Global asymptotic behavior of solutions of a semilinear parabolic equation

Zhang, Qi; Zhao, Z.

doi:10.1090/S0002-9939-98-04525-0

Global asymptotic behavior of solutions of a semilinear parabolic equation
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by Qi S. Zhang and Z. Zhao

Proc. Amer. Math. Soc. 126 (1998), 1491-1500

DOI: https://doi.org/10.1090/S0002-9939-98-04525-0

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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.

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