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March  2011, 31(1): 65-107. doi: 10.3934/dcds.2011.31.65

Classification of local asymptotics for solutions to heat equations with inverse-square potentials

1. 

Università di Milano Bicocca, Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano

2. 

ICMAT, Instituto de Ciencias Matemáticas, Campus Cantoblanco, Calle Nicolás Cabrera 13–15, 28049 Madrid, Spain

Received  March 2010 Revised  November 2010 Published  June 2011

Asymptotic behavior of solutions to heat equations with spatially singular inverse-square potentials is studied. By combining a parabolic Almgren type monotonicity formula with blow-up methods, we evaluate the exact behavior near the singularity of solutions to linear and subcritical semilinear parabolic equations with Hardy type potentials. As a remarkable byproduct, a unique continuation property is obtained.
Citation: Veronica Felli, Ana Primo. Classification of local asymptotics for solutions to heat equations with inverse-square potentials. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 65-107. doi: 10.3934/dcds.2011.31.65
References:
[1]

B. Abdellaoui, V. Felli and I. Peral, Existence and multiplicity for perturbations of an equation involving a Hardy inequality and the critical Sobolev exponent in the whole of $\mathbb{R}^{N}$, Adv. Differential Equations, 9 (2004), 481-508.

[2]

B. Abdellaoui, I. Peral and A. Primo, Influence of the Hardy potential in a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 897-926. doi: 10.1017/S0308210508000152.

[3]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," National Bureau of Standards Applied Mathematics Series, 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.

[4]

G. Alessandrini and S. Vessella, Remark on the strong unique continuation property for parabolic operators, Proc. Amer. Math. Soc., 132 (2004), 499-501. doi: 10.1090/S0002-9939-03-07142-9.

[5]

F. J. Almgren Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, in "Minimal Submanifolds and Geodesics" (Proc. Japan-United States Sem., Tokyo, 1977), 1-6, North-Holland, Amsterdam-New York, 1979.

[6]

P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. doi: 10.1090/S0002-9947-1984-0742415-3.

[7]

L. A. Caffarelli, A. L. Karakhanyan and F.-H. Lin, The geometry of solutions to a segregation problem for nondivergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351. doi: 10.1007/s11784-009-0110-0.

[8]

X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630. doi: 10.1007/s002080050202.

[9]

L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127. doi: 10.1215/S0012-7094-00-10415-2.

[10]

L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60. doi: 10.1007/BF02384566.

[11]

L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.

[12]

L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, Decay at infinity of caloric functions within characteristic hyperplanes, Math. Res. Lett., 13 (2006), 441-453.

[13]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II, Indiana Univ. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.

[14]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0.

[15]

V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, Journal of the European Mathematical Society, 13 (2011), 119-174. doi: 10.4171/JEMS/246.

[16]

V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods, preprint, arXiv:1007.4434.

[17]

V. Felli, A. Ferrero and S. Terracini, On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials, preprint, arXiv:1004.3949

[18]

V. Felli, E. M. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete and Continuous Dynamical Systems, 21 (2008), 91-119. doi: 10.3934/dcds.2008.21.91.

[19]

F. J. Fernández, Unique continuation for parabolic operators. II, Comm. Partial Differential Equations, 28 (2003), 1597-1604. doi: 10.1081/PDE-120024523.

[20]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. doi: 10.1006/jdeq.1997.3375.

[21]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.

[22]

G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1952.

[23]

F.-H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.

[24]

F.-H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308. doi: 10.1002/cpa.3160440303.

[25]

A. D. MacDonald, Properties of the confluent hypergeometric function, J. Math. Physics, 28 (1949), 183-191.

[26]

I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal., 129 (1995), 201-224. doi: 10.1007/BF00383673.

[27]

P. Poláčik and E. Yanagida, Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation, J. Dynam. Differential Equations, 21 (2009), 329-341. doi: 10.1007/s10884-009-9136-7.

[28]

C.-C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539.

[29]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[30]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Monographs and Studies in Mathematics, 1, Pitman, London-San Francisco, Calif.-Melbourne, 1977.

[31]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[32]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938. doi: 10.1090/S0002-9947-04-03769-9.

[33]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.

[34]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.

[35]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

show all references

References:
[1]

B. Abdellaoui, V. Felli and I. Peral, Existence and multiplicity for perturbations of an equation involving a Hardy inequality and the critical Sobolev exponent in the whole of $\mathbb{R}^{N}$, Adv. Differential Equations, 9 (2004), 481-508.

[2]

B. Abdellaoui, I. Peral and A. Primo, Influence of the Hardy potential in a semilinear heat equation, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 897-926. doi: 10.1017/S0308210508000152.

[3]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," National Bureau of Standards Applied Mathematics Series, 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964.

[4]

G. Alessandrini and S. Vessella, Remark on the strong unique continuation property for parabolic operators, Proc. Amer. Math. Soc., 132 (2004), 499-501. doi: 10.1090/S0002-9939-03-07142-9.

[5]

F. J. Almgren Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents, in "Minimal Submanifolds and Geodesics" (Proc. Japan-United States Sem., Tokyo, 1977), 1-6, North-Holland, Amsterdam-New York, 1979.

[6]

P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. doi: 10.1090/S0002-9947-1984-0742415-3.

[7]

L. A. Caffarelli, A. L. Karakhanyan and F.-H. Lin, The geometry of solutions to a segregation problem for nondivergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351. doi: 10.1007/s11784-009-0110-0.

[8]

X.-Y. Chen, A strong unique continuation theorem for parabolic equations, Math. Ann., 311 (1998), 603-630. doi: 10.1007/s002080050202.

[9]

L. Escauriaza, Carleman inequalities and the heat operator, Duke Math. J., 104 (2000), 113-127. doi: 10.1215/S0012-7094-00-10415-2.

[10]

L. Escauriaza and F. J. Fernández, Unique continuation for parabolic operators, Ark. Mat., 41 (2003), 35-60. doi: 10.1007/BF02384566.

[11]

L. Escauriaza, F. J. Fernández and S. Vessella, Doubling properties of caloric functions, Appl. Anal., 85 (2006), 205-223. doi: 10.1080/00036810500277082.

[12]

L. Escauriaza, C. E. Kenig, G. Ponce and L. Vega, Decay at infinity of caloric functions within characteristic hyperplanes, Math. Res. Lett., 13 (2006), 441-453.

[13]

L. Escauriaza and L. Vega, Carleman inequalities and the heat operator. II, Indiana Univ. Math. J., 50 (2001), 1149-1169. doi: 10.1512/iumj.2001.50.1937.

[14]

M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of the heat equation, Nonlinear Anal., 11 (1987), 1103-1133. doi: 10.1016/0362-546X(87)90001-0.

[15]

V. Felli, A. Ferrero and S. Terracini, Asymptotic behavior of solutions to Schrödinger equations near an isolated singularity of the electromagnetic potential, Journal of the European Mathematical Society, 13 (2011), 119-174. doi: 10.4171/JEMS/246.

[16]

V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods, preprint, arXiv:1007.4434.

[17]

V. Felli, A. Ferrero and S. Terracini, On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials, preprint, arXiv:1004.3949

[18]

V. Felli, E. M. Marchini and S. Terracini, On the behavior of solutions to Schrödinger equations with dipole type potentials near the singularity, Discrete and Continuous Dynamical Systems, 21 (2008), 91-119. doi: 10.3934/dcds.2008.21.91.

[19]

F. J. Fernández, Unique continuation for parabolic operators. II, Comm. Partial Differential Equations, 28 (2003), 1597-1604. doi: 10.1081/PDE-120024523.

[20]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476. doi: 10.1006/jdeq.1997.3375.

[21]

N. Garofalo and F.-H. Lin, Monotonicity properties of variational integrals, $A_p$ weights and unique continuation, Indiana Univ. Math. J., 35 (1986), 245-268. doi: 10.1512/iumj.1986.35.35015.

[22]

G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities," 2nd edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1952.

[23]

F.-H. Lin, A uniqueness theorem for parabolic equations, Comm. Pure Appl. Math., 43 (1990), 127-136. doi: 10.1002/cpa.3160430105.

[24]

F.-H. Lin, Nodal sets of solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 44 (1991), 287-308. doi: 10.1002/cpa.3160440303.

[25]

A. D. MacDonald, Properties of the confluent hypergeometric function, J. Math. Physics, 28 (1949), 183-191.

[26]

I. Peral and J. L. Vázquez, On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term, Arch. Rational Mech. Anal., 129 (1995), 201-224. doi: 10.1007/BF00383673.

[27]

P. Poláčik and E. Yanagida, Convergence of anisotropically decaying solutions of a supercritical semilinear heat equation, J. Dynam. Differential Equations, 21 (2009), 329-341. doi: 10.1007/s10884-009-9136-7.

[28]

C.-C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations, 21 (1996), 521-539.

[29]

J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations, 66 (1987), 118-139. doi: 10.1016/0022-0396(87)90043-X.

[30]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations," Monographs and Studies in Mathematics, 1, Pitman, London-San Francisco, Calif.-Melbourne, 1977.

[31]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.

[32]

D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938. doi: 10.1090/S0002-9947-04-03769-9.

[33]

C. D. Sogge, A unique continuation theorem for second order parabolic differential operators, Ark. Mat., 28 (1990), 159-182. doi: 10.1007/BF02387373.

[34]

S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations, 1 (1996), 241-264.

[35]

J. L. Vazquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

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