| Abstract: |
| We investigate the vibrating string equation
$\begin{equation*}
\frac{\partial^2 u}{\partial t^2}= a(t) \frac{\partial^{2}u}{\partial \xi^{2}} + g\biggl(t,\xi,u, \frac{\partial u}{\partial t} \biggr), \quad t \in[0,b], \xi \in (0,1),
\end{equation*}$
where the tension coefficient $a$ varies with time. Our strategy consists in transforming the PDE into the equivalent semilinear ODE
$\begin{equation*}
\ddot x(t)=a(t)A x(t) + f(t,x(t),\dot x(t)), \quad t \in [0,b],
\end{equation*}$
in the Banach space $ L^p([0,1]), $ having, as linear part, a multiplicative time-dependent perturbation of the spatial second derivative operator $ A $, generating a cosine family. Our technique is based on the reduction to the associated first order problem, whose linear part consists into a multiplicative time-dependent perturbation of a semigroup generator. The aim of our talk is providing sufficient conditions guaranteeing that such perturbations respectively generates a fundamental and an evolution system, finding an explicit formula in both cases. |
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