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 Free Access
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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.
V. Alexiades, Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems, Arch. Rational Mech. Anal. 79, 325–350 (1982)
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)
S. B. Chae, Lebesgue Integration, Marcel Dekker, New York, 1980, pp. 227–228
C. Y. Chan, New results in quenching, Proceedings of the First World Congress of Nonlinear Analysts, Walter de Gruyter & Co. (to appear)
C. Y. Chan and C. S. Chen, A numerical method for semilinear singular parabolic quenching problems, Quart. Appl. Math. 47, 45–57 (1989)
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)
C. Y. Chan and S. S. Cobb, Critical lengths for semilinear singular parabolic mixed boundary-value problems, Quart. Appl. Math. 49, 497–506 (1991)
C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math. Anal. 20, 558–566 (1989)
C. Y. Chan and B. M. Wong, Periodic solutions of singular linear and semilinear parabolic problems, Quart. Appl. Math. 47, 405–428 (1989)
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)
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
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)
K. Knopp, Theory and Application of Infinite Series, Hafner Publishing Company, New York, 1928, pp. 146, 337, and 346
J. Lamperti, A new class of probability theorems, J. Math. Mech. 11, 749–772 (1962)
N. W. McLachlan, Bessel Functions for Engineers, 2nd ed., Oxford University Press, London, 1955, pp. 26, 102–104, and 116
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, pp. 168–170
H. L. Royden, Real Analysis, 2nd ed., Macmillan Publishing Co., New York, 1968, pp. 84, 88, and 269–270
A. D. Solomon, Melt time and heat flux for a simple PCM body, Solar Energy 22, 251–257 (1979)
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., The Macmillan Company, New York, 1944, pp. 490–492, and 506
H. F. Weinberger, A First Course in Partial Differential Equations, Xerox College Publishing, Lexington, 1965, p. 73
V. Alexiades, Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems, Arch. Rational Mech. Anal. 79, 325–350 (1982)
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)
S. B. Chae, Lebesgue Integration, Marcel Dekker, New York, 1980, pp. 227–228
C. Y. Chan, New results in quenching, Proceedings of the First World Congress of Nonlinear Analysts, Walter de Gruyter & Co. (to appear)
C. Y. Chan and C. S. Chen, A numerical method for semilinear singular parabolic quenching problems, Quart. Appl. Math. 47, 45–57 (1989)
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)
C. Y. Chan and S. S. Cobb, Critical lengths for semilinear singular parabolic mixed boundary-value problems, Quart. Appl. Math. 49, 497–506 (1991)
C. Y. Chan and H. G. Kaper, Quenching for semilinear singular parabolic problems, SIAM J. Math. Anal. 20, 558–566 (1989)
C. Y. Chan and B. M. Wong, Periodic solutions of singular linear and semilinear parabolic problems, Quart. Appl. Math. 47, 405–428 (1989)
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)
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
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)
K. Knopp, Theory and Application of Infinite Series, Hafner Publishing Company, New York, 1928, pp. 146, 337, and 346
J. Lamperti, A new class of probability theorems, J. Math. Mech. 11, 749–772 (1962)
N. W. McLachlan, Bessel Functions for Engineers, 2nd ed., Oxford University Press, London, 1955, pp. 26, 102–104, and 116
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer-Verlag, New York, 1984, pp. 168–170
H. L. Royden, Real Analysis, 2nd ed., Macmillan Publishing Co., New York, 1968, pp. 84, 88, and 269–270
A. D. Solomon, Melt time and heat flux for a simple PCM body, Solar Energy 22, 251–257 (1979)
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., The Macmillan Company, New York, 1944, pp. 490–492, and 506
H. F. Weinberger, A First Course in Partial Differential Equations, Xerox College Publishing, Lexington, 1965, p. 73
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Article copyright:
© Copyright 1995
American Mathematical Society
