Abstract
We consider second-order parabolic equations describing diffusion with degeneration and diffusion on singular and combined structures. We give a united definition of a solution of the Cauchy problem for such equations by means of semigroup theory in the space L 2 with a suitable measure. We establish some weight estimates for solutions of Cauchy problems. Estimates of Nash–Aronson type for the fundamental solution follow from them. We plan to apply these estimates to known asymptotic diffusion problems, namely, to the stabilization of solutions and to the “central limit theorem.”
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References
D. G. Aronson, “Bounds for the fundamental solutions of a parabolic equation,” Bull. Am. Math. Soc., 890–896 (1967).
D. G. Aronson, “Non-negative solutions of linear parabolic equations,” Ann. Scuola Norm. Sup. Pisa (3), 4, No. 8, 607–694 (1968).
E. B. Bavies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge (1989).
S. D. Ehjdel’man and F. O. Porper, “Two-sided estimates of fundamental solutions of second-order parabolic equations, and some applications,” Russ. Math. Surv., 39, No. 3, 119–178 (1984).
E. B. Fabes, C. E. Kenig, and R. P. Serapioni, “The local regularity of solutions of degenerate elliptic equations,” Commun. Partial Differ. Equ., 7, 77–116 (1982).
A. Grigor’yan, “Gaussian upper bounds for the heat kernel on arbitrary manifolds,” J. Differ. Geom., 45, 33–52 (1997).
A. K. Gushchin, “Uniform stabilization of solutions of the second initial-boundary problem for a parabolic equation,” Mat. Sb. 119, 451–508 (1982).
B. Muckenhoupt, “Weighted norm inequalities for Hardy maximal functions,” Trans. Am. Math. Soc., 165, 207–226 (1972).
V. V. Zhikov, “Asymptotic problems connected with the heat equation in perforated domains,” Math. USSR-Sb., 71, No. 1, 125–147 (1992).
V. V. Zhikov, “On weighted Sobolev spaces,” Sb. Math., 189, Nos. 7–8, 1139–1170 (1998).
V. V. Zhikov, “On an extension of the method of two-scale convergence and its applications,” Sb. Math., 191, No. 7, 973–1014 (2000).
Zbl 1160.35360 V. V. Zhikov, “Estimates of Nash–Aronson type for a diffusion equation with asymmetric matrix and their applications to homogenization,” Sb. Math., 197, No. 12, 1775–1804 (2006).
V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin (1994).
V. V. Zhikov and A. L. Pyatniskii, “Homogenization of random singular structures and random measures,” Izv. Math., 70, No. 1, 19–67 (2006).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 39, Partial Differential Equations, 2011.
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Zhikov, V.V. Estimates of the Nash–Aronson type for degenerating parabolic equations. J Math Sci 190, 66–79 (2013). https://doi.org/10.1007/s10958-013-1246-4
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DOI: https://doi.org/10.1007/s10958-013-1246-4