Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation

This article is devoted to the study of the initial boundary value problem for a mixed pseudo-parabolic r ( x ) r\left(x) -Laplacian-type equation. First, by employing the imbedding theorems, the theory of potential wells, and the Galerkin method, we establish the existence and uniqueness of global solutions with subcritical initial energy, critical initial energy, and supercritical initial energy, respectively. Then, we obtain the decay estimate of global solutions with sub-sharp-critical initial energy, sharp-critical initial energy, and supercritical initial energy, respectively. For supercritical initial energy, we also need to analyze the properties of ω \omega -limits of solutions. Finally, we discuss the finite-time blowup of solutions with sub-sharp-critical initial energy and sharp-critical initial energy, respectively.