We assume that $u$ satisfies the heat equation $\partial_t u-\Delta u=f$ in some parabolic cylinder $Q_R=B_R\times(-R^2,0]$. What is the so-called bootstrap arguments to get the estimates of the following norms
$$||u||_{L^{\infty}(Q_{\frac{1}{2}R})}+||\nabla u||_{L^{\infty}(Q_{\frac{1}{2}R})}+||\nabla^2 u||_{L^{\infty}(Q_{\frac{1}{2}R})}+\sup_{-\frac{1}{4}R^2\leq t\leq 0}||\partial_t u(t)||_{L^{p}(B_{\frac{1}{2}R})}$$
by a constant relates to $f$ ?
Any suggestions are welcome! Thanks.