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 \(\begin{aligned} \vert y_t\vert ^{r}y_{tt}{} & {} +\gamma \Delta ^2 y_{tt}+\Delta ^2y-\left( a+b\vert \vert \nabla y\vert \vert ^2+c (\nabla y, \nabla y_t)\right) \Delta y\\{} & {} \quad +\Delta ^2 y_t+ d(x)h(y_t)+f(y)=0, \end{aligned}\) $$\begin{array}{cc}\hfill |y_t{|}^r{y}_{\mathit{tt}}& +\gamma {\mathrm{\Delta }}^2{y}_{\mathit{tt}}+{\mathrm{\Delta }}^{2}y-\left(a,+,{b||\mathrm{\nabla }y||}^2,+,c,(\mathrm{\nabla }y,\mathrm{\nabla }{y}_t)\right)\mathrm{\Delta }y\hfill \\ \hfill & +{\mathrm{\Delta }}^2{y}_t+d(x)h(y_t)+f(y)=0,\hfill \end{array}$$ in a bounded domain of \(\mathbb {R}^N\) ${\mathbb{R}}^N$ , with clamped boundary conditions in either cases: when \(r=\gamma =0\) $r=\gamma =0$ or else when r and \(\gamma \) $\gamma$ are positive. We prove, in both cases, the existence of solutions and the exponential decay of energy.