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

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 F(w₀) < d, D(w₀) > 0 or F(w₀) = d, D(w₀) ≥ 0, the existence and exponential decay result of weak solutions are given. For F(w₀) < d, D(w₀) < 0 or F(w₀) = d, D(w₀) < 0, we obtain the blow-up behavior at a finite time for weak solutions. For F(w₀) > 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 $g(w)$.