Indirect stabilization of coupled abstract evolution equations with memory: Different propagation speeds

Abstract

Listen

Translate article

We consider the Cauchy problem for coupled second‐order evolution equations in a Hilbert space with indirect memory damping, which arises from the theory of viscoelasticity. Here, the coefficients of the main operators in the system are non‐equal (the case of different speeds of propagation). Under much weaker assumptions on memory kernels, we obtain a polynomial decay rate, whereas it is known that exponential stability does not hold. This work essentially generalizes and improves the related ones in the literature. As an application of our abstract results, we also establish for the Timoshenko system the so far best decay rate under such weak assumptions on the memory kernel in the case of different speeds of propagation.