Abstract
Cauchy problem and initial boundary value problem for nonlinear parabolic equation inCB([0,T):L p ) orL q (0,T; L p ) type space are considered. Similar to wave equation and dispersive wave equation, the space-time means for linear parabolic equation are shown and a series of nonlinear estimates for some nonlinear functions are obtained by space-time means. By Banach fixed point principle and usual iterative technique a local mild solution of Cauchy problem or IBV problem is constructed for a class of nonlinear parabolic equations inCB([0,T);L p orL q (0,T; L p ) with ϕ(x)∈L r . In critical nonlinear case it is also proved thatT can be taken as infinity provided that ||ϕ(x)||r is sufficiently small, where (p,q,r) is an admissible triple.
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