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March  2021, 20(3): 1213-1227. doi: 10.3934/cpaa.2021017

Existence of solution and asymptotic behavior for a class of parabolic equations

Anderson L. A. de Araujo 1, and  Marcelo Montenegro 2,,

1. 

Universidade Federal de Viçosa, Departamento de Matemática, Avenida Peter Henry Rolfs, s/n, CEP 36570-900, Viçosa, MG, Brasil
2. 

Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651, CEP 13083-859, Campinas, SP, Brasil

* Corresponding author

Received  July 2020 Revised  December 2020 Published  March 2021 Early access  February 2021

Fund Project: The authors have been supported by FAPESP and CNPq

We prove existence and uniqueness of a positive solution for a class of quasilinear parabolic equations. We also show some maximum principles on the derivatives of the solution and study the asymptotic behavior of the solution near the maximal time of existence.

Keywords: Parabolic equation, existence of solution, uniqueness, maximum principle, asymptotic behavior.
Mathematics Subject Classification: Primary: 35K10, 35B50; Secondary: 35K90, 35K67.
Citation: Anderson L. A. de Araujo, Marcelo Montenegro. Existence of solution and asymptotic behavior for a class of parabolic equations. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1213-1227. doi: 10.3934/cpaa.2021017
References:
[1]

S. Altschuler, S. B. Angenent and Y. Giga, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358.  doi: 10.1007/BF02921800.  Google Scholar

[2]

S. Angenent, Parabolic equations for curves on surfaces: part I. Curves with $p$–integrable curvature, Ann. Math., 132 (1990), 451-483.  doi: 10.2307/1971426.  Google Scholar

[3]

M. Athanassenas, Behaviour of singularities of the rotationally symmetric, volume–preserving mean curvature flow, Calc. Var. PDE, 17 (2003), 1-16.  doi: 10.1007/s00526-002-0098-4.  Google Scholar

[4] K. A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 2015.   Google Scholar
[5]

J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. AMS, 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.  Google Scholar

[6]

J. Escher and B. V. Matioc, Neck pinching for periodic mean curvature flows, Analysis, 30 (2010), 253-260.  doi: 10.1524/anly.2010.1039.  Google Scholar

[7]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differ. Geom., 33 (1991), 635-681.   Google Scholar

[8]

M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96.   Google Scholar

[9]

Z. Gang and I. M. Sigal, Neck pinching dynamics under mean curvature flow, J. Geom. Anal., 19 (2009), 36-80.  doi: 10.1007/s12220-008-9050-y.  Google Scholar

[10]

Y. Giga, Y. Seki and N. Umeda, Mean curvature flow, closes open ends of noncompact surfaces of rotation, Comm. Part. Diff. Eq., 34 (2009), 1508-1529.  doi: 10.1080/03605300903296926.  Google Scholar

[11]

M. A. Grayson, The shape of afigure eight under the curve shortening flow, Invent. Math., 96 (1989), 177-180.  doi: 10.1007/BF01393973.  Google Scholar

[12]

G. Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differ. Equ., 77 (1989), 369-378.  doi: 10.1016/0022-0396(89)90149-6.  Google Scholar

[13]

G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differ. Geom., 31 (1990), 285-299.   Google Scholar

[14]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946.  Google Scholar

[15]

I. Kim and D. Kwon, On mean curvature flow with forcing, Commun. Partial Differ. Equ., 45 (2020), 414-455.  doi: 10.1080/03605302.2019.1695262.  Google Scholar

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 2005. doi: 10.1142/3302.  Google Scholar

[17]

B. V. Matioc, value problems for rotationally symmetric mean curvature flows, Arch. Math., 89 (2007), 365-372.  doi: 10.1007/s00013-007-2141-3.  Google Scholar

[18]

J. A. McCoy, F. Y. Y. Mofarreh and G. H. Williams, Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions, Ann. Mat. Pura Appl., 193 (2014), 1443-1455.  doi: 10.1007/s10231-013-0337-7.  Google Scholar

[19]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, 1992.  Google Scholar

[20]

K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Manuscr. Math., 95 (1998), 225-236.  doi: 10.1007/s002290050025.  Google Scholar

[21]

H. M. Soner and P. E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Commun. Partial Differ. Equ., 18 (1993), 859-894.  doi: 10.1080/03605309308820954.  Google Scholar

show all references

References:
[1]

S. Altschuler, S. B. Angenent and Y. Giga, Mean curvature flow through singularities for surfaces of rotation, J. Geom. Anal., 5 (1995), 293-358.  doi: 10.1007/BF02921800.  Google Scholar

[2]

S. Angenent, Parabolic equations for curves on surfaces: part I. Curves with $p$–integrable curvature, Ann. Math., 132 (1990), 451-483.  doi: 10.2307/1971426.  Google Scholar

[3]

M. Athanassenas, Behaviour of singularities of the rotationally symmetric, volume–preserving mean curvature flow, Calc. Var. PDE, 17 (2003), 1-16.  doi: 10.1007/s00526-002-0098-4.  Google Scholar

[4] K. A. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 2015.   Google Scholar
[5]

J. Escher and G. Simonett, The volume preserving mean curvature flow near spheres, Proc. AMS, 126 (1998), 2789-2796.  doi: 10.1090/S0002-9939-98-04727-3.  Google Scholar

[6]

J. Escher and B. V. Matioc, Neck pinching for periodic mean curvature flows, Analysis, 30 (2010), 253-260.  doi: 10.1524/anly.2010.1039.  Google Scholar

[7]

L. C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Differ. Geom., 33 (1991), 635-681.   Google Scholar

[8]

M. E. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96.   Google Scholar

[9]

Z. Gang and I. M. Sigal, Neck pinching dynamics under mean curvature flow, J. Geom. Anal., 19 (2009), 36-80.  doi: 10.1007/s12220-008-9050-y.  Google Scholar

[10]

Y. Giga, Y. Seki and N. Umeda, Mean curvature flow, closes open ends of noncompact surfaces of rotation, Comm. Part. Diff. Eq., 34 (2009), 1508-1529.  doi: 10.1080/03605300903296926.  Google Scholar

[11]

M. A. Grayson, The shape of afigure eight under the curve shortening flow, Invent. Math., 96 (1989), 177-180.  doi: 10.1007/BF01393973.  Google Scholar

[12]

G. Huisken, Nonparametric mean curvature evolution with boundary conditions, J. Differ. Equ., 77 (1989), 369-378.  doi: 10.1016/0022-0396(89)90149-6.  Google Scholar

[13]

G. Huisken, Asymptotic behaviour for singularities of the mean curvature flow, J. Differ. Geom., 31 (1990), 285-299.   Google Scholar

[14]

G. Huisken and C. Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math., 183 (1999), 45-70.  doi: 10.1007/BF02392946.  Google Scholar

[15]

I. Kim and D. Kwon, On mean curvature flow with forcing, Commun. Partial Differ. Equ., 45 (2020), 414-455.  doi: 10.1080/03605302.2019.1695262.  Google Scholar

[16]

G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific, 2005. doi: 10.1142/3302.  Google Scholar

[17]

B. V. Matioc, value problems for rotationally symmetric mean curvature flows, Arch. Math., 89 (2007), 365-372.  doi: 10.1007/s00013-007-2141-3.  Google Scholar

[18]

J. A. McCoy, F. Y. Y. Mofarreh and G. H. Williams, Fully nonlinear curvature flow of axially symmetric hypersurfaces with boundary conditions, Ann. Mat. Pura Appl., 193 (2014), 1443-1455.  doi: 10.1007/s10231-013-0337-7.  Google Scholar

[19]

C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, 1992.  Google Scholar

[20]

K. Smoczyk, Starshaped hypersurfaces and the mean curvature flow, Manuscr. Math., 95 (1998), 225-236.  doi: 10.1007/s002290050025.  Google Scholar

[21]

H. M. Soner and P. E. Souganidis, Singularities and uniqueness of cylindrically symmetric surfaces moving by mean curvature, Commun. Partial Differ. Equ., 18 (1993), 859-894.  doi: 10.1080/03605309308820954.  Google Scholar

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