Global asymptotic behavior of solutions of a semilinear parabolic equation. (English) Zbl 0887.35074
Summary: 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.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
References:
| [1] | D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968), 607 – 694. · Zbl 0182.13802 |
| [2] | M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209 – 273. · Zbl 0459.60069 · doi:10.1002/cpa.3160350206 |
| [3] | F. Chiarenza, E. Fabes, and N. Garofalo, Harnack’s inequality for Schrödinger operators and the continuity of solutions, Proc. Amer. Math. Soc. 98 (1986), no. 3, 415 – 425. · Zbl 0626.35022 |
| [4] | Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for \?_{\?}=\Delta \?+\?^{1+\?}, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109 – 124 (1966). · Zbl 0163.34002 |
| [5] | Carlos E. Kenig and Wei-Ming Ni, An exterior Dirichlet problem with applications to some nonlinear equations arising in geometry, Amer. J. Math. 106 (1984), no. 3, 689 – 702. · Zbl 0559.35025 · doi:10.2307/2374291 |
| [6] | Fang-Hua Lin, On the elliptic equation \?\?[\?\?\?(\?)\?\?\?]-\?(\?)\?+\?(\?)\?^{\?}=0, Proc. Amer. Math. Soc. 95 (1985), no. 2, 219 – 226. · Zbl 0584.35031 |
| [7] | Wei Ming Ni, On the elliptic equation \Delta \?+\?(\?)\?^{(\?+2)/(\?-2)}=0, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493 – 529. · Zbl 0496.35036 · doi:10.1512/iumj.1982.31.31040 |
| [8] | Qi Zhang, On a parabolic equation with a singular lower order term, Trans. Amer. Math. Soc. 348 (1996), no. 7, 2811 – 2844. · Zbl 0860.35044 |
| [9] | Qi Zhang, Global existence and local continuity of solutions for semilinear parabolic equations, Comm. PDE, to appear 1997. · Zbl 0883.35061 |
| [10] | Qi Zhang, Linear parabolic equations with singular lower order coefficients, PhD Thesis, Purdue U. 1996. |
| [11] | Z. Zhao, On the existence of positive solutions of nonlinear elliptic equations — a probabilistic potential theory approach, Duke Math. J. 69 (1993), no. 2, 247 – 258. · Zbl 0793.35032 · doi:10.1215/S0012-7094-93-06913-X |
| [12] | Z. Zhao, Subcriticality, positivity and gaugeability of the Schrödinger operator, Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 513 – 517. · Zbl 0744.35042 |
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.
