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Qualitative analysis of the dynamic for the nonlinear Korteweg-de Vries equation with a boundary memory. (English) Zbl 1467.35278

Summary: This paper addresses the impact of the presence of a boundary memory term in the third-order Korteweg-de Vries equation in a bounded interval \([0,\ell]\). First, an overall literature review is provided. Indeed, a comprehensive discussion on the literature constitutes a survey part of the current paper. Thereafter, it is shown that the system under consideration possesses a unique solution under a smallness assumption on the initial data and an appropriate condition on the parameters and the kernel involved in the memory term. Last but not least, we demonstrate that the zero solution is exponentially stable as long as the length \(\ell\) is small enough by means of Lyapunov method, which permits to provide an estimate of the exponential decay rate. These findings improve and complement those of B.-Y. Zhang [ISNM, Int. Ser. Numer. Math. 118, 371–389 (1994; Zbl 0811.35133)] (resp. L. Baudouin et al. [IEEE Trans. Autom. Control 64, No. 4, 1403–1414 (2019; Zbl 1482.93462)]), where no memory term is present (resp. a delay occurs instead of memory).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B35 Stability in context of PDEs
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References:

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