On uniqueness classes of solutions of the first mixed problem for a quasi-linear second-order parabolic system in an unbounded domain

We study a quasi-linear parabolic system of divergence type having an energy inequality and satisfying monotonicity conditions. For such a system, the first mixed problem is considered in a cylindrical domain $\{t>0\}\times\Omega$ that is unbounded with respect to the spatial variables. Generally, the initial vector function $\varphi$ in the problem may not belong to $\mathbb L_2(\Omega)$. A uniqueness class close to that of Täcklind [3] is established for the solutions of this problem. Moreover, a uniqueness theorem is proved for a solution belonging to this class and having an initial vector function increasing at infinity.