The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization

Abstract

In this work, we consider a system of two wave equations coupled by velocities in a one-dimensional space, with one boundary fractional damping. First, we show that the system is strongly asymptotically stable if and only if the coupling parameter b of the two equations is outside a discrete set of exceptional real values. Next, we show that our system is not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using a frequency domain approach combined with the multiplier method, we prove that the energy decay rate is greatly influenced by the nature of the coupling parameter b, the arithmetic property of the wave propagation speed a and the order of the fractional damping $$\alpha $$ . Indeed, under the equal speed propagation condition, i.e., $$a=1$$ , we establish an optimal polynomial energy decay rate of type $$t^{-\frac{2}{{1-\alpha }}}$$ if the coupling parameter $$b\notin \pi {\mathbb {Z}}$$ and of type $$t^{-\frac{2}{{5-\alpha }}}$$ if the coupling parameter $$b\in \pi {\mathbb {Z}}$$ . Furthermore, when the wave propagates with different speeds, i.e., $$a\not =1$$ , we prove that, for any rational number $$\sqrt{a}$$ and almost all irrational numbers $$\sqrt{a}$$ , the energy of our system decays polynomially to zero like as $$t^{-\frac{2}{{5-\alpha }}}$$ . This result still holds if $$a\in {\mathbb {Q}}$$ , $$\sqrt{a}\notin {\mathbb {Q}}$$ and b small enough.