Existence and exponential stability of solutions for a Balakrishnan–Taylor quasilinear wave equation with strong damping and localized nonlinear damping

Abstract

In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - ( ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ ( ∇ ⁡ z , ∇ ⁡ z t ) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ ( x ) ⁢ f ⁢ ( z t ) + g ⁢ ( z ) = 0 |z_{t}|^{\alpha}z_{tt}-\Delta z_{tt}-\bigl{(}\xi_{1}+\xi_{2}\|\nabla z\|^{2}+% \sigma(\nabla z,\nabla z_{t})\bigr{)}\Delta z-\Delta z_{t}+\beta(x)f(z_{t})+g(% z)=0 in a bounded domain of ℝ n {\mathbb{R}^{n}} with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established.