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In this work we study the existence, uniqueness and decay of solutions to a class of viscoelastic equations in a separable Hilbert space $H$given by $$\begin{array}{c}{u}_{tt}+M\left(\left[u\right]\right)Au-{\int }_0^{t}g(t-\tau )N\left(\left[u\right]\right)Aud\tau =0,\text{in}{L}^2(0,T;H)\\ u\left(0\right)=u_0,u_t\left(0\right)=u_1\end{array}$$ where by$\left[u\left(t\right)\right]$we are denoting $$\left[u\left(t\right)\right]=\left((u\left(t\right),u_t\left(t\right),(Au\left(t\right),u_t\left(t\right)),\parallel A^{\frac{1}{2}}u\left(t\right){\parallel }^2,\parallel A^{\frac{1}{2}}{u}_t\left(t\right){\parallel }^2,\parallel Au\left(t\right){\parallel }^2\right)\in {ℝ}^5$$ $A:D\left(A\right)\subset H\to H$ is a nonnegative, self-adjoint operator, $M$, $N:{\mathbb{R}}^5\to \mathbb{R}$ are $C^2$- functions and $g:\mathbb{R}\to \mathbb{R}$ is a $C^3$-function with appropriates conditions. We show that there exists global solution in time for small initial data. When $\left[u\left(t\right)\right]={∥A^{\frac{1}{2}}u∥}^2$ and $N=1$, we show the global existence for large initial data $\left(u_0,u_1\right)$ taken in the space $D\left(A\right)D\left(A^{1/2}\right)$ provided they are close enough...

We study the thermoelastic system for material which are partially thermoelastic. That is, a material divided into two parts, one of them a good conductor of heat, so there exists a thermoelastic phenomenon. The other is a bad conductor of heat so there is not heat flux. We prove for such models that the solution decays exponentially as time goes to infinity. We also consider a nonlinear case.