Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Global asymptotic behavior of solutions
of a semilinear parabolic equation

Authors: Qi S. Zhang and Z. Zhao
Journal: Proc. Amer. Math. Soc. 126 (1998), 1491-1500
MSC (1991): Primary 35K57
MathSciNet review: 1458274
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the large time behavior of solutions for the semilinear parabolic equation $ \Delta u + Vu^{p} - u_{t} =0$. Under a general and natural condition on $V= V(x)$ and the initial value $u_{0}$, we show that global positive solutions of the parabolic equation converge pointwise to positive solutions of the corresponding elliptic equation. As a corollary of this, we recapture the global existence results on semilinear elliptic equations obtained by Kenig and Ni and by F.H. Lin and Z. Zhao. Our method depends on newly found global bounds for fundamental solutions of certain linear parabolic equations.

References [Enhancements On Off] (What's this?)

  • [A] D.G. Aronson, Non-negative solutions of linear parabolic equations, Annali della Scuola Norm. Sup. Pisa 22 (1968), 607-694. MR 55:8553
  • [AS] M. Aizenman and B. Simon, Brownian motion and Harnack's inequality for Schrödinger operators, Comm. Pure Appl. Math 35 (1982), 209-271. MR 84a:35062
  • [CFG] F. Chiarenza, E. Fabes and N. Garofalo, Harnack's inequality for Schrödinger operators and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), 415-425. MR 88a:35037
  • [F] H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}= \Delta u + u^{1 + \alpha }$, J. Fac. Sci. Univ. Tokyo, Sect I 13 (1966), 109-124. MR 35:5761
  • [KN] C. Kenig and W. M. Ni, An exterior Dirichlet problem with applications to some equations arising in geometry, Amer. J. Math. 106 (1984), 689-702. MR 85j:35072
  • [L] F. H. Lin, On the elliptic equation $D_{i}[a_{ij}(x)D_{j} U] - k(x) U^{p} =0$, Proc, AMS 95 (1985), 219-266. MR 86k:35041
  • [N] W. M. Ni, On the elliptic equation $\Delta v + K(x) u^{(n+2)/(n-2)} = 0$, its generalizations and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493-529. MR 84e:35049
  • [Zhang1] Qi Zhang, On a parabolic equation with a singular lower order term, Transactions of AMS 348 (1996), 2811-2844. MR 96k:35073
  • [Zhang2] Qi Zhang, Global existence and local continuity of solutions for semilinear parabolic equations, Comm. PDE, to appear 1997.
  • [Zhang3] Qi Zhang, Linear parabolic equations with singular lower order coefficients, PhD Thesis, Purdue U. 1996.
  • [Zhao1] Z. Zhao, On the existence of positive solutions of nonlinear elliptic equations- A probabilistic potential theory approach, Duke Math J. 69 (1993), 247-258. MR 94c:35090
  • [Zhao2] Z. Zhao, Subcriticality, positivity, and guageability of the Schrödinger operator, Bull. AMS 23 (1990), 513-517. MR 91h:35104

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35K57

Retrieve articles in all journals with MSC (1991): 35K57

Additional Information

Qi S. Zhang
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Z. Zhao
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Received by editor(s): November 6, 1996
Communicated by: Jeffrey B. Rauch
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society