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This paper concerns the initial-boundary value problem for a mixedpseudo-parabolic $p$-Laplacian type equation. By constructing a family ofpotential wells, we first present the explicit expression for the depth ofpotential well, and then prove the existence, uniqueness and decay estimate ofglobal solutions and the blowup phenomena of solutions with subcritical initialenergy. Next, we extend parallelly these results to the critical initialenergy. Lastly, the existence, uniqueness and asymptotic behavior of globalsolutions with supercritical initial energy are proved by further analyzing theproperties of $\omega$-limits of solutions.

Global existence and finite time blowup for a mixed pseudo-parabolic $p$-Laplacian type equation