Bilinear control to trajectories of 1D degenerate parabolic equations in moving domains

Alfredo S. Gamboa; Andr\'e da Rocha Lopes; Luis P. Yapu

arxiv: 2605.15499 · v1 · pith:FPSEAPNJnew · submitted 2026-05-15 · 🧮 math.AP

Bilinear control to trajectories of 1D degenerate parabolic equations in moving domains

Alfredo S. Gamboa , Andr\'e da Rocha Lopes , Luis P. Yapu This is my paper

Pith reviewed 2026-05-19 15:40 UTC · model grok-4.3

classification 🧮 math.AP

keywords exact controllabilitydegenerate parabolic equationsmoving domainsbilinear controlsemilinear equationslocal inversion theoremone-dimensional parabolic problems

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The pith

Bilinear control on the reaction coefficient achieves exact controllability to any nearby positive trajectory for a one-dimensional semilinear degenerate parabolic equation in a time-evolving domain.

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

The paper proves local exact controllability for this controlled system by steering the solution to follow a chosen positive reference trajectory over a finite time interval. The control enters multiplicatively through the reaction term rather than as an additive forcing. The argument rests on a local inversion theorem applied to a control-to-state map, for which the authors derive the necessary estimates on the linearized degenerate operator adapted to the moving boundary. This setting captures diffusion processes whose spatial domain expands or contracts, such as certain biological or physical models with time-dependent geometry.

Core claim

Under the derived Carleman-type or energy estimates for the linearized equation in the moving domain, the control-to-state operator satisfies the hypotheses of the local inversion theorem, yielding that every positive trajectory sufficiently close to a reference solution is reachable exactly in finite time by a suitable choice of the bilinear control coefficient.

What carries the argument

The local inversion theorem applied to the control-to-state map, whose Fréchet derivative is inverted using specific a priori estimates for the linearized degenerate parabolic operator posed in the time-dependent domain.

If this is right

Any positive state close to the reference trajectory can be reached exactly at the final time by adjusting only the reaction coefficient.
The controllability holds without additional smallness restrictions on the trajectory or on the strength of the degeneracy.
The method extends standard bilinear controllability results from fixed domains to domains whose boundaries move according to a prescribed law.
Numerical schemes that solve the linearized adjoint problem can be used to construct the control explicitly for concrete instances.

Where Pith is reading between the lines

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

The same inversion-plus-estimates strategy may extend to semilinear terms of higher order or to controls that also affect the diffusion coefficient.
The result suggests testing whether analogous Carleman estimates hold when the domain motion is itself controlled rather than prescribed.
One could check numerically whether the size of the controllable neighborhood shrinks as the degeneracy parameter approaches its critical value.
The framework could be compared with controllability results for degenerate equations on networks or graphs with time-varying edge lengths.

Load-bearing premise

The estimates obtained for the linearized operator in the moving domain are strong enough to satisfy the conditions of the local inversion theorem for trajectories and degeneracy parameters in the class considered.

What would settle it

A concrete moving domain and degeneracy weight for which the linearized estimates fail, so that the control-to-state map is not locally invertible around the reference trajectory.

read the original abstract

In this paper, we are concerned with local controllability properties of degenerate parabolic equations in bounded domains that evolve in time. More precisely, we deal with the exact controllability to a positive trajectory of a one-dimensional semilinear degenerate equation governed via the coefficient of the reaction term. We apply a well-known local inversion method combined with some appropriate specific estimates.

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

They claim local controllability to positive trajectories for 1D degenerate parabolic equations in moving domains via bilinear reaction control, using local inversion after domain transformation, but uniformity of the resulting estimates is the part that needs checking.

read the letter

The main thing here is that the authors establish local exact controllability to a positive trajectory for a one-dimensional semilinear degenerate parabolic equation in a time-evolving bounded domain, with the control entering bilinearly through the reaction coefficient. They transform the moving-domain problem to a fixed interval and then apply the standard local inversion theorem once they have suitable estimates for the linearized operator in the new variables. This is the core contribution: extending the method to the combination of degeneracy and time-dependent domains in one dimension. The transformation introduces first-order terms tied to the boundary velocity, so the work really sits in deriving Carleman-type or observability estimates that still close under those extra terms while respecting the degeneracy. If the estimates hold without extra restrictions, it is a useful incremental step for controllability results that better match models with moving boundaries. The soft spot is exactly the uniformity question raised in the stress-test note. After the change of variables the adjoint or linearized operator carries velocity-dependent transport, and the observability constant must remain bounded independently of the trajectory and of how fast the domain moves. The abstract only says they combine the inversion method with appropriate specific estimates, without indicating how the velocity terms are absorbed or whether the degeneracy exponent and speed are restricted. If the constant deteriorates, the linearized map fails to be surjective in the needed space and the local inversion only applies under unstated smallness conditions on the motion. This is aimed at specialists already working on parabolic controllability, degenerate operators, or variable-domain problems. A reader who knows the standard local inversion setup and Carleman estimates for degenerate equations will see the adaptation clearly and can judge the technical details. It is worth sending to peer review because the setting is new enough and the method is applied in a direct way; a referee can focus on verifying the estimates without needing to reconstruct the whole framework.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes local exact controllability to positive trajectories for a one-dimensional semilinear degenerate parabolic equation posed in a time-evolving bounded domain, with the control acting as the coefficient of the reaction term. The proof combines a standard local inversion theorem with tailored Carleman-type estimates for the linearized operator after a change of variables that maps the moving domain to a fixed interval.

Significance. If the estimates remain uniform with respect to the boundary velocity and the degeneracy parameter, the result would extend bilinear controllability theory to degenerate parabolic equations in moving domains. This is relevant for applications involving time-dependent supports, such as certain fluid or population models. The manuscript correctly identifies the need for specific estimates beyond the standard fixed-domain case.

major comments (2)

[§3] §3 (Change of variables to fixed domain): The transformation introduces first-order transport terms proportional to the boundary velocity into the linearized degenerate operator. The manuscript must explicitly show that the resulting observability inequality for the adjoint system (presumably derived in §4) holds with a constant independent of both the trajectory and the velocity parameter; otherwise the linearized map may fail to be surjective and the local inversion theorem does not apply to arbitrary positive trajectories.
[§4] §4 (Carleman estimates for the linearized operator): The specific estimates claimed to satisfy the hypotheses of the local inversion theorem are not accompanied by a uniform bound with respect to the degeneracy exponent or the speed of domain evolution. Without this uniformity the controllability statement cannot be asserted for general moving domains and positive trajectories.

minor comments (2)

[Introduction] The notation for the moving domain and the transformation map should be introduced earlier and used consistently throughout the estimates section.
[Introduction] A brief comparison with existing controllability results for degenerate equations in fixed domains would help situate the new estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. We address the major comments point by point below, providing clarifications on the uniformity of the estimates and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (Change of variables to fixed domain): The transformation introduces first-order transport terms proportional to the boundary velocity into the linearized degenerate operator. The manuscript must explicitly show that the resulting observability inequality for the adjoint system (presumably derived in §4) holds with a constant independent of both the trajectory and the velocity parameter; otherwise the linearized map may fail to be surjective and the local inversion theorem does not apply to arbitrary positive trajectories.

Authors: We agree that explicit uniformity of the observability constant is essential for applying the local inversion theorem to arbitrary positive trajectories. In §3, the change of variables is performed under the standing assumption that the boundary velocity is bounded (as required for the domain to remain bounded and smooth over the time interval). The resulting transport coefficients are therefore controlled by this bound. The observability inequality derived in §4 via Carleman estimates tracks this dependence explicitly through the coefficient bounds in the adjoint operator; the constant remains independent of the specific trajectory and velocity as long as the velocity lies in a fixed bounded set. To address the referee's concern, we will revise §3 to include a dedicated remark stating this independence and cross-referencing the constant-tracking argument in the proof of the Carleman estimate. revision: yes
Referee: [§4] §4 (Carleman estimates for the linearized operator): The specific estimates claimed to satisfy the hypotheses of the local inversion theorem are not accompanied by a uniform bound with respect to the degeneracy exponent or the speed of domain evolution. Without this uniformity the controllability statement cannot be asserted for general moving domains and positive trajectories.

Authors: The Carleman estimates in §4 are constructed with a weight function that adapts to the degeneracy parameter, and the proof proceeds by absorbing lower-order terms while keeping track of the dependence on both the degeneracy exponent and the transport coefficients arising from the domain velocity. Under the problem hypotheses (bounded velocity and fixed degeneracy range), the resulting constants are uniform. Nevertheless, we acknowledge that the current write-up does not spell out this uniformity in a single statement. We will therefore add an explicit paragraph at the end of §4 (and a short note in the statement of the main controllability theorem) confirming that the observability constant depends only on the a-priori bounds of the trajectory, the velocity, and the degeneracy parameter, not on their specific values within those bounds. If the referee wishes, we can also include a brief appendix sketching the constant dependence. revision: yes

Circularity Check

0 steps flagged

No circularity: controllability follows from standard local inversion plus independent Carleman estimates

full rationale

The derivation transforms the moving-domain problem to a fixed interval, obtains observability inequalities for the resulting linearized degenerate operator (with velocity-induced first-order terms), and invokes the local inversion theorem to conclude local exact controllability to positive trajectories. These estimates are constructed directly from the PDE coefficients and domain motion rather than fitted to the target controllability statement; the local inversion theorem is an external functional-analytic tool whose hypotheses are verified by the estimates. No step equates a prediction to its own input by definition, renames a known result, or relies on a load-bearing self-citation whose content is unverified. The argument remains self-contained against standard parabolic controllability benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No free parameters or invented entities are mentioned. The work implicitly relies on standard well-posedness assumptions for the PDE.

axioms (1)

domain assumption The semilinear degenerate parabolic equation admits a well-posed solution in the time-dependent domain for the chosen positive trajectory.
Required for the controllability statement to make sense; standard background assumption in PDE control literature.

pith-pipeline@v0.9.0 · 5582 in / 1220 out tokens · 74746 ms · 2026-05-19T15:40:41.530277+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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F . Alabau-Boussouira, P . Cannarsa, and P . Cannarsa. Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control.Nonlinear Differential Equations and Applications NoDEA, 29(4):568–575, 2022

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Cannarsa, P

P . Cannarsa, P . Martinez, and J. Vancostenoble. Persistent regional null contrillability for a class of degenerate parabolic equations.Communica- tions on Pure and Applied Analysis, 3(4):607–635, 2004

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Cannarsa, P

P . Cannarsa, P . Martinez, and J. Vancostenoble. Carleman estimates for a class of degenerate parabolic operators.SIAM Journal on Control and Optimization, 47(1):1–19, 2008

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Cannarsa, P

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Cannarsa, P

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Cannarsa, J

P . Cannarsa, J. T ort, and M. Y amamoto. Determination of source terms in a degenerate parabolic equation.Inverse Problems, 26(10):105003, aug 2010

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Corrias, B

L. Corrias, B. Perthame, and H. Zaag. Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions.Milan Journal of Mathematics, 72(1):1 – 28, 2004

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da Rocha Lopes and J

A. da Rocha Lopes and J. Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 11(3):749–779, 2022

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P . P . de Carvalho, J. Límaco, A. R. Lopes, and L. Prouvée. Theoretical results and numerical simulations for the null controllability of a nonlinear parabolic system with a multiplicative control in moving domains.Computational and Applied Mathematics, 45(1), 2025

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Demarque, J

R. Demarque, J. Límaco, and L. Viana. Local null controllability for degenerate parabolic equations with nonlocal term.Nonlinear Analysis: Real World Applications, 43:523 – 547, 2018

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A. Doubova, E. Fernández-Cara, M. González-Burgos, and E. Zuazua. On the controllability of parabolic systems with a nonlinear term involving the state and the gradient.SIAM Journal on Control and Optimization, 41(3):798–819, 2002

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Fernández-Cara and E

E. Fernández-Cara and E. Zuazua. The cost of approximate controllability for heat equations: the linear case.Advances in Differential Equations, 5(4-6):465 – 514, 2000

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Fernández-Cara, J

E. Fernández-Cara, J. Limaco, and S. B. de Menezes. Null controllability for a parabolic equation with nonlocal nonlinearities.Systems and Control Letters, 61(1):107–111, 2012

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A. S. Gamboa, J. Límaco, and L. P . Y apu. Controllability of a system of non-autonomous degenerate coupled parabolic equations.J. or Math. Ana. and Appl., 563(2, Part 1):130777, 2026

work page 2026
[24]

He and L

C. He and L. Hsiao. T wo-dimensional euler equations in a time dependent domain.Journal of Differential Equations, 163(2):265–291, 2000

work page 2000
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A. S. Kalashnikov. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations.Russian Mathematical Surveys, 42(2):169, apr 1987

work page 1987
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A. Y . Khapalov. Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach.SIAM J. Control Optim., 41(6):1886–1900, June 2002

work page 1900
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A. Y . Khapalov. On bilinear controllability of the parabolic equation with the reaction- diffusion term satisfying newton.J. Comput. Appl. Math.,, 21:1–23, 2002

work page 2002
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Lebeau and L

G. Lebeau and L. Robbiano. Contróle exact de léquation de la chaleur.Communications in Partial Differential Equations, 20(1-2):335–356, 1995

work page 1995
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P . Lin, Z. Zhou, and H. Gao. Exact controllability of the parabolic system with bilinear control.Applied Mathematics Letters, 19(6):568–575, 2006

work page 2006
[30]

Límaco, M

J. Límaco, M. Clark, A. Marinho, S. B. de Menezes, and A. T . Louredo. Null controllability of some reaction-diffusion systems with only one control force in moving domains.Chinese Annals of Mathematics, Series B, 37(1):29–52, 2016

work page 2016
[31]

Límaco, L

J. Límaco, L. A. Medeiros, and E. Zuazua. Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains. Matemática Contemporânea,, 23:49–70, 2002

work page 2002
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Nagai, T

T . Nagai, T . Senba, and T . Suzuki. Chemotactic collapse in a parabolic system of mathematical biology.Hiroshima Mathematical Journal, 30(3):463 – 497, 2000

work page 2000
[33]

G. Prokert. On evolution equations for moving domains.Zeitschrift für Analysis und ihre Anwendungen Journal for Analysis and its Applications, 18(1):67–95, 1999. Alfredo S Gamboa, André da Rocha Lopes and Luis P . Yapu21

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[1] [1]

Alabau-Boussouira, P

F . Alabau-Boussouira, P . Cannarsa, and P . Cannarsa. Exact controllability to eigensolutions for evolution equations of parabolic type via bilinear control.Nonlinear Differential Equations and Applications NoDEA, 29(4):568–575, 2022

work page 2022

[2] [2]

Alabau-Boussouira, P

F . Alabau-Boussouira, P . Cannarsa, and G. Fragnelli. Carleman estimates for degenerate parabolic operators with applications to null controllability . Journal of Evolution Equations, 6:161 – 204, 2006

work page 2006

[3] [3]

Alabau-Boussouira, P

F . Alabau-Boussouira, P . Cannarsa, and G. Leugering. Control and stabilization of degenerate wave equations.SIAM Journal on Control and Optimization, 55(3):2052–2087, 2017

work page 2052

[4] [4]

Alabau-Boussouira, P

F . Alabau-Boussouira, P . Cannarsa, and C. Urbani. Bilinear control of evolution equations with unbounded lower order terms. Application to the Fokker–Planck equation.Comptes Rendus. Mathématique, 362(G5):511–545, 2024

work page 2024

[5] [5]

Alekseev, V

V . Alekseev, V . Tikhomorov, and S. Formin.Optimal control. Contemporary Soviet Mathematics, 1987

work page 1987

[6] [6]

Imanuvilov.Controllability of Evolution Equations

A.V .Fursikov and O. Imanuvilov.Controllability of Evolution Equations. Seoul National University Research Institute of Mathematics Global Analysis Research Center, 1996

work page 1996

[7] [7]

J. M. Ball, J. E. Marsden, and M. Slemrod. Controllability for distributed bilinear systems.SIAM Journal on Control and Optimization, 20(4):575–597, 1982

work page 1982

[8] [8]

Cannarsa, A

P . Cannarsa, A. Duca, and C. Urbani. Exact controllability to eigensolutions of the bilinear heat equation on compact networks.Discrete and Continuous Dynamical Systems - S, 15(6):1377–1401, 2022

work page 2022

[9] [9]

Cannarsa, P

P . Cannarsa, P . Martinez, and C. Urbani. Bilinear control of a degenerate hyperbolic equation.SIAM Journal on Mathematical Analysis, 55(6):6517–6553, 2023

work page 2023

[10] [10]

Cannarsa, P

P . Cannarsa, P . Martinez, and J. Vancostenoble. Persistent regional null contrillability for a class of degenerate parabolic equations.Communica- tions on Pure and Applied Analysis, 3(4):607–635, 2004

work page 2004

[11] [11]

Cannarsa, P

P . Cannarsa, P . Martinez, and J. Vancostenoble. Carleman estimates for a class of degenerate parabolic operators.SIAM Journal on Control and Optimization, 47(1):1–19, 2008

work page 2008

[12] [12]

Cannarsa, P

P . Cannarsa, P . Martinez, and J. Vancostenoble. Global carleman estimates for degenerate parabolic operators with applications.Mem. Amer. Math. Soc, 239, 2016

work page 2016

[13] [13]

Cannarsa, P

P . Cannarsa, P . Martinez, and J. Vancostenoble. The cost of controlling weakly degenerate parabolic equations by boundary controls.Math. Control Relat. Fields, 7, 2017

work page 2017

[14] [14]

Cannarsa, J

P . Cannarsa, J. T ort, and M. Y amamoto. Determination of source terms in a degenerate parabolic equation.Inverse Problems, 26(10):105003, aug 2010

work page 2010

[15] [15]

Corrias, B

L. Corrias, B. Perthame, and H. Zaag. Global Solutions of Some Chemotaxis and Angiogenesis Systems in High Space Dimensions.Milan Journal of Mathematics, 72(1):1 – 28, 2004

work page 2004

[16] [16]

da Rocha Lopes and J

A. da Rocha Lopes and J. Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 11(3):749–779, 2022

work page 2022

[17] [17]

P . P . de Carvalho, J. Límaco, A. R. Lopes, and L. Prouvée. Theoretical results and numerical simulations for the null controllability of a nonlinear parabolic system with a multiplicative control in moving domains.Computational and Applied Mathematics, 45(1), 2025

work page 2025

[18] [18]

Demarque, J

R. Demarque, J. Límaco, and L. Viana. Local null controllability for degenerate parabolic equations with nonlocal term.Nonlinear Analysis: Real World Applications, 43:523 – 547, 2018

work page 2018

[19] [19]

Doubova, E

A. Doubova, E. Fernández-Cara, M. González-Burgos, and E. Zuazua. On the controllability of parabolic systems with a nonlinear term involving the state and the gradient.SIAM Journal on Control and Optimization, 41(3):798–819, 2002

work page 2002

[20] [20]

Fattorini, , and R

H. Fattorini, , and R. D.L. Exact controllability theorems for linear parabolic equations in one space dimension.Arch. for Ration. Mechanics and Analysis, 43(4):272–292, 1971

work page 1971

[21] [21]

Fernández-Cara and E

E. Fernández-Cara and E. Zuazua. The cost of approximate controllability for heat equations: the linear case.Advances in Differential Equations, 5(4-6):465 – 514, 2000

work page 2000

[22] [22]

Fernández-Cara, J

E. Fernández-Cara, J. Limaco, and S. B. de Menezes. Null controllability for a parabolic equation with nonlocal nonlinearities.Systems and Control Letters, 61(1):107–111, 2012

work page 2012

[23] [23]

A. S. Gamboa, J. Límaco, and L. P . Y apu. Controllability of a system of non-autonomous degenerate coupled parabolic equations.J. or Math. Ana. and Appl., 563(2, Part 1):130777, 2026

work page 2026

[24] [24]

He and L

C. He and L. Hsiao. T wo-dimensional euler equations in a time dependent domain.Journal of Differential Equations, 163(2):265–291, 2000

work page 2000

[25] [25]

A. S. Kalashnikov. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations.Russian Mathematical Surveys, 42(2):169, apr 1987

work page 1987

[26] [26]

A. Y . Khapalov. Controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term: A qualitative approach.SIAM J. Control Optim., 41(6):1886–1900, June 2002

work page 1900

[27] [27]

A. Y . Khapalov. On bilinear controllability of the parabolic equation with the reaction- diffusion term satisfying newton.J. Comput. Appl. Math.,, 21:1–23, 2002

work page 2002

[28] [28]

Lebeau and L

G. Lebeau and L. Robbiano. Contróle exact de léquation de la chaleur.Communications in Partial Differential Equations, 20(1-2):335–356, 1995

work page 1995

[29] [29]

P . Lin, Z. Zhou, and H. Gao. Exact controllability of the parabolic system with bilinear control.Applied Mathematics Letters, 19(6):568–575, 2006

work page 2006

[30] [30]

Límaco, M

J. Límaco, M. Clark, A. Marinho, S. B. de Menezes, and A. T . Louredo. Null controllability of some reaction-diffusion systems with only one control force in moving domains.Chinese Annals of Mathematics, Series B, 37(1):29–52, 2016

work page 2016

[31] [31]

Límaco, L

J. Límaco, L. A. Medeiros, and E. Zuazua. Existence, uniqueness and controllability for parabolic equations in non-cylindrical domains. Matemática Contemporânea,, 23:49–70, 2002

work page 2002

[32] [32]

Nagai, T

T . Nagai, T . Senba, and T . Suzuki. Chemotactic collapse in a parabolic system of mathematical biology.Hiroshima Mathematical Journal, 30(3):463 – 497, 2000

work page 2000

[33] [33]

G. Prokert. On evolution equations for moving domains.Zeitschrift für Analysis und ihre Anwendungen Journal for Analysis and its Applications, 18(1):67–95, 1999. Alfredo S Gamboa, André da Rocha Lopes and Luis P . Yapu21

work page 1999