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

We investigate positive solutions of pseudoparabolic equations ∂tu−△∂tu=△u+V(x)up∂tu−△∂tu=△u+V(x)up in Rn×(0,∞)Rn×(0,∞), where p>1p>1 and VV 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=∣x∣σV=∣x∣σ where 0≤σ≤4n−2 if n≥3n≥3 and σ∈[0,∞)σ∈[0,∞) if n=1,2n=1,2, then the critical exponent of the Cauchy problem is 1+σ+2n. This generalizes the result in the case σ=0σ=0 by Cao et al. (2009).