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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 & Continuous Dynamical Systems - A, 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 $\R^N$,, Adv. Differential Equations, 9 (2004), 481. Google Scholar

[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. doi: 10.1017/S0308210508000152. Google Scholar

[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 (1964). Google Scholar

[4]

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

[5]

F. J. Almgren Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents,, in, (1977), 1. Google Scholar

[6]

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

[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. doi: 10.1007/s11784-009-0110-0. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[13]

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

[14]

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

[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. doi: 10.4171/JEMS/246. Google Scholar

[16]

V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods,, preprint, (). Google Scholar

[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, (). Google Scholar

[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. doi: 10.3934/dcds.2008.21.91. Google Scholar

[19]

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

[20]

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

[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. doi: 10.1512/iumj.1986.35.35015. Google Scholar

[22]

G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", 2nd edition, (1952). Google Scholar

[23]

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

[24]

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

[25]

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

[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. doi: 10.1007/BF00383673. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,", Monographs and Studies in Mathematics, 1 (1977). Google Scholar

[31]

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

[32]

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

[33]

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

[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. Google Scholar

[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. doi: 10.1006/jfan.1999.3556. Google Scholar

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 $\R^N$,, Adv. Differential Equations, 9 (2004), 481. Google Scholar

[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. doi: 10.1017/S0308210508000152. Google Scholar

[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 (1964). Google Scholar

[4]

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

[5]

F. J. Almgren Jr., Dirichlet's problem for multiple valued functions and the regularity of mass minimizing integral currents,, in, (1977), 1. Google Scholar

[6]

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

[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. doi: 10.1007/s11784-009-0110-0. Google Scholar

[8]

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

[9]

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

[10]

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

[11]

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

[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. Google Scholar

[13]

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

[14]

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

[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. doi: 10.4171/JEMS/246. Google Scholar

[16]

V. Felli, A. Ferrero and S. Terracini, A note on local asymptotics of solutions to singular elliptic equations via monotonicity methods,, preprint, (). Google Scholar

[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, (). Google Scholar

[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. doi: 10.3934/dcds.2008.21.91. Google Scholar

[19]

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

[20]

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

[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. doi: 10.1512/iumj.1986.35.35015. Google Scholar

[22]

G. Hardy, J. E. Littlewood and G. Pólya, "Inequalities,", 2nd edition, (1952). Google Scholar

[23]

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

[24]

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

[25]

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

[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. doi: 10.1007/BF00383673. Google Scholar

[27]

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

[28]

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

[29]

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

[30]

R. E. Showalter, "Hilbert Space Methods for Partial Differential Equations,", Monographs and Studies in Mathematics, 1 (1977). Google Scholar

[31]

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

[32]

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

[33]

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

[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. Google Scholar

[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. doi: 10.1006/jfan.1999.3556. Google Scholar

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