Abstract
Motivated by the celebrated paper of Baras and Goldstein (Trans Am Math Soc 284:121–139, 1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, nonexistence and uniqueness of solutions and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.
Similar content being viewed by others
References
O. Arena, On a singular parabolic equation related to axially symmetric heat potentials. Ann. Mat. Pura Appl. (4) 105 (1975), 347–393.
P. Baras and J. Goldstein, The heat equation with a singular potential. Trans. Amer. Math. Soc. 284 (1984), 121–139.
L. R. Bragg, The radial heat equation with pole type data. Bull. Amer. Math. Soc. 73 (1967), 133–135.
X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 973–978.
F. Chiarenza and R. Serapioni, A remark on a Harnack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova 73 (1985), 179–190.
M. Fila, J. R. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation. J. Differential Equations 228 (2006), 339–356.
M. Fila, J. Takahashi and E. Yanagida, Solutions with moving singularities for equations of porous medium type. Nonlinear Anal. 179 (2019), 237–253.
Y. Fujishima and K. Ishige, Blowing up solutions for nonlinear parabolic systems with unequal elliptic operators. J. Dynam. Differential Equations 32 (2020), 1219–1231.
V. A. Galaktionov and I. V. Kamotski, On nonexistence of Baras-Goldstein type for higher-order parabolic equations with singular potentials. Trans. Amer. Math. Soc. 362 (2010), 4117–4136.
J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems. J. Differential Equations 144 (1998), 441–476.
D. T. Haimo, Functions with the Huygens property. Bull. Amer. Math. Soc. 71 (1965), 528–532.
D. Hirata and M. Tsutsumi, On the well-posedness of a linear heat equation with a critical singular potential. Differential Integral Equations 14 (2001), 1–18.
K. Ishige and A. Mukai, Large time behavior of solutions of the heat equation with inverse square potential. Discrete Contin. Dyn. Syst. 38 (2018), 4041–4069.
T. Kan and J. Takahashi, On the profile of solutions with time-dependent singularities for the heat equation. Kodai Math. J. 37 (2014), 568–585.
V. Liskevich, A. Shishkov and Z. Sobol, Singular solutions to the heat equations with nonlinear absorption and Hardy potentials. Commun. Contemp. Math. 14 (2012), 1250013, 28 pp.
C. Marchi, The Cauchy problem for the heat equation with a singular potential. Differential Integral Equations 16 (2003), 1065–1081.
J. Moser, A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13 (1960), 457–468.
J. Moser, On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 (1961), 577–591.
I. Okada and E. Yanagida On the heat equation with a dynamic Hardy-type potential: probabilistic approach, preprint.
S. Sato and E. Yanagida, Solutions with moving singularities for a semilinear parabolic equation. J. Differential Equations 246 (2009), 724–748.
J. Takahashi and E. Yanagida, Time-dependent singularities in the heat equation, Commun. Pure Appl. Anal. 14 (2015), 969–979.
J. Takahashi and E. Yanagida, Time-dependent singularities in a semilinear parabolic equation with absorption. Commun. Contemp. Math. 18 (2016), 1550077, 27 pp.
J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials. J. Funct. Anal. 254 (2008), 1864–1902.
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The first author was supported in part by Ministry of Science and Technology (MOST) of Taiwan (No. MOST 107-2115-M-008-005-MY3). The second author was supported by the National Research Foundation of Korea(NRF) grant funded by the Korean government(MSIT) (No. NRF-2018R1C1B5086492). The third author was supported in part by JSPS KAKENHI Early-Career Scientists (No. 19K14567). The fourth author was supported in part by JSPS KAKENHI Grant-in-Aid for Scientific Research (A) (No. 16K13769).
Rights and permissions
About this article
Cite this article
Chern, JL., Hwang, G., Takahashi, J. et al. On the evolution equation with a dynamic Hardy-type potential. J. Evol. Equ. 21, 2141–2165 (2021). https://doi.org/10.1007/s00028-021-00675-5
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-021-00675-5