# Regularity of parabolic PDEs for large $\lambda$

Let $$\Omega$$ be a sufficiently smooth domain, $$T>0$$, and $$L$$ be the following elliptic operator of the divergence form: $$Lu(t,x)=a^{ij}(t,x)u_{ij}(t,x),$$ such that $$a^{ij}\in C([0,T]\times\bar{\Omega})$$ and satisfies the uniform elliptic condition.

I was wondering whether there is a reference showing that, there exist constants $$C$$ and $$\lambda_0$$, such that for all $$\lambda\ge \lambda_0$$ and for all
$$u\in L^2((0,T);H^2(\Omega))\cap H^1((0,T);L^2(\Omega)):= H^{2,1}(\Omega_T)$$ with $$u(0,\cdot)=0$$ and $$u=0$$ on $$\partial \Omega$$, we have the estimate: $$\|u\|_{H^{2,1}(\Omega_T)}+\lambda\|u\|_{L^2(\Omega_T)}\le C\|\partial_t u-Lu+\lambda u\|_{L^2(\Omega_T)}.$$ where we denote $$\Omega_T=(0,T)\times \Omega$$.

For elliptic equations, a similar estimate has been shown in Theorem 9.14 of Gilbarg's book for $$Lu(x)=a^{ij}(x)u_{ij}(x)$$ under the assumption that $$a^{ij}\in C(\bar{\Omega})$$. For parabolic equation on the whole space $$\mathbb{R}^n$$, the estimate has been shown in Krylov's book. Hence I guess a similar estimate for parabolic equations on a bounded domain may already been shown somewhere.

I think it suffices to show $$\lambda\|u\|_{L^2(\Omega_T)}\le C\|\partial_t u-Lu+\lambda u\|_{L^2(\Omega_T)}.$$ Since then by letting $$f=\partial_t u-Lu+\lambda u$$, we can estimate the $$H^{2,1}$$-norm by $$\|u\|_{H^{2,1}(\Omega_T)}\le C\|\partial_t u-Lu\|_{L^2(\Omega_T)}\le C(\|f\|_{L^2(\Omega_T)}+\lambda \|u\|_{L^2(\Omega_T)})\le C\|f\|_{L^2(\Omega_T)}.$$ However, I am not sure how to derive the $$L^2$$ estimate, since $$L$$ is of the nondegenerate form.