A qualitative study of the generalized dispersive systems with time-delay: The unbounded case
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| Format: | Preprint |
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2023
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| author | Filho, Roberto de A. Capistrano Gallego, Fernando Komornik, Vilmos |
| author_facet | Filho, Roberto de A. Capistrano Gallego, Fernando Komornik, Vilmos |
| contents | We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +λ_0(x) u(x,t)+λ(x) u(x,t-τ)=0 \end{equation*} where \begin{equation*} Au=(-1)^{j+1}\partial_x^{2j+1}u+(-1)^m\partial_x^{2m}u+ \frac{1}{p+1}\partial_xu^{p+1} \end{equation*} with $m\le j$ and $1\le p<2j$. Under suitable assumptions on the time delay coefficients, we prove that the system is exponentially stable if the coefficient of the delay term is bounded from below by a suitable positive constant, without any assumption on the sign of the coefficient of the undelayed feedback. Additionally, in the absence of delay, general results of stabilization are established in $H^s(\mathbb{R})$ for $s\in[0,2j+1]$. Our results generalize several previous theorems for the Korteweg-de Vries type delayed systems in the literature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_06693 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A qualitative study of the generalized dispersive systems with time-delay: The unbounded case Filho, Roberto de A. Capistrano Gallego, Fernando Komornik, Vilmos Analysis of PDEs Optimization and Control 35Q53, 93D15, 93D30, 93C20 We study the asymptotic behavior of the solutions of the time-delayed higher-order dispersive nonlinear differential equation \begin{equation*} u_t(x,t)+Au(x,t) +λ_0(x) u(x,t)+λ(x) u(x,t-τ)=0 \end{equation*} where \begin{equation*} Au=(-1)^{j+1}\partial_x^{2j+1}u+(-1)^m\partial_x^{2m}u+ \frac{1}{p+1}\partial_xu^{p+1} \end{equation*} with $m\le j$ and $1\le p<2j$. Under suitable assumptions on the time delay coefficients, we prove that the system is exponentially stable if the coefficient of the delay term is bounded from below by a suitable positive constant, without any assumption on the sign of the coefficient of the undelayed feedback. Additionally, in the absence of delay, general results of stabilization are established in $H^s(\mathbb{R})$ for $s\in[0,2j+1]$. Our results generalize several previous theorems for the Korteweg-de Vries type delayed systems in the literature. |
| title | A qualitative study of the generalized dispersive systems with time-delay: The unbounded case |
| topic | Analysis of PDEs Optimization and Control 35Q53, 93D15, 93D30, 93C20 |
| url | https://arxiv.org/abs/2303.06693 |