Stability for the beam equation with memory in non‐cylindrical domains

Abstract

SUMMARYIn this paper, we prove the exponential decay as time goes to inÿnity of regular solutions of the problem for the beam equation with memory and weak damping uttwhereQ is a non-cylindrical domains of R n+1 (n¿1) with the lateral boundaryˆ and is a positive constant.

Stability for the Kirchhoff Plates Equations with Viscoelastic Boundary Conditions in Noncylindrical Domains

“…The above method was introduced by Dal Passo and Ughi [15] for studying a certain class of parabolic equations in noncylindrical domains. This idea was used in [11,13,14,16,17]. We will use (19) and (20) to estimate the values B 1 and B 2 on Γ 1 .…”

“…Proof. This idea was used in [11,13,14,16,17]. To show the existence in noncylindrical domains, we return to our original problem in the noncylindrical domains by using the change variable given in (14) by ( , ) = ( , ), ( , ) ∈̂.…”

“…In [9][10][11][12][13][14][15][16][17], the authors considered the global existence and the uniform decay of solution in noncylindrical domains. Dal Passo and Ughi [15] investigated a certain class of parabolic equations in noncylindrical domains.…”

Energy Decay for the Strongly Damped Nonlinear Beam Equation and Its Applications in Moving Boundary

“…In a fixed domain, it is well known that the energy of the system (1.2) also decays to zero, see [8,13]. But in a moving domain, see [2,[5][6][7]17], energy decay were more difficult to obtain than the result in fixed domain. And in this case, the transverse deflection u(x, t) of a beam which changed its configuration at each instant of time increases its deformation, and hence increases its tension.…”

Global attractor for a nonlinear plate equation with supported boundary conditions

“…A larger class of stability of beam is in papers [9,18,19] and references therein. The existence of the absorbing set and the inertial manifolds for Eq.…”