Instantaneous blow-up versus global well-posedness and asymptotic behavior for linear pseudoparabolic equations

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.