Global and blow-up solutions of superlinear pseudoparabolic equations with unbounded coefficient

We investigate positive solutions of pseudoparabolic equations partial derivative(t)u - Delta partial derivative(t)u = Delta u+ V (x)u(p) in R-n x (0, infinity), where p > 1 and V is a (possibly unbounded or singular) potential. Under some rather weak assumptions on the potential, we establish the existence of solutions, both locally and globally in time, within weighted Lebesgue spaces for the Cauchy problem. Blow-up behavior is also derived using the test function method. As a consequence, we show that if V = vertical bar x vertical bar(sigma) where 0 <= sigma <= 4/n-2 if n >= 3 and sigma is an element of [0, infinity) if n = 1, 2, then the critical exponent of the Cauchy problem is 1 + sigma+2/n. This generalizes the result in the case sigma = 0 by Cao et al. (2009). (C) 2015 Elsevier Ltd. All rights reserved.