On the evolution equation with a dynamic Hardy-type potential

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.

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Correspondence to Gyeongha Hwang.

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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).

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Chern, JL., Hwang, G., Takahashi, J. et al. On the evolution equation with a dynamic Hardy-type potential. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-021-00675-5

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Keywords

  • Heat equation
  • Initial value problem
  • Singularity
  • Hardy potential
  • Existence
  • Uniqueness

Mathematics Subject Classification

  • Primary 35K15
  • 35K67
  • Secondary 35K05
  • 35A01
  • 35A02