Time-Space Estimates of Solutions to General Semilinear Parabolic Equations

Abstract

We study the Cauchy problem and the initial boundary value problem (IBVP) for nonlinear parabolic equations in $\mathcal{C}_b([0,T);L^p)$ and $L^q(0,T;L^p)$. We give a unified method to construct local mild solutions of the Cauchy problem or IBVP for a class of nonlinear parabolic equations in $\mathcal{C}_b([0,T);L^p)$ or $L^q(0,T;L^p)$ by introducing admissible triplet, generalized admissible triplet and establishing time space estimates for the solutions to the linear parabolic equations. Moreover, using our method, we also obtain the existence of global small solutions to the nonlinear parabolic equations.