Abstract
A sharp result on global small solutions to the Cauchy problem $u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $
In Rn is obtained under the the assumption that f is C1+r for r>2/n and ‖u0‖C2(Rn) +‖u0‖W ${}_{1}^{2}$ (Rn) is small. This implies that the assumption that f is smooth and ‖u0 ‖W ${}_{1}^{k}$ (Rn)+‖u0‖W ${}_{2}^{k}$ (Rn) is small for k large enough, made in earlier work, is unnecessary.
Citation
Chen Zhimin. "Long time small solutions to nonlinear parabolic equations." Ark. Mat. 28 (1-2) 371 - 381, 1990. https://doi.org/10.1007/BF02387387
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