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Optimized uniform decay estimate of the solution to Petrovsky equation with memory. (English) Zbl 1470.35063

Summary: In this paper, we investigate uniform decay estimate of the solution to the Petrovsky equation with memory \[ u_{tt}+\Delta^2u-\int_0^t g(t-s)\Delta^2u(s)ds=0 \] with initial conditions and boundary conditions, where \(g\) is a memory kernel function. The related energy has been shown to decay exponentially or polynomially as \(t\rightarrow+\infty\) by the theorem established under the assumption \(g'(t)\leqslant-kg^{1+\frac{1}{p}}(t)\) with \(p\in (2,\infty)\) and \(k>0\) in the reference [F. Alabau-Boussouira et al., J. Funct. Anal. 254, No. 5, 1342–1372 (2008; Zbl 1145.35025)]. Using the ideas introduced by by I. Lasiecka and X. Wang [Springer INdAM Ser. 10, 271–303 (2014; Zbl 1476.35146)], we prove the optimized uniform general decay result under the assumption \(g'(t)+H(g(t))\leq 0\), where the function \(H(\cdot)\in C^1(\mathbb{R}^1)\) is positive, increasing and convex with \(H(0)=0\), which is introduced for the first time by F. Alabau-Boussouira and P. Cannarsa [C. R., Math., Acad. Sci. Paris 347, No. 15–16, 867–872 (2009; Zbl 1179.35058)] and studied systematically by Lasiecka and Wang [loc. cit.]. The exponential decay result and polynomial decay result in the reference [Alabau-Boussouira et al., loc. cit.] are the special cases of this paper by choosing special \(H(\cdot)\).

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35R09 Integro-partial differential equations
Full Text: DOI

References:

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