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We study linear pseudoparabolic equations with unbounded and time-dependentcoefficients. We solve the case which has remained open in several recentstudies of pseudoparabolic equations with unbounded and time-dependentcoefficients. In this work we get a sharp condition for the existence ornon-existence of solutions. Conditions on the initial function and coefficientare provided so that every nontrivial positive solution blows upinstantaneously. When the coefficient and the initial function do not grow toorapidly, we establish the existence and uniqueness of global solutions, forboth time-independent and time-dependent potentials. This is done via theanalysis of the Bessel convolution multiplication operators. Asymptoticbehavior and comparison principles are also established. The globalwell-posedness results can be extended to the equation with convection.

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