Weak solutions of parabolic equations in non-cylindrical domains
HTML articles powered by AMS MathViewer
- by Russell M. Brown, Wei Hu and Gary M. Lieberman PDF
- Proc. Amer. Math. Soc. 125 (1997), 1785-1792 Request permission
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
- O. A. Ladyženskaja, V. A. Solonnikov, and N. N. Ural′ceva, Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa, Izdat. “Nauka”, Moscow, 1967 (Russian). MR 0241821
- Piermarco Cannarsa, Giuseppe Da Prato, and Jean-Paul Zolésio, Evolution equations in noncylindrical domains, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 83 (1989), 73–77 (1990) (English, with Italian summary). MR 1142441
- Gary M. Lieberman, Regularized distance and its applications, Pacific J. Math. 117 (1985), no. 2, 329–352. MR 779924, DOI 10.2140/pjm.1985.117.329
- Gary M. Lieberman, Intermediate Schauder theory for second order parabolic equations. I. Estimates, J. Differential Equations 63 (1986), no. 1, 1–31. MR 840590, DOI 10.1016/0022-0396(86)90052-5
- J.-L. Lions, Sur les problèmes mixtes pour certains systèmes paraboliques dans des ouverts non cylindriques, Ann. Inst. Fourier (Grenoble) 7 (1957), 143–182 (French). MR 102666, DOI 10.5802/aif.69
- Jiong Min Yong, Weak solutions of second order parabolic equations in noncylindrical domains, J. Partial Differential Equations 2 (1989), no. 2, 76–86. MR 1010775
Additional Information
- Russell M. Brown
- MR Author ID: 259097
- Email: rbrown@ms.uky.edu
- Wei Hu
- Affiliation: (R. M. Brown and W. Hu) Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
- Email: weihu@ms.uky.edu
- Gary M. Lieberman
- Affiliation: (G. M. Lieberman) Department of Mathematics, Iowa State University, Ames, Iowa 50011
- Email: lieb@iastate.edu
- 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 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1785-1792
- MSC (1991): Primary 35K15; Secondary 35D05
- DOI: https://doi.org/10.1090/S0002-9939-97-03759-3
- MathSciNet review: 1372024