Summary: Let \(T>0\), \(\Omega\in\mathbb R^n\) be a unbounded domain, \(\Omega_{\tau}= \{(x,t): x\in\Omega,t=\tau\}\), \(Q_{t_1,t_2}=\Omega\times(t_1;t_2),{\mathcal K}\subset L^p(0,T,W_{\text{loc}}^{1,p}(\overline\Omega))\cap L^2_{\text{loc}}(\overline{Q_{0,T}})\) be a closure convex subset, \(p\in(1;2)\). We seek the function \(u\in{\mathcal K}\cap C([0;T];L_{\text{loc}}^{2}(\overline\Omega))\) such that \(u\) satisfies the parabolic variational inequality \[ \int_{Q_{0,\tau}}\left[v_t(v-u)\psi+\sum_{i=1}^{n}a_i| u_{x_i}| ^{p-2} u_{x_i}[(v-u)\psi]_{x_i}+(vu-f)(v-u)\psi+\frac{1}{2}\psi_t| v-u| ^2\right]dx~dt\geq \]
\[ \frac{1}{2}\int_{\Omega_{\tau}}| v-u| ^2\psi~dx -\frac{1}{2}\int_{\Omega_{0}}| v-u| ^2\psi~dx \] for all test function \(v\) and for all \(\tau\in(0,T]\) and arbitrary test functions \(\psi\geq0,v\). We suppose that \(u_0\in W_{\text{loc}}^{2,2}(\overline{\Omega})\), \(f,f_t\in L^2_{\text{loc}}(\overline{Q_{0,T}})\), coefiicients \(a_1\dots,a_n\) may increase if \(| x| \to\infty\). If some additional conditions are satisfied, then we prove that our variational inequality has a unique solution.