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

Dear colleague,

The headline is that this is the first rigorous treatment of the spatially diffusive EDG system in arbitrary dimension, and the strategy is genuinely promising. But as written, the proof of Theorem 1.2 drops the diffusion coefficient. In Definition 1.1 the diffusion term is d_i ∫∇ξ·∇f_i, while Lemma 3.4 and the final passage to Λ→∞ produce ∫∇f_I·∇ξ without the d_I factor. The earlier renormalized equation in Proposition 2.9 does carry d_I, and the factor presumably can be carried through the η→0 and Λ→∞ limits, so the gap is likely repairable, but the central claim as stated is not established for diffusion coefficients not all equal to 1.

What is genuinely new: the EDG source lacks the sign-definite monomer structure that makes coagulation tractable, so the Fisher-information/renormalized-solution route is a reasonable fresh approach. The model is well motivated by migration and wealth-exchange applications, and the kernel class (b_i a_j with linear donor growth and sublinear receiver growth) is a natural separable class. The entropy–entropy dissipation identity, using the recursively defined weights Q_j, is the right tool and gives a uniform Fisher information bound that is the backbone of the compactness argument. Credit also for the explicit renormalized formulation and the careful handling of the truncation-to-identity functions.

Soft spots, in order of size. The missing d_I is the main one. Second, several key estimates are cited rather than proved: Proposition 2.1 is dismissed with "proof follows along the same lines as [7,13]"; Theorem 2.3 is a one-line reference to [27]; and Lemma 2.5 is quoted. For a paper whose whole point is a new existence theorem, the truncated-system existence deserves a real proof or a precise appendix. Third, the epsilon and N limit passages contain a few places where the indexing is careless (e.g., sums over N−1 vs N, and the measure estimate in Prop 2.6 has a slight mismatch in the factor (N+1) in front of the j-sum). These look like typos, not fatal. The citation pattern is honest: the dependence on [16], [18], [27], [13] is clearly stated, and the authors' own previous work appears only in auxiliary comparisons.

Bottom line: for a reader working on reaction-diffusion systems or EDG, this is worth a careful look once the d_I issue is fixed. It deserves peer review rather than desk rejection: a serious referee can verify the fix and tighten the cited-estimate issue. I would not cite it in its current form, but I would send it to a qualified referee.

Referee Report

3 major / 5 minor

Summary. The paper studies the diffusive exchange-driven growth (EDG) system on a bounded smooth domain in arbitrary dimension, with homogeneous Neumann boundary conditions and infinitely many species. The authors assume a separable kernel K_{i,j}=b_i a_j, with b_i growing at most linearly and a_j growing sublinearly with exponent α∈(0,1), and diffusion coefficients uniformly bounded above and below. Under an L^2 moment condition and a weighted entropy condition on the initial data, Theorem 1.2 claims global-in-time existence of nonnegative weak solutions. The strategy is to regularize a finite-species truncated system, derive a uniform Fisher information bound from an entropy-entropy dissipation identity, pass to the limit in the damping parameter to obtain renormalized solutions of the truncated system, and then use compactness to pass to the infinite-species limit.

Significance. If the main theorem is correct, this would be the first global existence result for the diffusive EDG system in arbitrary space dimension for a class of unbounded separable kernels. The entropy-entropy dissipation identity (2.7), the recursively defined weights Q_j, and the uniform Fisher information bound (2.9) are the central technical novelties, and they are well matched to the structure of the problem. The structural discussion of why the monomer equation is not sign-definite is also useful. However, the proof as written contains a concrete algebraic omission: the final limiting equations drop the diffusion coefficient d_i, so the argument actually establishes only the case d_i≡1 rather than Theorem 1.2 with general coefficients satisfying (1.8). Because this issue is load-bearing, the paper needs major revision before the claimed theorem is established.

major comments (3)

[Section 3, Lemma 3.4 and Proof of Theorem 1.2] Definition 1.1 requires the diffusion contribution d_i ∫_0^T∫_Ω ∇ξ·∇f_i dxdt with the coefficient d_i from (1.8). Proposition 2.9 contains the factor d_I in the diffusion bracket, and the proof of Lemma 3.3 also writes the limiting diffusion term with d_I. However, the statements of Lemma 3.3 and Lemma 3.4 omit d_I, and the final passage Λ→∞ in the proof of Theorem 1.2 yields −∫ f_{I,0}ξ(0)−∫ f_I ∂_t ξ−∫ Q_I ξ+∫ ∇f_I·∇ξ = 0. This equation is the heat equation with unit diffusivity, not the weak formulation of (1.1). Since (1.8) permits d_i ≠ 1, Theorem 1.2 as stated is not proved; the factor d_I must be carried through Lemmas 3.3–3.4 and the final limit.
[Section 2, Proposition 2.1 and Theorem 2.3] The uniform L^2 estimates (2.2) and the existence of the truncated weak solution f^N are asserted with only references: Proposition 2.1 says the proof is the same as in [7,13] and omits it, and Theorem 2.3 is proved in one sentence by invoking [27] and Fatou's lemma. These statements are the foundation of every subsequent compactness argument. The authors should either provide the estimates and the ε→0 passage, or state explicitly which theorem of [27] applies to the damped system (2.1) and verify its hypotheses, including the global Lipschitz property of the regularized nonlinearity and the L^2 regularity of the initial data.
[Section 3, Lemma 3.4] The passage η→0+ in Lemma 3.3 is not proved, only described as similar to Lemma 3.3. In particular, the sequence of Radon measures ν^Λ_{I,η} must be shown to have a weak limit ν^Λ_I with total variation tending to zero as Λ→∞; the displayed inequality in Lemma 3.3 only gives that the full expression is O(η^{1/2}) for fixed Λ. Without this step, the existence of the measure in Lemma 3.4 is an assertion rather than a proved limit, and the final compactness argument is incomplete.

minor comments (5)

[Section 1.1] The heading 'Notaion' should be 'Notation'.
[Theorem 1.3] The hypothesis 'F_i^in ∈ L^2((0,T)×Ω)' is stated for initial data; it should be F_i^in ∈ L^2(Ω), as is used in the displayed bound on the same line.
[Theorem 2.3 and equations (2.10a), Lemma 3.2] Theorem 2.3 states 'for every 0≤i<N' but the truncated system (1.11) contains N+1 species, so the bound should be 0≤i≤N. Also, in (2.10a) and in the first estimate of Lemma 3.2 the right-hand side is written with a truncated sum over i=0,...,N; it should be the full sum over i∈N∪{0}.
[Proof of Theorem 1.2] In the calculation of the limit of the second term as Λ→∞, the sentence following the estimate for J_2 again concludes 'J_3=0'; it should conclude J_2=0.
[Section 3, Lemma 3.3] The notation ∫ ν^Λ_{I,η} ξ dxdt is used for a Radon measure; for consistency with Definition 1.1 and Lemma 2.7, the term should be written as ∫ ξ dν^Λ_{I,η}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all load-bearing estimates are imported from external works with independent content.

full rationale

The derivation chain goes: (i) regularization/damping of the truncated finite system (2.1); (ii) a weighted entropy functional E_N with weights Q_j defined recursively from the kernel coefficients (2.5); (iii) a uniform Fisher-information bound (2.9) via an entropy production identity; (iv) a renormalized formulation using the framework of [18]; and (v) passage to the limit N→∞, then η→0 and Λ→∞. The decisive estimates are external to this paper: Theorem 1.3 is taken from [13], the L^2 estimates of Proposition 2.1 follow from [7,13], the entropy production inequality D_N≥0 is imported from [16, Lemma 5], the truncation-to-identity and renormalized-solution machinery comes from [18], and the compactness lemma comes from [4,6,27]. None of these inputs presuppose existence of solutions to the infinite diffusive EDG system; they are results for finite reaction-diffusion systems or for discrete fragmentation-type systems. The self-citations [10,11] by the present authors appear only in auxiliary literature lists and analogous-estimate remarks, not in the load-bearing argument for Theorem 1.2. The entropy condition (1.10) and the weights Q_j are hypotheses and definitions, not consequences of the conclusion. Thus there is no self-definitional step, no fitted input renamed as a prediction, and no uniqueness or ansatz smuggled in through self-citation. The apparent omission of the factor d_I in the final limit of Lemmas 3.4 and the proof of Theorem 1.2 is a correctness concern, not a circularity concern, and does not affect this score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The theorem is supported by four families of hypotheses: admissible kernel bounds, uniformly bounded diffusion, well-prepared initial data, and a set of imported analytic tools (entropy production, parabolic compactness, renormalized solution machinery). No new physical entities are postulated and no constants are fitted to data.

assumptions (7)

domain assumption Kernel bounds (1.6): K_{i,j}=b_i a_j with b_*(i+1) <= b_i <= b^*(i+1), 0 < a_j <= a^*(j+1)^alpha, alpha in (0,1).
Defines the admissible unbounded kernel class; alpha<1 ensures the weighted sum of Q_j is finite and the decay condition (1.7) holds.
domain assumption Uniform diffusion bounds (1.8): d_* <= d_i <= d^* for every species i.
Uniform ellipticity is used in the L2 estimates, the Fisher information bound, and the renormalized equations.
domain assumption Initial data satisfy (1.9)-(1.10): sum_i (i+1) f_i,0 in L2(Omega) and the Q-weighted entropy is finite.
These hypotheses make the entropy functional bounded below and produce the uniform Fisher information bound (2.9).
standard math Entropy production inequality D_N >= 0 for the damped exchange term, imported from [16, Lemma 5].
Used to derive the entropy-entropy dissipation identity (2.7) and the Fisher information bound (2.9).
standard math Quadratic reaction-diffusion estimates and parabolic compactness from [13, 27, 4, 6]: Theorem 1.3, Lemma 2.5, Lemma 3.1.
Provides the L2 duality estimate, the truncation energy estimate, and the compactness of the heat semigroup used in the N to infinity limit.
standard math Renormalized solution framework of Fischer [18], including Lemma 2.7 and the truncation-to-identity functions of Section 2.1.
Supplies the chain-rule identities and measure-correction formalism used for the truncated and limiting systems.
domain assumption Omega is a bounded smooth domain with homogeneous Neumann boundary conditions.
The boundary condition is part of the model and the smoothness is used by the parabolic theory.

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Cite this review

Pith. "Pith review of The Diffusive Exchange Driven Growth Model with unbounded kernels." pith.science (2026). https://pith.science/paper/BA7RRT3T

@misc{pith2026260809229,
  author       = {Pith},
  title        = {Pith review of: The Diffusive Exchange Driven Growth Model with unbounded kernels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BA7RRT3T}},
  note         = {Machine review of arXiv:2608.09229}
}

abstract

We study the discrete diffusive exchange-driven growth (EDG) equations on a bounded smooth domain of arbitrary dimension subject to homogeneous Neumann boundary conditions. The system belongs to the class of infinite systems of semilinear partial differential equations with nonlinear source terms of quadratic type. Global-in-time existence of solutions is established for separable exchange kernels of the form $K_{i,j}=b_i a_j$, where the donor rates exhibit at most linear growth while the receiver rates are sublinear. The analysis is based on a uniform Fisher information estimate obtained from an entropy-entropy dissipation identity. This estimate yields renormalized solutions to a truncated system with finitely many species. A compactness argument then enables passage to the limit in the exchange operator, leading to the existence of global-in-time solutions for the full system.

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Works this paper leans on

35 extracted references · 35 canonical work pages

[27]

M. Pierre. Global existence in reaction-diffusion systems with control of mass: a survey.Milan Journal of Mathematics, 78(2):417–455, 2010

work page 2010
[16]

Esenturk and J

E. Esenturk and J. J. L. Vel´ azquez. Large time behavior of exchange-driven growth.Discrete and Continuous Dynamical Systems, 41(2):747–775, 2021

work page 2021
[18]

J. Fischer. Global existence of renormalized solutions to entropy-dissipating reaction–diffusion sys- tems.Archive for Rational Mechanics and Analysis, 218(1):553–587, 2015

work page 2015
[13]

Desvillettes, K

L. Desvillettes, K. Fellner, M. Pierre, and J. Vovelle. Global existence for quadratic systems of reaction-diffusion.Advanced Nonlinear Studies, 7(3):491–511, 2007

work page 2007
[1]

Amann et al.Linear and quasilinear parabolic problems, volume 1

H. Amann et al.Linear and quasilinear parabolic problems, volume 1. Springer, 1995

work page 1995
[2]

Amann and C

H. Amann and C. Walker. Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion.Journal of Differential Equations, 218(1):159–186, 2005

work page 2005
[3]

Banasiak, W

J. Banasiak, W. Lamb, and Ph. Lauren¸ cot.Analytic methods for coagulation-fragmentation models. CRC Press, 2019

work page 2019
[4]

Baras and M

P. Baras and M. Pierre. Problems paraboliques semi-lineaires avec donnees measures.Applicable Analysis, 18(1-2):111–149, 1984

work page 1984

Show all 35 references

[5]

Ben-Naim and P

E. Ben-Naim and P. L. Krapivsky. Exchange-driven growth.Physical Review E, 68(3):031104, 2003

2003
[6]

Bothe and M

D. Bothe and M. Pierre. Quasi-steady-state approximation for a reaction–diffusion system with fast intermediate.Journal of Mathematical Analysis and Applications, 368(1):120–132, 2010

2010
[7]

Canizo, L

J. Canizo, L. Desvillettes, and K. Fellner. Absence of gelation for models of coagulation-fragmentation with degenerate diffusion.Il Nuovo cimento della Societ` a italiana di fisica. C, 33(1):79, 2010

2010
[8]

J. A. Ca˜ nizo, L. Desvillettes, and K. Fellner. Regularity and mass conservation for dis- crete coagulation-fragmentation equations with diffusion.Annales de l’IHP Analyse non lin´ eaire, 27(2):639–654, 2010

2010
[9]

J. A. Canizo, L. Desvillettes, and K. Fellner. Improved duality estimates and applications to reaction- diffusion equations.Communications in Partial Differential Equations, 39(6):1185–1204, 2014

2014
[10]

S. Das. Existence of solution of a triangular degenerate reaction–diffusion system: S. das.Journal of Evolution Equations, 25(2):51, 2025

2025
[11]

Das and R

S. Das and R. G. Jaiswal. Existence for the discrete nonlinear fragmentation equation with degenerate diffusion.arXiv preprint arXiv:2602.14070, 2026

2026
[12]

Desvillettes and K

L. Desvillettes and K. Fellner. Duality and entropy methods for reversible reaction-diffusion equations with degenerate diffusion.Mathematical Methods in the Applied Sciences, 38(16):3432–3443, 2015

2015
[14]

Eichenberg and A

C. Eichenberg and A. Schlichting. Self-similar behavior of the exchange-driven growth model with product kernel.Communications in Partial Differential Equations, 46(3):498–546, 2021. 33

2021
[15]

Esenturk

E. Esenturk. Mathematical theory of exchange-driven growth.Nonlinearity, 31(8):3460–3483, 2018

2018
[17]

Fellner, J

K. Fellner, J. Morgan, and B. Q. Tang. Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions.Discrete & Continuous Dynamical Systems- Series S, 14(2), 2021

2021
[19]

W. E. Fitzgibbon, J. J. Morgan, B. Q. Tang, and H.-M. Yin. Reaction-diffusion-advection systems with discontinuous diffusion and mass control.SIAM Journal on Mathematical Analysis, 53(6):6771– 6803, 2021

2021
[20]

Ispolatov, P

S. Ispolatov, P. L. Krapivsky, and S. Redner. Wealth distributions in models of capital exchange. The European Physical Journal B, 2:267–276, 1998

1998
[21]

Ke and Z

J. Ke and Z. Lin. Kinetics of migration-driven aggregation processes with birth and death.Physical Review E, 67(3):031103, 2003

2003
[22]

Lauren¸ cot and S

P. Lauren¸ cot and S. Mischler. The continuous coagulation-fragmentation equations with diffusion. Archive for rational mechanics and analysis, 162(1):45–99, 2002

2002
[23]

Lauren¸ cot and S

P. Lauren¸ cot and S. Mischler. Global existence for the discrete diffusive coagulation-fragmentation equations inl 1.Rev. Mat. Iberoamericana, 18(1):731–745, 2002

2002
[24]

Leyvraz and S

F. Leyvraz and S. Redner. Scaling theory for migration-driven aggregate growth.Physical Review Letters, 88(6):068301, 2002

2002
[25]

List and J

R. List and J. R. Gillespie. Evolution of raindrop spectra with collision-induced breakup.Journal of the Atmospheric Sciences, 33(10):2007–2013, 1976

2007
[26]

M. Pierre. Weak solutions and supersolutions in for reaction-diffusion systems.Journal of Evolution Equations, 3(1):153–168, 2003

2003
[28]

Quittner and P

P. Quittner and P. Souplet.Superlinear parabolic problems: blow-up, global existence and steady states. Springer, 2007

2007
[29]

Rothe.Global solutions of reaction-diffusion systems

F. Rothe.Global solutions of reaction-diffusion systems. Springer, 2006

2006
[30]

V. S. Safronov.Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets. Israel Program for Scientific Translations, Jerusalem, 1972. English translation of the 1969 Russian edition

1972
[31]

Schlichting

A. Schlichting. The exchange-driven growth model: Basic properties and longtime behavior.Journal of Nonlinear Science, 30(3):793–830, 2020

2020
[32]

Si and A

S. Si and A. K. Giri. Existence and non-existence for exchange-driven growth model.Nonlinearity, 38:125014, 2025

2025
[33]

C. Walker. The discrete diffusive coagulation–fragmentation equations with scattering.Nonlinear Analysis: Theory, Methods & Applications, 58(1-2):121–142, 2004

2004
[34]

C. Walker. On a new model for continuous coalescence and breakage processes with diffusion.Ad- vances in Differential Equations, 10(2):121 – 152, 2005

2005
[35]

D. Wrzosek. Mass-conserving solutions to the discrete coagulation–fragmentation model with diffu- sion.Nonlinear Analysis: Theory, Methods & Applications, 49(3):297–314, 2002

2002

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [27]

M. Pierre. Global existence in reaction-diffusion systems with control of mass: a survey.Milan Journal of Mathematics, 78(2):417–455, 2010

work page 2010

[2] [16]

Esenturk and J

E. Esenturk and J. J. L. Vel´ azquez. Large time behavior of exchange-driven growth.Discrete and Continuous Dynamical Systems, 41(2):747–775, 2021

work page 2021

[3] [18]

J. Fischer. Global existence of renormalized solutions to entropy-dissipating reaction–diffusion sys- tems.Archive for Rational Mechanics and Analysis, 218(1):553–587, 2015

work page 2015

[4] [13]

Desvillettes, K

L. Desvillettes, K. Fellner, M. Pierre, and J. Vovelle. Global existence for quadratic systems of reaction-diffusion.Advanced Nonlinear Studies, 7(3):491–511, 2007

work page 2007

[5] [1]

Amann et al.Linear and quasilinear parabolic problems, volume 1

H. Amann et al.Linear and quasilinear parabolic problems, volume 1. Springer, 1995

work page 1995

[6] [2]

Amann and C

H. Amann and C. Walker. Local and global strong solutions to continuous coagulation–fragmentation equations with diffusion.Journal of Differential Equations, 218(1):159–186, 2005

work page 2005

[7] [3]

Banasiak, W

J. Banasiak, W. Lamb, and Ph. Lauren¸ cot.Analytic methods for coagulation-fragmentation models. CRC Press, 2019

work page 2019

[8] [4]

Baras and M

P. Baras and M. Pierre. Problems paraboliques semi-lineaires avec donnees measures.Applicable Analysis, 18(1-2):111–149, 1984

work page 1984