Global Well-posedness of Solutions for the $p$-Laplacian Hyperbolic Type Equation with Weak and Strong Damping Terms and Logarithmic Nonlinearity

Abstract

In this paper, we consider the p-Laplacian hyperbolic type equation with weak and strong damping terms and logarithmic nonlinearity. By using the potential well method and a logarithmic Sobolev inequality, we prove global existence, infinite time blow up and asymptotic behavior of solutions in two cases E(0) < d and E(0) = d. Furthermore, the infinite time blow up of solutions for the problem with E(0) > 0 (ω = 0) is studied.