Once again on evolution equations with monotone operators in Hilbert spaces and applications

We prove the existence of $W^{1}_{2}$-solutions of uniformly nondegenerate parabolic equations
$$ \partial_{t}u= D_i(a^{ij}_{t}D_{j}u_t+\beta^i_tu_t)+b^{i}_{t}D_{i} u_t +c_tu_t+f_{t} $$
in case that $f_{\cdot}\in (L_{2}+L_{1}) ([0,T],L_{2}(\mathbb{R}^{d}))$, $b=b^{M}+b^{B}$ and for some $r$ satisfying $2<r\leq d$ and sufficiently small constant $\hat b$
$$ \Big(\int_{B_{\rho}}\!\!\!\!\!\!\!\!\!\!-\quad|b^{M}_{t}|^{r}\,dx \Big)^{1/r}\leq \hat b\rho^{-1},\quad \rho\leq \rho_{0}, \quad \int_{0}^{T}\sup_{x}|b^{B}_{t}|^{2}\,dt<\infty. $$
Similar conditions are imposed on $\beta$ and $c$, so that $|b_{t}|=|\beta_{t}|=\varepsilon/|x|$, $|c_{t}|=\varepsilon/|x|^{2}$ are allowed. Even the case of $b^{M}=0$, $\beta=0,c=0$, $f\in L_{1}([0,T], L_{2}(\mathbb{R}^{d}))$ seems to be new. Functions $b\in L_{q}(L_{p}(\mathbb{R}^{d}))$ with $p>d,d/p+2/q=1$ are in the above described class.
Joint work with I. Gyöngy.