Exponential stability of extensible beams equation with Balakrishnan–Taylor, strong and localized nonlinear damping

We study a nonlinear Cauchy problem modeling the motion of an extensible beam

$$\begin{aligned} \vert y_t\vert ^{r}y_{tt}{} & {} +\gamma \Delta ^2 y_{tt}+\Delta ^2y-\left( a+b\vert \vert \nabla y\vert \vert ^2+c (\nabla y, \nabla y_t)\right) \Delta y\\{} & {} \quad +\Delta ^2 y_t+ d(x)h(y_t)+f(y)=0, \end{aligned}$$

in a bounded domain of \(\mathbb {R}^N\), with clamped boundary conditions in either cases: when \(r=\gamma =0\) or else when r and \(\gamma \) are positive. We prove, in both cases, the existence of solutions and the exponential decay of energy.