# Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question:

Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$: $$\frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t),$$ with smooth initial data $f_0 \colon \Omega \to \mathbb{R}^n$. Suppose we know that there exists a smooth solution $f_t$ for all times $t \in [0, \infty)$. Can one find a time $t_0 > 0$ such that the $W^{2,2}$-norm of $f_{t_0}$ (on some smaller domain) is bounded by an expression depending only on $\|f_0\|_{W^{1,2}}$ and on the coefficients of $Q$? Here $t_0$ may depend freely on $f_0$, however.

Motivation:

Due to my complete lack of knowledge on the field of PDE, the question is probably very ill-posed, so here's a bit of background. Consider the Eells-Sampson heat flow equation, which in local coordinates reads: $$\tag{1} \frac{\partial f_t^{\alpha}}{\partial{t}} = - \Delta f_t^\alpha + {}^N\Gamma_{\beta\gamma}^\alpha \frac{\partial f^\beta}{\partial x^i}\frac{\partial f^\gamma}{\partial x^j}g^{ij} -{}^M\Gamma_{ij}^k \frac{\partial f^\alpha}{\partial x^k}g^{ij}$$ (here, from the global point of view, we are in Corlette's setting, that is, we are given a representation $\rho \colon \pi_1(M) \to G$ and we are considering $\pi_1(M)$-equivariant maps $f\colon \tilde M \to N = G/K$). We know that for every $\pi_1(M)$-equivariant initial data $f = f_0$ solutions of (1) exist for all time, and that the $L^2$-norm of $d f_t \colon \tilde M \to N$ is decreasing. I would like to use the method above to find another equivariant map $f'= f_{t_0}$ with explicit bounds on the norms of higher derivatives (at least on the $W^{2,2}$ norm) depending only on $E(f) = \|d f_0\|_{L^2}^2$.

Corlette, in his original paper of 1988, in order to construct the solution gives a bound on the $W^{1,2}$ norm of $df$ ($\theta$, in his notations), but this is not what I need, since it also depends on the $L^2$-norm of the tension field $\tau(f)$ ($\Phi$, in his notations); in fact, he considers estimates that hold true for every $t$, while I am really interested in improving the properties of $f_0$.

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