Existence and Asymptotic Behaviors to a Nonlinear Fourth-order Parabolic Equation with a General Source

Abstract

The existence and asymptotic behavior of solutions a fourth-order partial differential equation with a $p$-Laplacian diffusion and a nonlinear source are studied by using potential well theory. When the initial functionals satisfy $\mathcal{F}(w_{0}) \lt d$, $\mathcal{D}(w_{0}) \gt 0$ or $\mathcal{F}(w_{0}) = d$, $\mathcal{D}(w_{0}) \geq 0$, the existence and exponential decay result of weak solutions are given. For $\mathcal{F}(w_{0}) \lt d$, $\mathcal{D}(w_{0}) \lt 0$ or $\mathcal{F}(w_{0}) = d$, $\mathcal{D}(w_{0}) \lt 0$, we obtain the blow-up behavior at a finite time for weak solutions. For $\mathcal{F}(w_{0}) \gt d$, we show the global existence for small initial datum and blow-up for big initial datum. Moreover, the uniqueness holds for bounded solutions. In addition, we show that the $p$-Laplacian term has an essential effect to the source function so that we add some growth conditions to $g(w)$.