  We study linear pseudoparabolic equations with unbounded and time-dependent
coefficients. We solve the case which has remained open in several recent
studies of pseudoparabolic equations with unbounded and time-dependent
coefficients. In this work we get a sharp condition for the existence or
non-existence of solutions. Conditions on the initial function and coefficient
are provided so that every nontrivial positive solution blows up
instantaneously. When the coefficient and the initial function do not grow too
rapidly, we establish the existence and uniqueness of global solutions, for
both time-independent and time-dependent potentials. This is done via the
analysis of the Bessel convolution multiplication operators. Asymptotic
behavior and comparison principles are also established. The global
well-posedness results can be extended to the equation with convection.
