Well-posedness and Asymptotic Stability in Nonlinear Fourth-order Parabolic Evolution Equations

A 4th-order parabolic equation is researched through the combined application of potential well theory and energy estimation techniques. When $q \gt p-1$, the well-posedness and asymptotic behavior for $J(u_{0}) \lt d$, $I(u_{0}) \gt 0$ or $J(u_{0}) = d$, $I(u_{0}) \geq 0$ and finite-time blow-up phenomena for $J(u_{0}) \lt d$, $I(u_{0}) \lt 0$ or $J(u_{0}) = d$, $I(u_{0}) \lt 0$ are obtained respectively. For $J(u_{0}) \gt d$, based on different initial energy levels, we establish sufficient conditions for the global existence of solutions as well as blow-up phenomena. When $q \leq p-1$, we establish the global existence for $q \lt p-1$ with any $k \gt 0$ or $q = p-1$ with small $k$.