# zbMATH — the first resource for mathematics

On the asymptotic behavior of solutions of parabolic equations. (English) Zbl 0534.35051
We consider the asymptotic behavior of solutions u(x,t) of $$-u_ t+L_ xu=-f(x,t)$$ in the cylinder $$\Omega\times(0,\infty)$$ where $$L_ x$$ is an elliptic operator, and show that under the assumption $$\int_{\Omega}| f(x,t)|^ p\quad dx\to 0$$ as $$t\to \infty$$ for p sufficiently large, the solutions satisfying homogeneous Dirichlet or Robin boundary conditions tend uniformly to zero as $$t\to \infty$$; while the solutions satisfying a homogeneous Neumann boundary condition tend uniformly to constants a $$t\to \infty$$ provided that $$\lim_{t\to \infty}\int^{t}_{0}\int_{\Omega}f(x,\tau)dx d\tau$$ exists.

##### MSC:
 35K20 Initial-boundary value problems for second-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
Full Text:
##### References:
 [1] R. A. Adams: Sobolev Spaces. Academic Press, New York, 1975. · Zbl 0314.46030 [2] R. Arima: On general boundary value problems for parabolic equations. J. Math. Kyoto Univ. 4-1 (1964), 207-243. · Zbl 0143.13902 [3] A. Friedman: Partial Differential Equations of Parabolic Type. Prentice Hall Inc., Englewood Cliffs, 1964. · Zbl 0144.34903 [4] A. Friedman: Partial Differential Equations. Holt, Rinehart and Winston Inc., New York, 1969. · Zbl 0224.35002 [5] E. Gagliardo: Proprietà di alcune classi di funzioni in pui variabili. Ricerche di Mat. 7 (1958), 102-137. · Zbl 0089.09401 [6] E. Gagliardo: Ulteriori propriétà di alcune classi di funzioni in pui variabili. Ricerche di Mat. 8 (1959), 24-51. · Zbl 0199.44701 [7] R. A. Gardner: Asymptotic behavior of semilinear reaction-diffusion systems with Dirichlet boundary conditions. Indiana Univ. Math. J. 29 (1980), 161-190. · Zbl 0428.35011 [8] R. A. Gardner: Global stability of stationary solutions of reaction-diffusion systems. J. Diff. Eqs. 37 (1980), 60-69. · Zbl 0413.34058 [9] D. Gilbarg, N. S. Trudinger: Elliptic Partial Differential Equations of Second Order. Springer Verlag, Berlin, 1977. · Zbl 0361.35003 [10] G. H. Hardy J. E. Littlewood, G. Polya: Inequalities. 2nd, Cambridge University Press, Cambridge, 1952. [11] A. M. Il’in A. S. Kalashnikov, O. A. Oleinek: Second order linear equations of parabolic type. Russian Math. Surveys 17, No. 3 (1962), 1-143. · Zbl 0108.28401 [12] S. Itô: A boundary value problem of partial differential equations of parabolic type. Duke Math. J. 24, (1957), 299-312. · Zbl 0084.30202 [13] S. Itô: Fundamental solutions of parabolic differential equations and boundary value problems. Japan J. Math. 27 (1957), 55-102. · Zbl 0092.31101 [14] C. S. Kahane: On a system of nonlinear parabolic equations arising in chemical engineering. J. Math. Anal. Appl. 53 (1976), 343-358. · Zbl 0326.35044 [15] O. A. Ladyzhenskaja V. A. Solonnikov, N. N. Ural’ceva: Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, 1968. [16] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations. Springer Verlag, New York, 1966. · Zbl 0142.38701 [17] J. Nash: Continuity of the solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931-954. · Zbl 0096.06902 [18] L. Nirenberg: On elliptic partial differential equations. Ann. Scuo. Norm. Pisa 13 (3) (1959), 1-48. · Zbl 0088.07601 [19] W. Pogorzelski: Etude d’un function de Green et du problème aux limites pour l’équation parabolique normale. Ann. Polon. Math. 4 (1958), 288-307. · Zbl 0084.30102 [20] G. F. Webb: A determinitic diffusive epidemic model with an incubation period. Proceedings of the Functional Differential and Integral Equations Conference held at West Virginia University, June 18-20, 1979
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.