Dynamics of reaction-diffusion equations with nonlocal boundary conditions
Author:
C. V. Pao
Journal:
Quart. Appl. Math. 53 (1995), 173-186
MSC:
Primary 35K57; Secondary 35B40, 35Q72
DOI:
https://doi.org/10.1090/qam/1315454
MathSciNet review:
MR1315454
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Abstract: The purpose of this paper is to investigate the existence, uniqueness, and dynamics of a nonlinear reaction-diffusion equation with a nonlocal boundary condition which is motivated by a model problem arising from quasi-static thermoelasticity. The method of upper and lower solutions is used to obtain some existence-comparison results for both the time-dependent problem and its corresponding steady-state problem. A sufficient condition for the uniqueness of a steady-state solution is given. The comparison and uniqueness results are used to show the dynamical behavior of time-dependent solutions as well as their monotone convergence to a steady-state solution. Also given is a necessary and sufficient condition for the convergence of time-dependent solutions in relation to a steady-state solution which is not explicitly known. These results lead to the global stability of a steady-state solution for some special cases, including the model problem from thermoelasticity.
J. W. Bebernes and R. Ely, Comparison techniques and the method of lines for a parabolic functional equation, Rocky Mountain J. Math. 12, 723–733 (1982)
V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math. 46, 431–450 (1988)
W. A. Day, Extensions of a property of the 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)
P. De Mottoni, E. Orlandi, and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, J. Nonlinear Analysis 3, 663–675 (1979)
K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math. 50, 517–522 (1992)
A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44, 401–407 (1986)
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)
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992
C. V. Pao, Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory, J. Math. Anal. Appl. 166, 591–600 (1992)
C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87, 165–198 (1982)
C. V. Pao, Asymptotic stability and nonexistence of global solutions for a semilinear parabolic equation, Pacific J. Math. 84, 191–197 (1979)
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary-value problems, Indiana Univ. Math. J. 21, 979–1000 (1972)
J. W. Bebernes and R. Ely, Comparison techniques and the method of lines for a parabolic functional equation, Rocky Mountain J. Math. 12, 723–733 (1982)
V. Capasso and K. Kunisch, A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math. 46, 431–450 (1988)
W. A. Day, Extensions of a property of the 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)
P. De Mottoni, E. Orlandi, and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection, J. Nonlinear Analysis 3, 663–675 (1979)
K. Deng, Comparison principle for some nonlocal problems, Quart. Appl. Math. 50, 517–522 (1992)
A. Friedman, Monotonic decay of solutions of parabolic equations with nonlocal boundary conditions, Quart. Appl. Math. 44, 401–407 (1986)
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)
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992
C. V. Pao, Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory, J. Math. Anal. Appl. 166, 591–600 (1992)
C. V. Pao, On nonlinear reaction-diffusion systems, J. Math. Anal. Appl. 87, 165–198 (1982)
C. V. Pao, Asymptotic stability and nonexistence of global solutions for a semilinear parabolic equation, Pacific J. Math. 84, 191–197 (1979)
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967
D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary-value problems, Indiana Univ. Math. J. 21, 979–1000 (1972)
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Article copyright:
© Copyright 1995
American Mathematical Society
