Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. (English) Zbl 0920.35030

The author investigates the asymptotic behavior of solutions to a semilinear parabolic equation on \(\Omega\times \mathbb{R}^+\) with given initial data and nonlocal boundary condition of the form \(Bu(x,t)=\)
\(\int_\Omega K(x,y)u(t,y)dy\), where \(Bu= \alpha_0\partial u/\partial\nu+ u\) and \(\alpha_0\geq 0\) (nonlocal Dirichlet or Robin condition). Under suitable assumptions on \(K\) and the nonlinearity the solution displays corresponding asymptotic behavior. For \(K\geq 0\) and \(\widehat K(x)= \int_\Omega K(x,y)dy\geq 1\), for instance, the solution can blow up in finite time.
Reviewer: B.Kawohl (Köln)


35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
Full Text: DOI


[1] Day, W. A., Existence of a property of the heat equation to linear thermoelasticity and other theories, Quart. Appl. Math., 40, 319-330 (1982) · Zbl 0502.73007
[2] Day, W. A., Heat Conduction within Linear Thermoelasticity (1985), Springer: Springer New York · Zbl 0577.73009
[3] Deng, K., Comparison principle for some nonlocal problems, Quart. Appl. Math., 50, 517-522 (1992) · Zbl 0777.35006
[4] Deng, K., Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions, J. Math. Anal. Appl., 44, 401-407 (1986)
[5] Kawohl, B., Remarks on a paper by W.A. Day on a maximum principle under nonlocal boundary conditions, Quart. Appl. Math., 44, 741-752 (1987) · Zbl 0617.35064
[7] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044
[8] Pao, C. V., Dynamics of reaction-diffusion equations with nonlocal boundary conditions, Quart. Appl. Math., 53, 173-186 (1995) · Zbl 0822.35070
[9] Pao, C. V., Reaction diffusion equations with nonlocal boundary and nonlocal initial conditions, J. Math. Anal. Appl., 195, 702-718 (1995) · Zbl 0851.35063
[10] Pao, C. V., Dynamics of weakly coupled parabolic systems with nonlocal boundary conditions, (Advances in Nonlinear Dynamics: Stability and Control, vol. 5 (1997)), 319-327 · Zbl 0923.35072
[11] Protter, H. M.; Weinberger, H. F., Maximum Principles in Differential Equations (1967), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602
[12] Yin, Y. F., On nonlinear parabolic equations with nonlocal boundary condition, J. Math. Anal. Appl., 185, 161-174 (1994) · Zbl 0820.35085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.