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| In this work we study decay estimates for the energy and the $L^2$-norm of solutions for several models associated to the following abstract second order evolution equation
$$A_1u_{tt}(t, x) + A_2u_t(t, x) + A_3u(t, x) = 0$$
with initial conditions
$$u(0, x) = u_0(x) \quad \mbox{and} \quad u_t(0, x) = u_1(x); $$
where $t \in R^{+}$, $x \in R^{n}$ and $A_i$ ($i = 1, 2, 3$) are positive self adjoint di fferential operators with symbols given by functions $P_i(y)$ ($i = 1, 2, 3$). To obtain decay estimates to the above problem the idea is to work with the associated problem in the Fourier space. To this end we take the Fourier transform of the above problem to obtain the following initial problem
$$P_1(y) v_{tt}(t, y) + P_2(y) v_t(t, y) + P_3(y)v(t, y) = 0;$$
$$v(0, y) = v_0(y) \quad \mbox{and} \quad v_t(0,y ) = v_1(y)$$
where $v$ is the Fourier transform of $u$. This problem is an initial value one to a linear ordinary di fferential equation of second order with coefficients depending on a parameter $y \in R^n$. Thus, we obtain estimates for an abstract initial value problem in Fourier space with diff erent conditions in $P_1(y)$, $P_2(y)$, $P_3(y)$ and the initial data. Then, the decay rates of the $L^2$-norm and of the energy associated to the abstract problem will be consequences of these results. In particular, we can use such results to get several estimates for initial problems associated to the wave equation, plate equation, IBq equation and more. The framework used in this work is supported on the energy method in the Fourier space combined with the integrability of functions of type $|y|^{-\alpha} f(y)$ for suitable $f(y)$ depending on the symbols $P_i(y)$ ($i = 1, 2, 3$) and the Haraux-Komornik Lemma. |
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