Decay character and global existence for weakly coupled system of semilinear σ$\sigma$‐evolution damped equations with time‐dependent damping

Cung The Anh, Phan Duc An and Pham Trieu Duong

Mathematische Nachrichten, 2025, vol. 298, issue 2, 478-510

Abstract: In this article, we investigate the existence and decay rate of the global solution to the coupled system of semilinear structurally damped σ$\sigma$‐evolution equations with time‐dependent damping in the so‐called effective cases utt+(−Δ)σ1u+b1(t)(−Δ)δ1ut=|vt|p,vtt+(−Δ)σ2v+b2(t)(−Δ)δ2vt=|ut|q.$$\begin{equation*} \hspace*{7pc}{\begin{cases} u_{t t}+(-\Delta)^{\sigma _1} u+b_1(t) (-\Delta)^{\delta _1} u_t=|v_t|^p, \\ v_{t t}+(-\Delta)^{\sigma _2} v+b_2(t) (-\Delta)^{\delta _2} v_t=|u_t|^q. \end{cases}} \end{equation*}$$We obtain conditions for the existence and the decay estimates of the global (in time) solution that are expressed in terms of the decay character of the initial data u0(x)=u(0,x),v0(x)=v(0,x)$u_0(x)=u(0, x), \nobreakspace v_0(x)=v(0, x)$ and u1(x)=ut(0,x),v1(x)=vt(0,x)$u_1(x)=u_t(0, x),\nobreakspace v_1(x)=v_t(0, x)$. Furthermore, the blow‐up results for solutions to the semilinear problem also presented.

Date: 2025
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