2012
DOI: 10.1002/mma.2629
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Asymptotic stability and blow up of solutions for a Petrovsky inverse source problem with dissipative boundary condition

Abstract: Communicated by S. ChenThis paper is concerned with global in time behavior of solutions for a quasilinear Petrovsky inverse source problem with boundary dissipation. We establish a stability result when the integral constraint vanishes as time goes to infinity. We also show that the smooth solutions blow up when the data is chosen appropriately.(1.5) has been considered by Amroun and Benaissa in [1], where the authors proved: global existence of solutions by means of the stable set method in H 2 0 . / combine… Show more

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Cited by 10 publications
(6 citation statements)
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“…They found conditions on data guaranteeing global nonexistence of solutions when šœ™(t) ā‰” 1, also established a stability result with the opposite sign on the power type nonlinearity and b(x, t, u, āˆ‡ u) ā‰” 0. Next Tahamtani and Shahrouzi 4 extend previous results to a Petrovsky inverse source problem (see also Tahamtani and Shahrouzi 5 ). Shahrouzi 6 studied the following damped viscoelastic inverse problem:…”
Section: Introductionsupporting
confidence: 65%
“…They found conditions on data guaranteeing global nonexistence of solutions when šœ™(t) ā‰” 1, also established a stability result with the opposite sign on the power type nonlinearity and b(x, t, u, āˆ‡ u) ā‰” 0. Next Tahamtani and Shahrouzi 4 extend previous results to a Petrovsky inverse source problem (see also Tahamtani and Shahrouzi 5 ). Shahrouzi 6 studied the following damped viscoelastic inverse problem:…”
Section: Introductionsupporting
confidence: 65%
“…Therefore, we have Proof of Theorem 2.2 Inspired by the idea in [24], we define F (t) = ME(t) + N (Ļˆ 1 (t) + Ī¾Ļˆ 2 (t)), (3.11) where…”
Section: )mentioning
confidence: 99%
“…For more information about inverse problems, the interested reader is referred to the papers [1,6,11,12,15,20,23] Our objective in the present work is to extend the results of [3,24] by considering problem (1.1)- (1.4) in the presence of m-Laplacian operator and boundary conditions. Motivated by the aforementioned works, our result here is twofold: First, we consider a = 1 and show that if we take initial data and parameters in the appropriately domain, then solutions of (1.1)-(1.4) are asymptotically stable when Ļ†(t) tends to zero as time goes to infinity.…”
Section: Introductionmentioning
confidence: 97%
“…There are some papers devoted to the study of existence and uniqueness of solutions of inverse problems for various parabolic type equations with unknown source functions [1,3,5]. In [6,7,8,9,10] authors studied global nonexistence and blow-up solution to fourth-order equations.…”
Section: (A3)mentioning
confidence: 99%