Well‐posedness of solutions for a class of quasilinear wave equations with strong damping and logarithmic nonlinearity
Hang Ding, Jun Zhou
Abstract
Abstract This paper investigates the well‐posedness of solutions for the following quasilinear wave equation with strong damping and logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary: , where . By virtue of the classical Faedo–Galerkin method and some technical efforts, we first establish the local well‐posedness of solutions. Then we discuss the dynamical behaviors of solutions in detail: When and , we show that the solutions exist globally with subcritical and critical initial energy, where denotes the Nehari functional with the initial value u 0 . Especially, under further suitable assumptions about the initial data, we show that the energy functional decays exponentially. When and , we show that the solutions blow up in finite time with subcritical and critical initial energy. Moreover, by removing the restriction , we prove that the solutions may blow up in finite time with arbitrary high initial energy. In particular, we derive the upper and lower bounds of the blow‐up time. When and , we show that the maximal existence time of solutions can be extended to infinity and the solutions blow up at infinity with subcritical, critical, and arbitrary high initial energy.