Uniqueness and nonuniqueness of solutions to parabolic problems with singular coefficients

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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

Abstract
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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 and 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.
[2]	R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'' Springer-Verlag, Berlin, 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), 47-92. 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), 93-112. doi: 10.1007/s002050100171.
[5]	S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'' Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin, 2001.
[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.
[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.
[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.
[9]	A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
[10]	A. Friedman, "Stochastic Differential Equations and Applications,'' I, II, Probability and Mathematical Statistics, 28, Academic Press, New York - London, 1976.
[11]	I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'' MIR Publishers, Moscow, 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), 135-249. 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), 1-144. 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), 101-120.
[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.
[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.
[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.
[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.
[19]	L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'' Pure and Applied Mathematics, 283, Chapman & Hall/CRC, Boca Raton, FL, 2007.
[20]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$, Studia Math., 128 (1998), 171-198.
[21]	O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'' Amer. Math. Soc., Plenum Press, New York - London, 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), 353-386. 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), 117-137. 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), 2001-2024.
[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.
[26]	F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Rend. Lincei Mat. Appl., 21 (2010), 1-38.
[27]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, New York, 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), 651-713. doi: 10.1002/cpa.3160250603.
[29]	K. Taira, "Diffusion Processes and Partial Differential Equations,'' Academic Press, Inc., Boston, MA, 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), 503-523. doi: 10.1112/jlms/jdm104.
[2]	R. Benedetti and C. Petronio, "Lecture on Hyperbolic Geometry,'' Springer-Verlag, Berlin, 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), 47-92. 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), 93-112. doi: 10.1007/s002050100171.
[5]	S. Cerrai, "Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach,'' Lecture Notes in Mathematics 1762, Springer-Verlag, Berlin, 2001.
[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.
[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.
[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.
[9]	A. Friedman, "Partial Differential Equations of Parabolic Type,'' Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
[10]	A. Friedman, "Stochastic Differential Equations and Applications,'' I, II, Probability and Mathematical Statistics, 28, Academic Press, New York - London, 1976.
[11]	I. Guikhman and A. Skorokhod, "Introduction à la Théorie des Processus Aléatoires,'' MIR Publishers, Moscow, 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), 135-249. 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), 1-144. 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), 101-120.
[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.
[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.
[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.
[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.
[19]	L. Lorenzi and M. Bertoldi, "Analytical Methods for Markov Semigroups,'' Pure and Applied Mathematics, 283, Chapman & Hall/CRC, Boca Raton, FL, 2007.
[20]	A. Lunardi, Schauder theorems for linear elliptic and parabolic problems with unbounded coefficients in $IR^n$, Studia Math., 128 (1998), 171-198.
[21]	O. A. Oleinik and E. V. Radkevic, "Second Order Equations with Nonnegative Characteristic Form,'' Amer. Math. Soc., Plenum Press, New York - London, 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), 353-386. 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), 117-137. 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), 2001-2024.
[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.
[26]	F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Rend. Lincei Mat. Appl., 21 (2010), 1-38.
[27]	M. H. Protter and H. F. Weinberger, "Maximum Principles in Differential Equations,'' Springer-Verlag, New York, 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), 651-713. doi: 10.1002/cpa.3160250603.
[29]	K. Taira, "Diffusion Processes and Partial Differential Equations,'' Academic Press, Inc., Boston, MA, 1988.

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