No Access

AN ILL-POSED PROBLEM FOR THE HEAT EQUATION

L. E. PAYNE

Department of Mathematics, Cornell University, Ithaca, NY 14853, USA

Search for more papers by this author

and

G. A. PHILIPPIN

Département de Mathématiques et de Statistique, Université Laval Québec, Canada G1K 7P4, Canada

Search for more papers by this author

https://doi.org/10.1142/S0218202509003917Cited by:1 (Source: Crossref)

Abstract

The Cauchy problem for the heat equation in which Cauchy data are prescribed on the outer boundary of a domain with cavity and no data are given on the inner boundary is known to be ill-posed. By a slight modification of the boundary conditions a new problem is introduced whose solution depends continuously on the data in L₂.

Dedicated to the memory of K. A. Ames

Keywords:

AMSC: 35B30, 35K05, 35R25

References

O. M. Alifanov , Inverse Heat Transfer Problems ( Springer-Verlag , 1995 ) . Google Scholar
O. M. Alifanov , E. A. Artiukhin and S. V. Rumiantsev , Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems ( Begell House Publishers , 1995 ) . Crossref, Google Scholar
K. A. Ames and L. E. Payne, MMAS 8, 187 (1998). Web of Science, Google Scholar
K. A. Ames, L. E. Payne and P. W. Schaefer, Proc. Roy. Soc. Edinburgh A 134, 1 (2004), DOI: 10.1017/S0308210500003036. Crossref, Web of Science, Google Scholar
K. A. Ames, L. E. Payne and P. W. Schaefer, Appl. Anal. 83, 125 (2004), DOI: 10.1080/00036810310001620108. Crossref, Google Scholar
K. A. Ames, L. E. Payne and J. C. Song, JMAA 311, 254 (2005). Web of Science, Google Scholar
K. A. Ames and B. Straughan , Non-Standard and Improperly-Posed Problems , Mathematics in Sciences and Engineering 194 ( Academic Press , 1997 ) . Google Scholar
J. V. Beck , B. Blackwell and C. R. St. Clair Jr. , Inverse heat conduction: Ill-posed problems ( Wiley-Interscience , 1985 ) . Google Scholar
J. B. Bell, SIAM J. Math. Anal. 12, 759 (1981), DOI: 10.1137/0512064. Crossref, Web of Science, Google Scholar
J. B. Bell, SIAM J. Math. Anal. 12, 778 (1981), DOI: 10.1137/0512065. Crossref, Web of Science, Google Scholar
J. R. Cannon , The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications 23 ( Cambridge University Press , 1984 ) . Crossref, Google Scholar
G. Clark and S. Oppenheimer, Quasireversibility methods for non-well-posed problems, Electron. J. Diff. Eqns. (1994), #8 (approx. 9 pp.) . Google Scholar
R. E. Ewing and R. Falk, Math. Comp. 33, 1125 (1979), DOI: 10.2307/2006451. Crossref, Web of Science, Google Scholar
D. N. Hào , Methods for Inverse Heat Conduction Problems ( Peter Lang Verlag , Frankfurt am Main , 1998 ) . Google Scholar
V. Isakov , Inverse Problems in Partial Differential Equations ( Springer-Verlag , 1998 ) . Crossref, Google Scholar
F. John, Comm. Pure Appl. Math. 8, 494 (1955), DOI: 10.1002/cpa.3160080409. Google Scholar
P. Knabner and S. Vessella, Ann. Mat. Pura Appl. 149, 393 (1987), DOI: 10.1007/BF01773944. Crossref, Web of Science, Google Scholar
P. Knabner and S. Vessella, Inverse and Ill-Posed Problems, eds. H. W. Engl and C. W. Groetsch (Academic Press, 1987) pp. 351–368. Crossref, Google Scholar
P. Knabner and S. Vessella, Math. Meth. Appl. Sci. 10, 575 (1988), DOI: 10.1002/mma.1670100507. Crossref, Web of Science, Google Scholar
M. G. Krein, Dokl. Akad. Nauk. SSSR 39, 195 (1944). Google Scholar
R. Lattès and J. L. Lions , Applications in Partial Differential Equations ( Elsevier , 1969 ) . Google Scholar
M. M. Lavrentiev, Izv. Akad. Nauk SSSR, Ser. Math. 20, 819 (1956). Google Scholar
P. Manselli and K. Miller, Ann. Mat. Pura Appl. 123, 161 (1980), DOI: 10.1007/BF01796543. Crossref, Google Scholar
S. G. Michlin , Partielle Differentialgleichungen in der mathematischen Physik ( Akademie Verlag , 1978 ) . Google Scholar
L. E. Payne , Improperly Posed Problems in Partial Differential Equations , Regional Conference Series in Applied Mathematics 22 ( SIAM , 1975 ) . Crossref, Google Scholar
L. E. Payne and H. F. Weinberger, J. Math. Phys. 33, 291 (1955). Crossref, Web of Science, Google Scholar
C. Pucci, Rend. Acad. Naz. Lincei 18, 473 (1955). Google Scholar
I. M. Rapaport, Dokl. Akad. Nauk SSSR 28, 305 (1940). Google Scholar
F. Rellich, Math. Zeitschrift 46, 635 (1940), DOI: 10.1007/BF01181459. Crossref, Google Scholar
R. E. Showalter, Trends in the Theory and Practice of Nonlinear Analysis (Elsevier, 1985) pp. 421–425. Crossref, Google Scholar
V. G. Sigillito , Explicit a priory inequalities with applications to boundary value problems , Pitman Res. Notes Math. 13 ( Pitman Press , 1977 ) . Google Scholar
V. N. Sretenskii, Izv. Akad. Nauk SSSR, Ser. Math. 2, 551 (1938). Google Scholar
A. N. Tikhonov, Dokl. Akad. Nauk SSSR 39, 195 (1944). Google Scholar