Global well-posedness and grow-up rate of solutions for a sublinear pseudoparabolic equation

We study positive solutions of the pseudoparabolic equation with a sublinear source in RnRn. In this work, the source coefficient (or potential) can be unbounded and time-dependent. Global existence of solutions to the Cauchy problem is established within weighted continuous spaces by approximation and a monotonicity argument. Every solution with a non-zero initial value is shown to exhibit a certain lower grow-up and radial growth bound, depending only upon the sublinearity and the potential. We prove the key comparison principle, using the lower grow-up and growth bound, and then settle the uniqueness of solutions for the problem with a non-zero initial value. For the zero initial-valued problem, we classify all non-trivial solutions in terms of the maximal solution. Finally, when the initial value has a power radial growth at infinity, we can derive the precise grow-up rate of solutions and obtain the critical growth exponent.