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Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients
1. | Dipartimento di Matematica "G. Castelnuovo", Università di Roma "La Sapienza", P.le A. Moro 5, I-00185 Roma, Italy |
2. | Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro 5, I-00185 Roma |
References:
[1] |
C. Bandle, V. Moroz and W. Reichel, "Boundary blowup'' type sub-solutions to semilinear elliptic equations with Hardy potential,, J. London Math. Soc., 77 (2008), 503.
doi: 10.1112/jlms/jdm104. |
[2] |
R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'', Springer-Verlag, (1992).
|
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains,, Comm. Pure Appl. Math., 47 (1994), 47.
doi: 10.1002/cpa.3160470105. |
[4] |
M. Bertsch, R. Dal Passo and R. Van Deer Hout, Nonuniqueness for the heat flow of harmonic maps on the disk,, Arch. Rat. Mech. Anal., 161 (2002), 93.
doi: 10.1007/s002050100171. |
[5] |
S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'', Lecture Notes in Mathematics {\bf 1762}, 1762 (1762).
|
[6] |
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc., 27 (1992), 1.
doi: 10.1090/S0273-0979-1992-00266-5. |
[7] |
S. D. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity,, Asympt. Anal., 22 (2000), 349.
|
[8] |
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, 5 (1956), 1.
|
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964).
|
[10] |
A. Friedman, "Stochastic Differential Equations and Applications,'' I, II,, Probability and Mathematical Statistics, 28 (1976).
|
[11] |
I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'', MIR Publishers, (1980).
|
[12] |
A. Grigoryan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,, Bull. Amer. Math. Soc., 36 (1999), 135.
doi: 10.1090/S0273-0979-99-00776-4. |
[13] |
A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type,, Russian Math. Surveys, 17 (1962), 1.
doi: 10.1070/RM1962v017n03ABEH004115. |
[14] |
H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,, Funkc. Ekv., 38 (1995), 101.
|
[15] |
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Differential Equations, 83 (1990), 26.
doi: 10.1016/0022-0396(90)90068-Z. |
[16] |
S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279.
|
[17] |
S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium,, Comm. Pure Appl. Math., 33 (1981), 831.
doi: 10.1002/cpa.3160340605. |
[18] |
R. Z. Khas'minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain,, Th. Prob. Appl., 4 (1958), 400.
doi: 10.1137/1103033. |
[19] |
L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'', Pure and Applied Mathematics, 283 (2007).
|
[20] |
A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$,, Studia Math., 128 (1998), 171.
|
[21] |
O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'', Amer. Math. Soc., (1973).
|
[22] |
M. A. Pozio, F. Punzo and A. Tesei, Criteria for well-posedness of degenerate elliptic and parabolic problems,, J. Math. Pures Appl., 90 (2008), 353.
doi: 10.1016/j.matpur.2008.06.001. |
[23] |
M. A. Pozio and A. Tesei, On the uniqueness of bounded solutions to singular parabolic problems,, Discr. Cont. Dyn. Syst., 13 (2005), 117.
doi: 10.3934/dcds.2005.13.117. |
[24] |
F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients,, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 26 (2009), 2001.
|
[25] |
F. Punzo and A. Tesei, On the refined maximum principle for degenerate elliptic and parabolic problems,, Nonlinear Anal., 70 (2009), 3047.
doi: 10.1016/j.na.2008.12.032. |
[26] |
F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential,, Rend. Lincei Mat. Appl., 21 (2010), 1. Google Scholar |
[27] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Springer-Verlag, (1984).
|
[28] |
D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associate diffusions,, Comm. Pure Appl. Math., 25 (1972), 651.
doi: 10.1002/cpa.3160250603. |
[29] |
K. Taira, "Diffusion Processes and Partial Differential Equations,'', Academic Press, (1988).
|
show all references
References:
[1] |
C. Bandle, V. Moroz and W. Reichel, "Boundary blowup'' type sub-solutions to semilinear elliptic equations with Hardy potential,, J. London Math. Soc., 77 (2008), 503.
doi: 10.1112/jlms/jdm104. |
[2] |
R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'', Springer-Verlag, (1992).
|
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and the maximum principle for second-order elliptic operators in general domains,, Comm. Pure Appl. Math., 47 (1994), 47.
doi: 10.1002/cpa.3160470105. |
[4] |
M. Bertsch, R. Dal Passo and R. Van Deer Hout, Nonuniqueness for the heat flow of harmonic maps on the disk,, Arch. Rat. Mech. Anal., 161 (2002), 93.
doi: 10.1007/s002050100171. |
[5] |
S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'', Lecture Notes in Mathematics {\bf 1762}, 1762 (1762).
|
[6] |
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc., 27 (1992), 1.
doi: 10.1090/S0273-0979-1992-00266-5. |
[7] |
S. D. Eidelman, S. Kamin and F. Porper, Uniqueness of solutions of the Cauchy problem for parabolic equations degenerating at infinity,, Asympt. Anal., 22 (2000), 349.
|
[8] |
G. Fichera, Sulle equazioni differenziali lineari ellittico-paraboliche del secondo ordine,, Atti Accad. Naz. Lincei. Mem. Cl. Sci. Fis. Mat. Nat. Sez. I, 5 (1956), 1.
|
[9] |
A. Friedman, "Partial Differential Equations of Parabolic Type,'', Prentice-Hall, (1964).
|
[10] |
A. Friedman, "Stochastic Differential Equations and Applications,'' I, II,, Probability and Mathematical Statistics, 28 (1976).
|
[11] |
I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'', MIR Publishers, (1980).
|
[12] |
A. Grigoryan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds,, Bull. Amer. Math. Soc., 36 (1999), 135.
doi: 10.1090/S0273-0979-99-00776-4. |
[13] |
A. M. Il'in, A. S. Kalashnikov and O. A. Oleinik, Linear equations of the second order of parabolic type,, Russian Math. Surveys, 17 (1962), 1.
doi: 10.1070/RM1962v017n03ABEH004115. |
[14] |
H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,, Funkc. Ekv., 38 (1995), 101.
|
[15] |
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations,, J. Differential Equations, 83 (1990), 26.
doi: 10.1016/0022-0396(90)90068-Z. |
[16] |
S. Kamin, R. Kersner and A. Tesei, On the Cauchy problem for a class of parabolic equations with variable density,, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur., 9 (1998), 279.
|
[17] |
S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium,, Comm. Pure Appl. Math., 33 (1981), 831.
doi: 10.1002/cpa.3160340605. |
[18] |
R. Z. Khas'minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain,, Th. Prob. Appl., 4 (1958), 400.
doi: 10.1137/1103033. |
[19] |
L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'', Pure and Applied Mathematics, 283 (2007).
|
[20] |
A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$,, Studia Math., 128 (1998), 171.
|
[21] |
O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'', Amer. Math. Soc., (1973).
|
[22] |
M. A. Pozio, F. Punzo and A. Tesei, Criteria for well-posedness of degenerate elliptic and parabolic problems,, J. Math. Pures Appl., 90 (2008), 353.
doi: 10.1016/j.matpur.2008.06.001. |
[23] |
M. A. Pozio and A. Tesei, On the uniqueness of bounded solutions to singular parabolic problems,, Discr. Cont. Dyn. Syst., 13 (2005), 117.
doi: 10.3934/dcds.2005.13.117. |
[24] |
F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients,, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 26 (2009), 2001.
|
[25] |
F. Punzo and A. Tesei, On the refined maximum principle for degenerate elliptic and parabolic problems,, Nonlinear Anal., 70 (2009), 3047.
doi: 10.1016/j.na.2008.12.032. |
[26] |
F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential,, Rend. Lincei Mat. Appl., 21 (2010), 1. Google Scholar |
[27] |
M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'', Springer-Verlag, (1984).
|
[28] |
D. Stroock and S. R. S. Varadhan, On degenerate elliptic-parabolic operators of second order and their associate diffusions,, Comm. Pure Appl. Math., 25 (1972), 651.
doi: 10.1002/cpa.3160250603. |
[29] |
K. Taira, "Diffusion Processes and Partial Differential Equations,'', Academic Press, (1988).
|
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