Abstract
The small-amplitude motion of a thin elastic membrane is investigated in n-dimensional bounded and unbounded domains, with n 2 N. Existence and uniqueness of the solutions are established. Asymptotic behavior of the solutions is proved too.
Highlights
The one-dimensional equation of motion of a thin membrane fixed at both ends and undergoing cylindrical bending can be written as utt(x, t) − ζ0 + ζ1 |ux(t)|2dx + σ ux(t)uxt(t)dx uxx(x, t)
The investigation of existence of a solution for the Cauchy problem associated with equation (1.1) in n -dimensional bounded and unbounded domain will be made by the application of diagonalization theorem of Dixmier & Von Neumann
The use of the diagonalization theorem in the study of Cauchy problem associated with the Kirchhoff equation was initially utilized by Matos [22] to prove existence of a local solution
Summary
The one-dimensional equation of motion of a thin membrane fixed at both ends and undergoing cylindrical bending can be written as utt(x, t) − ζ0 + ζ1 |ux(t)|2dx + σ ux(t)uxt(t)dx uxx(x, t). Equation (1.4) has been extensively studied by several authors in both {1, 2, · · · , n} -dimensional cases and general mathematical models in a Hilbert space H Both local and global solutions have been shown to exist in several physical-mathematical contexts. The investigation of existence of a solution for the Cauchy problem associated with equation (1.1) in n -dimensional bounded and unbounded domain will be made by the application of diagonalization theorem of Dixmier & Von Neumann. The use of the diagonalization theorem in the study of Cauchy problem associated with the Kirchhoff equation was initially utilized by Matos [22] to prove existence of a local solution.
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