Comparison principle for some nonlocal problems
Author:
Keng Deng
Journal:
Quart. Appl. Math. 50 (1992), 517-522
MSC:
Primary 35K60; Secondary 35B05
DOI:
https://doi.org/10.1090/qam/1178431
MathSciNet review:
MR1178431
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Abstract: In this paper, for the parabolic equation ${u_t} = \Delta u + g\left ( {x, u} \right ), \left ( {x, t} \right ) \in \\ \Omega \times \left ( {0, T} \right )$, with nonlocal boundary conditions $u\left | {_{\partial \Omega }} \right . = \int _{\Omega } f\left ( {x, y} \right )u\left ( {y, t} \right )dy$, we establish the comparison theorem and local existence of the solution. We also discuss its long time behavior.
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N. W. Bazley and J. Weyer, Explicitly resolvable equations with functional non-linearities, Math. Methods Appl. Sci. 10, 477–485 (1988)
L. Byszewski, Strong maximum and minimum principles for parabolic problems with non-local inequalities, Z. Angew. Math. Mech. 70, 202–206 (1990)
J. Chabrowski, On non-local problems for parabolic equations, Nagoya Math. J. 93, 109–131 (1984)
W. A. Day, Extensions of a property of heat equation to linear thermoelasticity and other theories, Quart. Appl. Math. 40, 319–330 (1982)
W. A. Day, A decreasing property of solutions of parabolic equations with applications to thermoelasticity, Quart. Appl. Math. 40, 468–475 (1983)
K. Deng, Behavior of solutions of Burgers’ equation with nonlocal boundary conditions, preprint
A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44, 401–407 (1986)
A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1983
J. Graham-Eagle, Remarks on the theory of upper and lower solutions for boundary value problems of non-local type, preprint
B. Kawohl, Remarks on a paper by W. A. Day on a maximum principle under nonlocal boundary conditions, Quart. Appl. Math. 44, 751–752 (1987)
A. A. Kerefov, Non-local boundary value problems for parabolic equation, Differentsial’nye Uravneniya 15, 74–78 (1979) (Russian)
B. Straughan, R. E. Ewing, P. G. Jacobs, and M. J. Djomehri, Nonlinear instability for a modified form of Burger’s equation, Numer. Meth. for Partial Differential Equations 3, 51–64 (1987)
P. N. Vabishchevich, Non-local parabolic problems and the inverse heat-conduction problem, Differentsial’nye Uravneniya 17, 1193–1199 (1981) (Russian)
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Article copyright:
© Copyright 1992
American Mathematical Society
