Initial and Boundary Value Problems for a Class of Nonlinear Metaparabolic Equations

This paper is devoted to the initial and boundary value problems for a class of nonlinear metaparabolic equations ut − βuxx − k uxxt + γuxxxx = f(ux)x. At low initial energy level (J(u0) < d), we not only prove the existence of global weak solutions for these problems by the combination of the Galerkin approximation and potential well methods but also obtain the finite time blow-up result by adopting the potential well and improved concavity skills. Finally, we also discussed the finite time blow-up phenomenon for certain solutions of these problems with high initial energy.