  We study positive solutions of the pseudoparabolic equation with a sublinear
source in $\mathbb{R}^n$. In this work, the source coefficient could be
unbounded and time-dependent. Global existence of solutions to the Cauchy
problem is established within weighted continuous spaces by approximation and
monotonicity arguments. Every solution with non-zero initial value is shown to
exhibit a certain lower grow-up and radial growth bound, depending only upon
the sublinearity and the unbounded, time-dependent potential. Using the lower
grow-up/growth bound, we can prove the key comparison principle. Then we settle
the uniqueness of solutions for the problem with non-zero initial condition by
employing the comparison principle. For the problem with the zero initial
condition, we can classify the non-trivial solutions in terms of the maximal
solutions. When the initial condition has a power radial growth, we can derive
the precise asymptotic grow-up rate of solutions and obtain the critical growth
exponent.
