Global Existence and Finite Time Blow-Up for the Laplacian Equation with Variable Exponent Sources

In this paper, we study a class of semilinear m(.)-Laplacian equations with variable exponent sources. By using the potential well method, we discuss this problem at three different initial energy levels. When the initial energy is sub-critical, we obtain the blow-up result and estimate the lower and upper bounds of the blow-up time. In the case of critical initial energy, we prove global existence, asymptotic behavior, and finite-time blow-up and determine the lower bound of the blow-up time. For super-critical initial energy, we establish the finite-time blow-up and estimate the lower and upper bounds of the blow-up time.