On the existence of a weak solutions of a parabolic initial-boundary value problem in a class of repidly increasing functions

The initial-boundary value problem
\begin{gather*} \mathscr Lu\equiv\frac{\partial u}{\partial t}-\sum_{i, j=1}^n\frac{\partial}{\partial x_i}(a_{ij}(x, t)u_{x_j})+\sum_{i=1}^na_iu_{x_i}+au=f-\sum_{i=1}^n\frac{\partial f_i}{\partial x_i},\\ u|_{t=0}=\varphi(x),\quad u|_{\partial\Omega}=0, \end{gather*}
i s considered in an unbounded domain $\Omega\subset\mathbb R^n$. It is proved that this problem possesses the unique weak solution whose $W^{1, 0}_2(Q_{r, T})$-norm does not exceed $C_1e^{\lambda r^2}$, $\forall r>0$, $Q_{r, T}=\Omega_r\times(0, T)$, $\Omega_r=\{x\in\Omega:|x|<r\}$.