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 $E(0) \lt d$ and $E(0) = d$. Furthermore, the infinite time blow up of solutions for the problem with $E(0) \gt 0$ ($\omega = 0$) is studied.