Abstract
We show that the large-time behavior of solutions to the Cauchy problem for the linear heat equation with the inverse square potential is described by explicit self-similar solutions.
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References
Baras P., Goldstein J.A.: The heat equation with a singular potential. Trans. Am. Math. Soc. 284, 121–139 (1984)
Chiarenza F.M., Serapioni R.P.: A remark on a Harnack inequality for degenerate parabolic equations. Rend. Sem. Mat. Univ. Padova 73, 179–190 (1985)
Dziubański J., Hulanicki A.: On semigroups generated by left-invariant positive differential operators on nilpotent Lie groups. Studia Math. 94, 81–95 (1989)
Duoandikoetxea J., Zuazua E.: Moments, masses de Dirac et décomposition de fonctions. C. R. Acad. Sci. Paris, Sér. I 315, 693–698 (1992)
L. C. Evans, Partial Differential Equations, Amer. Math. Soc., Providence 19, (1998).
M.-H. Giga, Y. Giga, J. Saal, Progress in Nonlinear Differential Equations and Their Applications: asymptotic behavior of solutions and self-similar solutions, Boston [etc.], Birkhäuser, vol. 79.
Goldstein J.A., Kombe I.: Instantaneous blow up. Contemp. Math. 327, 141–149 (2003)
Liskevich V., Sobol Z.: Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients. Potential Anal. 18, 359–390 (2003)
Milman P.D., Yu. A.: Semenov, Global heat kernel bounds via desingularizing weights. J. Funct. Analysis 212, 373–398 (2004)
Moschini L., Tesei A.: Parabolic Harnack inequality for the heat equation with inverse–square potential, Forum Math. 19, 407–427 (2007)
E. M. Ouhabaz, Analysis of heat equation on domains, Princeton University Press, 2004.
Simon B.: Schrödinger semigroups. Bull. Amer. Math. Soc. (N. S.) 7, 447–526 (1982)
Vázquez J.L., Zuazua E.: The Hardy inequality and the asymptotic behavior of the heat equation with an inverse–square potential. J. Funct. Anal. 173, 103–153 (2000)
Zhang Qi S.: Global bounds of Schrödinger heat kernels with negative potentials. J. Funct. Anal. 182, 344–370 (2001)
Zhang Qi S.: Global lower bound for the heat kernel of \({-\Delta + \frac{c}{|x|^2}}\). Proc. Am. Math. Soc. 129, 1105–1112 (2000)
Acknowledgments
The author wishes to express her gratitude to the anonymous referee for several helpful comments and corrections which allowed the author to improve the previous version of the paper. This work was supported by the MNiSzW grant No. N N201 418839, and is a part of the author PhD dissertation written under supervision of Grzegorz Karch.
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Pilarczyk, D. Self-similar asymptotics of solutions to heat equation with inverse square potential. J. Evol. Equ. 13, 69–87 (2013). https://doi.org/10.1007/s00028-012-0169-8
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DOI: https://doi.org/10.1007/s00028-012-0169-8