Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Existence of classical solutions for singular parabolic problems

Authors: C. Y. Chan and Benedict M. Wong
Journal: Quart. Appl. Math. 53 (1995), 201-213
MSC: Primary 35K20; Secondary 35K65
DOI: https://doi.org/10.1090/qam/1330648
MathSciNet review: MR1330648
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ Lu \equiv {u_{xx}} + b{u_x}/x - {u_t}$ with $ b$ a constant less than 1. Its Green's function corresponding to first boundary conditions is constructed by eigenfunction expansion. With this, a representation formula is established to obtain existence of a classical solution for the linear first initial-boundary value problem. Uniqueness of a solution follows from the strong maximum principle. Properties of Green's function and of the solution are also investigated.

References [Enhancements On Off] (What's this?)

  • [1] V. Alexiades, Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems, Arch. Rational Mech. Anal. 79, 325-350 (1982)
  • [2] H. Brezis, W. Rosenkrantz, and B. Singer, with an appendix by P. D. Lax, On a degenerate elliptic-parabolic equation occurring in the theory of probability, Comm. Pure Appl. Math. 24, 395-416 (1971)
  • [3] S. B. Chae, Lebesgue Integration, Marcel Dekker, New York, 1980, pp. 227-228
  • [4] C. Y. Chan, New results in quenching, Proceedings of the First World Congress of Nonlinear Analysts, Walter de Gruyter & Co. (to appear)
  • [5] C. Y. Chan and C. S. Chen, A numerical method for semilinear singular parabolic quenching problems, Quart. Appl. Math. 47, 45-57 (1989)
  • [6] C. Y. Chan and C. S. Chen, Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems, Quart. Appl. Math. 47, 661-671 (1989)
  • [7] C. Y. Chan and S. S. Cobb, Critical lengths for semilinear singular parabolic mixed boundary-value problems, Quart. Appl. Math. 49, 497-506 (1991)
  • [8] C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math. Anal. 20, 558-566 (1989)
  • [9] C. Y. Chan and B. M. Wong, Periodic solutions of singular linear and semilinear parabolic problems, Quart. Appl. Math. 47, 405-428 (1989)
  • [10] C. Y. Chan and B. M. Wong, Computational methods for time-periodic solutions of singular semilinear parabolic problems, Appl. Math. Comput. 42, 287-312 (1991)
  • [11] N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Selfadjoint Operators in Hilbert Space, Interscience Publishers, New York, 1963, pp. 1640-1641
  • [12] H. Kawarada, On solutions of initial boundary problem for $ {u_t} = {u_{xx}} + 1/\left( 1 - u \right)$, Publ. Res. Inst. Math. Sci. 10, 729-736 (1975)
  • [13] K. Knopp, Theory and Application of Infinite Series, Hafner Publishing Company, New York, 1928, pp. 146, 337, and 346
  • [14] J. Lamperti, A new class of probability theorems, J. Math. Mech. 11, 749-772 (1962)
  • [15] N. W. McLachlan, Bessel Functions for Engineers, 2nd ed., Oxford University Press, London, 1955, pp. 26, 102-104, and 116
  • [16] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, pp. 168-170
  • [17] H. L. Royden, Real Analysis, 2nd ed., Macmillan Publishing Co., New York, 1968, pp. 84, 88, and 269-270
  • [18] A. D. Solomon, Melt time and heat flux for a simple PCM body, Solar Energy 22, 251-257 (1979)
  • [19] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., The Macmillan Company, New York, 1944, pp. 490-492, and 506
  • [20] H. F. Weinberger, A First Course in Partial Differential Equations, Xerox College Publishing, Lexington, 1965, p. 73

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35K20, 35K65

Retrieve articles in all journals with MSC: 35K20, 35K65

Additional Information

DOI: https://doi.org/10.1090/qam/1330648
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society