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Pointwise properties of $ L^p $-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms
General initial data for a class of parabolic equations including the curve shortening problem
1. | The Chinese University of Hong Kong, Hong Kong |
2. | Northern Illinois University, DeKalb, IL60115, U.S.A |
The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When $ m\in(1,2] $, it is shown that there exists a smooth solution for $ t>0 $ when the initial data belongs to $ L_{\text{loc}}^p, p>1 $. When $ m>2 $, the same results holds when the initial data belongs to $ W_{\text{loc}}^{1,p}, p\geq m-1 $. An example is displayed to show that a smooth solution may not exist when the initial data is merely in $ L^p_{\text{loc}}, p>1 $. Solvability of the weak solution is also studied.
References:
[1] |
S. Angenent,
Parabolic equations for curves on surfaces. I. Curves with p-integrable curvature, Ann. of Math., 132 (1990), 451-483.
doi: 10.2307/1971426. |
[2] |
S. Angenent,
Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math., 133 (1991), 171-215.
doi: 10.2307/2944327. |
[3] |
D. G. Aronson,
Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 25 (1971), 221-228.
|
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D. G. Aronson and L. A. Caffarelli,
The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc, 280 (1983), 351-366.
doi: 10.1090/S0002-9947-1983-0712265-1. |
[5] |
Ph. Benilan, M. G. Crandall and M. Pierre,
Solutions of the porous medium equationin $\mathbb R^n$ under optimal conditions on initial values, Indiana Univ. Math J., 33 (1984), 51-87.
doi: 10.1512/iumj.1984.33.33003. |
[6] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
[7] |
K. S. Chou and Y. C. Kwong,
On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth, Calc. Var. Partial Differential Equations, 12 (2001), 281-315.
doi: 10.1007/PL00009915. |
[8] |
K. S. Chou and X. P. Zhu, The Curve Shortening Problem, Chapman and Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420035704. |
[9] |
B. E. Dahlberg and C. E. Kenig,
Non-negative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Soc., 1 (1988), 401-412.
doi: 10.1090/S0894-0347-1988-0928264-9. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[11] |
K. Ecker,
An interior gradient bound for solutions of equations of capillary type, Arch. Rational Mech. Anal., 92 (1986), 137-151.
doi: 10.1007/BF00251254. |
[12] |
K. Ecker and G. Huisken,
Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991), 547-569.
doi: 10.1007/BF01232278. |
[13] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
[14] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature, II, Trans. Amer. Math. Soc., 330 (1992), 321-332.
doi: 10.1090/S0002-9947-1992-1068927-8. |
[15] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature, III, J. Geom. Anal., 2 (1992), 121-150.
doi: 10.1007/BF02921385. |
[16] |
A. Friedman, Partial Differential Equations, Holt, Rinehart, Winston, 1969. |
[17] |
M. Gage and R. S. Hamilton,
The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
doi: 10.4310/jdg/1214439902. |
[18] |
C. Gerhardt,
Evolutionary surfaces of prescribed mean curvature, J. Differential Equations, 36 (1980), 139-172.
doi: 10.1016/0022-0396(80)90081-9. |
[19] |
Y. Giga,
Interior derivative blow-up for quailinear parabolic equations, Discrete Contin. Dynam. Systems, 1 (1995), 449-461.
doi: 10.3934/dcds.1995.1.449. |
[20] |
M. A. Grayson,
The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
doi: 10.4310/jdg/1214441371. |
[21] |
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, , vol. 23, Translations of Mathematical Monographs, American Mathematical Society, Providence, 1968. |
[22] |
G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009.
doi: 10.1090/gsm/105. |
[23] |
X.-J. Wang,
Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.
doi: 10.1007/PL00004604. |
[24] |
D. V. Widder,
Positive temperatures on an infintie rod, Trans. Amer. Math. Soc., 55 (1944), 85-95.
doi: 10.1090/S0002-9947-1944-0009795-2. |
[25] |
Z. Wu, J. Yin, H. Li and J. Zhao, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001.
doi: 10.1142/4782. |
[26] |
J. Zhao,
On the Cauchy problem and initial traces for evolution p-Laplacian equation with strongly nonlinear sources, J. Differential Equations, 121 (1995), 329-383.
doi: 10.1006/jdeq.1995.1132. |
show all references
References:
[1] |
S. Angenent,
Parabolic equations for curves on surfaces. I. Curves with p-integrable curvature, Ann. of Math., 132 (1990), 451-483.
doi: 10.2307/1971426. |
[2] |
S. Angenent,
Parabolic equations for curves on surfaces. II. Intersections, blow-up and generalized solutions, Ann. of Math., 133 (1991), 171-215.
doi: 10.2307/2944327. |
[3] |
D. G. Aronson,
Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa, 25 (1971), 221-228.
|
[4] |
D. G. Aronson and L. A. Caffarelli,
The initial trace of a solution of the porous medium equation, Trans. Amer. Math. Soc, 280 (1983), 351-366.
doi: 10.1090/S0002-9947-1983-0712265-1. |
[5] |
Ph. Benilan, M. G. Crandall and M. Pierre,
Solutions of the porous medium equationin $\mathbb R^n$ under optimal conditions on initial values, Indiana Univ. Math J., 33 (1984), 51-87.
doi: 10.1512/iumj.1984.33.33003. |
[6] |
Y. G. Chen, Y. Giga and S. Goto,
Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom., 33 (1991), 749-786.
doi: 10.4310/jdg/1214446564. |
[7] |
K. S. Chou and Y. C. Kwong,
On quasilinear parabolic equations which admit global solutions for initial data with unrestricted growth, Calc. Var. Partial Differential Equations, 12 (2001), 281-315.
doi: 10.1007/PL00009915. |
[8] |
K. S. Chou and X. P. Zhu, The Curve Shortening Problem, Chapman and Hall/CRC, Boca Raton, FL, 2001.
doi: 10.1201/9781420035704. |
[9] |
B. E. Dahlberg and C. E. Kenig,
Non-negative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Soc., 1 (1988), 401-412.
doi: 10.1090/S0894-0347-1988-0928264-9. |
[10] |
E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-0895-2. |
[11] |
K. Ecker,
An interior gradient bound for solutions of equations of capillary type, Arch. Rational Mech. Anal., 92 (1986), 137-151.
doi: 10.1007/BF00251254. |
[12] |
K. Ecker and G. Huisken,
Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., 105 (1991), 547-569.
doi: 10.1007/BF01232278. |
[13] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature, I, J. Differential Geom., 33 (1991), 635-681.
doi: 10.4310/jdg/1214446559. |
[14] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature, II, Trans. Amer. Math. Soc., 330 (1992), 321-332.
doi: 10.1090/S0002-9947-1992-1068927-8. |
[15] |
L. C. Evans and J. Spruck,
Motion of level sets by mean curvature, III, J. Geom. Anal., 2 (1992), 121-150.
doi: 10.1007/BF02921385. |
[16] |
A. Friedman, Partial Differential Equations, Holt, Rinehart, Winston, 1969. |
[17] |
M. Gage and R. S. Hamilton,
The heat equation shrinking convex plane curves, J. Differential Geom., 23 (1986), 69-96.
doi: 10.4310/jdg/1214439902. |
[18] |
C. Gerhardt,
Evolutionary surfaces of prescribed mean curvature, J. Differential Equations, 36 (1980), 139-172.
doi: 10.1016/0022-0396(80)90081-9. |
[19] |
Y. Giga,
Interior derivative blow-up for quailinear parabolic equations, Discrete Contin. Dynam. Systems, 1 (1995), 449-461.
doi: 10.3934/dcds.1995.1.449. |
[20] |
M. A. Grayson,
The heat equation shrinks embedded plane curves to round points, J. Differential Geom., 26 (1987), 285-314.
doi: 10.4310/jdg/1214441371. |
[21] |
O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, , vol. 23, Translations of Mathematical Monographs, American Mathematical Society, Providence, 1968. |
[22] |
G. Leoni, A First Course in Sobolev Spaces, American Mathematical Society, Providence, 2009.
doi: 10.1090/gsm/105. |
[23] |
X.-J. Wang,
Interior gradient estimates for mean curvature equations, Math. Z., 228 (1998), 73-81.
doi: 10.1007/PL00004604. |
[24] |
D. V. Widder,
Positive temperatures on an infintie rod, Trans. Amer. Math. Soc., 55 (1944), 85-95.
doi: 10.1090/S0002-9947-1944-0009795-2. |
[25] |
Z. Wu, J. Yin, H. Li and J. Zhao, Nonlinear Diffusion Equations, World Scientific, Singapore, 2001.
doi: 10.1142/4782. |
[26] |
J. Zhao,
On the Cauchy problem and initial traces for evolution p-Laplacian equation with strongly nonlinear sources, J. Differential Equations, 121 (1995), 329-383.
doi: 10.1006/jdeq.1995.1132. |
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