On unique solvability of boundary value problems for semilinear parabolic equations in unbounded domains without conditions at infinity

The Cauchy problem, the first initial-boundary value problem, and the problem without initial conditions are considered for semilinear parabolic equations on unbounded domains. Conditions on the equation coefficients are established under which solutions to the problems are unique without restrictions on the behavior at infinity. Under these conditions the existence of solutions to the considered problems is proved without stipulating assumptions about the geometry of the domain and the growth of right-hand sides at infinity.