Aix-Marseille Université
Institut de Mathématiques de Marseille (I2M) - UMR 7373
3 place Victor Hugo
Case 19
13331 Marseille Cedex 3


Regularity of the semigroup associated with some interacting elastic systems

Date(s) : 28/05/2024   iCal
11h00 - 12h00

In this talk, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power $theta$, with $theta$ in $[-1, 1]$, of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for $theta$ in $(frac{1}{2},1]$, the underlying semigroup is not analytic, but is differentiable for $theta$ in $(0, 1)$; this is in sharp contrast with known results for a singlesimilarly damped elastic system, where the semigroup is analytic for $theta$ in $[frac{1}{2}, 1]$; this shows that the degeneracy dominates the dynamics of the interacting systems, preventing analyticity in that range. Next, we show that
for $theta$ in $(0, frac{1}{2}]$, the semigroup is of certain Gevrey classes. Finally, we show that the semigroup decays exponentially for $theta$ in $[0, 1]$, and polynomially for $theta$ in $[-1, 0)$. To prove our results, we use the frequency domain method, which relies on resolvent estimates. Optimality of our resolvent estimates is also established. Two examples of application are provided.

Salle de séminaire de l'I2M à St Charles


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