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August  2011, 30(3): 891-916. doi: 10.3934/dcds.2011.30.891

Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients

Maria Assunta Pozio 1, , Fabio Punzo 2, and  Alberto Tesei 1,

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

Received  April 2010 Revised  December 2010 Published  March 2011

Uniqueness and nonuniqueness of solutions to the first initial-boundary value problem for degenerate semilinear parabolic equations, with possibly unbounded coefficients, are studied. Sub- and supersolutions of suitable auxiliary problems, such as the first exit time problem, are used to determine on which part of the boundary Dirichlet data must be given. As an application of the general results, we study uniqueness and nonuniqueness of bounded solutions to a semilinear Cauchy problem in the hyperbolic space $\mathbb{H}^n$ using the Poincaré model.
Keywords: Degenerate parabolic equations; unbounded coefficients, well-posedness; sub - supersolutions..
Mathematics Subject Classification: Primary: 35K65, 35K20; Secondary: 35B3.
Citation: Maria Assunta Pozio, Fabio Punzo, Alberto Tesei. Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 891-916. doi: 10.3934/dcds.2011.30.891
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), 503-523. doi: 10.1112/jlms/jdm104.  Google Scholar

[2]

R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'' Springer-Verlag, Berlin, 1992.  Google Scholar

[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), 47-92. doi: 10.1002/cpa.3160470105.  Google Scholar

[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), 93-112. doi: 10.1007/s002050100171.  Google Scholar

[5]

S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'' Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin, 2001.  Google Scholar

[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[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), 349-358.  Google Scholar

[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), 1-30.  Google Scholar

[9]

A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[10]

A. Friedman, "Stochastic Differential Equations and Applications,'' I, II, Probability and Mathematical Statistics, 28, Academic Press, New York - London, 1976.  Google Scholar

[11]

I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'' MIR Publishers, Moscow, 1980.  Google Scholar

[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), 135-249. doi: 10.1090/S0273-0979-99-00776-4.  Google Scholar

[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), 1-144. doi: 10.1070/RM1962v017n03ABEH004115.  Google Scholar

[14]

H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkc. Ekv., 38 (1995), 101-120.  Google Scholar

[15]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[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), 279-298.  Google Scholar

[17]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 33 (1981), 831-852. doi: 10.1002/cpa.3160340605.  Google Scholar

[18]

R. Z. Khas'minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain, Th. Prob. Appl., 4 (1958), 400-419. doi: 10.1137/1103033.  Google Scholar

[19]

L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'' Pure and Applied Mathematics, 283, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar

[20]

A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$, Studia Math., 128 (1998), 171-198.  Google Scholar

[21]

O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'' Amer. Math. Soc., Plenum Press, New York - London, 1973.  Google Scholar

[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), 353-386. doi: 10.1016/j.matpur.2008.06.001.  Google Scholar

[23]

M. A. Pozio and A. Tesei, On the uniqueness of bounded solutions to singular parabolic problems, Discr. Cont. Dyn. Syst., 13 (2005), 117-137. doi: 10.3934/dcds.2005.13.117.  Google Scholar

[24]

F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 26 (2009), 2001-2024.  Google Scholar

[25]

F. Punzo and A. Tesei, On the refined maximum principle for degenerate elliptic and parabolic problems, Nonlinear Anal., 70 (2009), 3047-3055. doi: 10.1016/j.na.2008.12.032.  Google Scholar

[26]

F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Rend. Lincei Mat. Appl., 21 (2010), 1-38. Google Scholar

[27]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, New York, 1984.  Google Scholar

[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), 651-713. doi: 10.1002/cpa.3160250603.  Google Scholar

[29]

K. Taira, "Diffusion Processes and Partial Differential Equations,'' Academic Press, Inc., Boston, MA, 1988.  Google Scholar

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), 503-523. doi: 10.1112/jlms/jdm104.  Google Scholar

[2]

R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'' Springer-Verlag, Berlin, 1992.  Google Scholar

[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), 47-92. doi: 10.1002/cpa.3160470105.  Google Scholar

[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), 93-112. doi: 10.1007/s002050100171.  Google Scholar

[5]

S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'' Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin, 2001.  Google Scholar

[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), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar

[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), 349-358.  Google Scholar

[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), 1-30.  Google Scholar

[9]

A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.  Google Scholar

[10]

A. Friedman, "Stochastic Differential Equations and Applications,'' I, II, Probability and Mathematical Statistics, 28, Academic Press, New York - London, 1976.  Google Scholar

[11]

I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'' MIR Publishers, Moscow, 1980.  Google Scholar

[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), 135-249. doi: 10.1090/S0273-0979-99-00776-4.  Google Scholar

[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), 1-144. doi: 10.1070/RM1962v017n03ABEH004115.  Google Scholar

[14]

H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkc. Ekv., 38 (1995), 101-120.  Google Scholar

[15]

H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear second-order elliptic partial differential equations, J. Differential Equations, 83 (1990), 26-78. doi: 10.1016/0022-0396(90)90068-Z.  Google Scholar

[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), 279-298.  Google Scholar

[17]

S. Kamin and P. Rosenau, Propagation of thermal waves in an inhomogeneous medium, Comm. Pure Appl. Math., 33 (1981), 831-852. doi: 10.1002/cpa.3160340605.  Google Scholar

[18]

R. Z. Khas'minskii, Diffusion processes and elliptic differential equations degenerating at the boundary of the domain, Th. Prob. Appl., 4 (1958), 400-419. doi: 10.1137/1103033.  Google Scholar

[19]

L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'' Pure and Applied Mathematics, 283, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar

[20]

A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$, Studia Math., 128 (1998), 171-198.  Google Scholar

[21]

O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'' Amer. Math. Soc., Plenum Press, New York - London, 1973.  Google Scholar

[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), 353-386. doi: 10.1016/j.matpur.2008.06.001.  Google Scholar

[23]

M. A. Pozio and A. Tesei, On the uniqueness of bounded solutions to singular parabolic problems, Discr. Cont. Dyn. Syst., 13 (2005), 117-137. doi: 10.3934/dcds.2005.13.117.  Google Scholar

[24]

F. Punzo and A. Tesei, Uniqueness of solutions to degenerate elliptic problems with unbounded coefficients, Ann. Inst. H. Poincaré - Analyse Nonlinéaire, 26 (2009), 2001-2024.  Google Scholar

[25]

F. Punzo and A. Tesei, On the refined maximum principle for degenerate elliptic and parabolic problems, Nonlinear Anal., 70 (2009), 3047-3055. doi: 10.1016/j.na.2008.12.032.  Google Scholar

[26]

F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Rend. Lincei Mat. Appl., 21 (2010), 1-38. Google Scholar

[27]

M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, New York, 1984.  Google Scholar

[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), 651-713. doi: 10.1002/cpa.3160250603.  Google Scholar

[29]

K. Taira, "Diffusion Processes and Partial Differential Equations,'' Academic Press, Inc., Boston, MA, 1988.  Google Scholar

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