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Weak solutions of parabolic equations in non-cylindrical domains

Author(s): Russell M. Brown; Wei Hu; Gary M. Lieberman
Journal: Proc. Amer. Math. Soc. 125 (1997), 1785-1792.
MSC (1991): Primary 35K15; Secondary 35D05
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Abstract: In their classical work, Ladyzhenskaya and Ural$'$tseva gave a definition of weak solution for parabolic equations in cylindrical domains. Their definition was broad enough to guarantee the solvability of all such problems but narrow enough to guarantee the uniqueness of these solutions. We give here some alternative definitions which are appropriate to non-cylindrical domains, and we prove the unique solvability of such problems.

References:

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O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural$'$ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, R. I., 1967. MR 39:3159b

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P. Cannarsa, G. Da Prato, and J.-P. Zolèlsio, Evolution equations in non-cylindrical domains, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. (8) 88 (1990), 73-77. MR 93c:35059

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G. M. Lieberman, Regularized distance and its application, Pacific J. Math. 117 (1985), 329-352. MR 87j:35101

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G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations II. Existence, uniqueness, and regularity, J. Differential Equations 63 (1986), 32-57. MR 87m:35113b

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J.L. Lions, Sur les problèmes mixtes pour certains systèmes parboliques dans des ouverts non cylindriques, Ann. Inst. Fourier (Grenoble) (1957), 143-182. MR 21:1455

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Yong Jiongmin, Weak solutions of second order parabolic equations in noncylindrical domains, J. Partial Differential Equations 2 (2) (1989), 76-86. MR 90g:35068

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Additional Information:

Russell M. Brown
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: rbrown@ms.uky.edu

Wei Hu
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: weihu@ms.uky.edu

Gary M. Lieberman
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: lieb@iastate.edu

DOI: 10.1090/S0002-9939-97-03759-3
PII: S 0002-9939(97)03759-3
Received by editor(s): August 4, 1995
Received by editor(s) in revised form: January 2, 1996
Additional Notes: The first author was supported in part by the NSF and the Commonwealth of Kentucky through the NSF-EPSCoR program.
Communicated by: Jeffrey B. Rauch
Copyright of article: Copyright 1997, American Mathematical Society

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