General decay to a von Kármán system with memory

Raposo, Carlos A.; Santos, M. L.

doi:10.1016/j.na.2010.09.047

Cited by 34 publications

(21 citation statements)

References 7 publications

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“…To prove the regularity of the solution of the system , we introduce the following definition (see, e.g., . Definition We say that ( u 0 , u 1 , y 0 ) is 2‐regular if

u_{j} MathClass-rel\in H^{4 MathClass-bin- j} (Ω) MathClass-bin\cap W MathClass-punc, y_{j} MathClass-rel\in L^{2} (Γ_{1}) MathClass-punc, j MathClass-rel= 0 MathClass-punc, 1 MathClass-punc, 2 MathClass-punc, u_{3} MathClass-rel\in V MathClass-punc,

where u j is obtained by using the following recursive formula:

u_{j + 2} - h Δ u_{j + 2} = Δ^{2} u_{j} + g_{j} - f_{j} (0), [3 p t] h \frac{\partial u_{j + 2}}{\partial ν} = ℬ_{2} u_{j} + y_{j + 1} on Γ_{1}, u_{j + 1} + p y_{j + 1} + q y_{j} = 0 on Γ_{1}

and also

u_{j} MathClass-rel= \frac{\partial u_{j}}{∂ν} on Γ_{0} MathClass-punc, {scriptB}_{1} u_{j} MathClass-rel= 0 MathClass-punc, on Γ_{0} MathClass-rel\forall j MathClass-rel= 1 MathClass-punc, 2 MathClass-punc,

where …”

Section: Preliminariesmentioning

confidence: 99%

“…By standard Galerkin method , we can obtain the following existence result. Theorem Let the initial data ( u 0 , u 1 , y 0 ) be 2‐regular, h > 0, T > 0 and the aforementioned conditions on g , p and q hold.…”

Section: Preliminariesmentioning

confidence: 99%

“…Later, Santos and Soufyane generalized the decay result of Rivera et al . Recently, Raposo and Santos considered the general decay of the solutions to a von Kármán plate model under weaker condition on g , such as

\begin{matrix} eqnarrayright center left \\ eqnarray-1 g' MathClass-open(t MathClass-close) \leq - ζ MathClass-open(t MathClass-close) g MathClass-open(t MathClass-close), for t \geq 0, & eqnarray-2 & eqnarray-3 & eqnarray-4 (1.9) \end{matrix}

…”

Section: Introductionmentioning

confidence: 97%

See 2 more Smart Citations

Exponential decay for a von Kármán equations of memory type with acoustic boundary conditions

Kang

2014

Math Methods in App Sciences

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“…To prove the regularity of the solution of the system , we introduce the following definition (see, e.g., . Definition We say that ( u 0 , u 1 , y 0 ) is 2‐regular if

u_{j} MathClass-rel\in H^{4 MathClass-bin- j} (Ω) MathClass-bin\cap W MathClass-punc, y_{j} MathClass-rel\in L^{2} (Γ_{1}) MathClass-punc, j MathClass-rel= 0 MathClass-punc, 1 MathClass-punc, 2 MathClass-punc, u_{3} MathClass-rel\in V MathClass-punc,

where u j is obtained by using the following recursive formula:

u_{j + 2} - h Δ u_{j + 2} = Δ^{2} u_{j} + g_{j} - f_{j} (0), [3 p t] h \frac{\partial u_{j + 2}}{\partial ν} = ℬ_{2} u_{j} + y_{j + 1} on Γ_{1}, u_{j + 1} + p y_{j + 1} + q y_{j} = 0 on Γ_{1}

and also

u_{j} MathClass-rel= \frac{\partial u_{j}}{∂ν} on Γ_{0} MathClass-punc, {scriptB}_{1} u_{j} MathClass-rel= 0 MathClass-punc, on Γ_{0} MathClass-rel\forall j MathClass-rel= 1 MathClass-punc, 2 MathClass-punc,

where …”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

\begin{matrix} eqnarrayright center left \\ eqnarray-1 g' MathClass-open(t MathClass-close) \leq - ζ MathClass-open(t MathClass-close) g MathClass-open(t MathClass-close), for t \geq 0, & eqnarray-2 & eqnarray-3 & eqnarray-4 (1.9) \end{matrix}

…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

Exponential decay for a von Kármán equations of memory type with acoustic boundary conditions

Kang

2014

Math Methods in App Sciences

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“…16. Raposo and Santos 19 considered the general decay of the solutions to a von Kármán plate model under the condition (1.11). They showed that the energy decays with a similar rate of decay of the relaxation function, which is not necessarily decaying in a polynomial or exponential fashion.…”

Section: Introductionmentioning

confidence: 99%

Exponential decay for a von Kármán equations with memory

Kang

2013

Journal of Mathematical Physics

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“…For Viscoelastic plates with memory, J. E. M. Rivera et al [7,8] proved that the energy decays uniformly, exponentially or algebraically with the same rate of decay of the relaxation function. C. A. Raposo and M. L. Santos [9] gave a General Decay of solution for the memory case. In [10][11][12][13] the authors consider the von Kármán system with frictional dissipations effective in the whole plate, in a part of the plate or at the boundary.…”

Section: Introductionmentioning

confidence: 99%