Presented here is a study of long-term behavior of Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous boundary damping and boundary source terms. This work ...complements 28 which established local well-posedness of this problem, and global well-posedness when the boundary damping dominates the boundary sources (in an appropriate sense). The current paper develops the potential well theory for the RMT system: global existence for potential well solutions without restricting the boundary source exponents, and explicit energy decay rates dependent on the boundary damping exponents. This work along with 26–28 provides the fundamental well-posedness and stability theory for MT plates under the interplay of damping and source terms acting either in the interior or on the boundary of the plate.
This paper considers the adaptive observers of systems governed by partial differential equations with structured perturbations. As there are many combinations of control, observation, and parametric ...uncertainty and disturbances at the boundary, a classification of the various combinations is presented. In all cases considered, the parametric uncertainty entering the boundary is expressed as the product of the measured output with unknown parameters. Similarly, the disturbances are also entering at the same boundary as the parametric uncertainties. In all cases, the plant is written as an evolution equation in a Hilbert space and the interplay between the input, output and disturbances operators are presented, and for each case, the feasibility of the adaptive observer is discussed. Selecting the case where the boundary operator associated with the disturbance and the parametric uncertainty is co-located only with the output operator, the adaptive observer that estimates on line the process state, the disturbance and the unknown parameters is presented along with the stability and convergence results. Extensive numerical studies including the special case of nonlinear terms at the boundary are presented.
An effective homogenization of control rods is studied by considering neutron leakage as a fixed boundary source. Good numerical convergence is seen in the iterative calculation of effective ...homogenized cross sections that preserve integrated reaction rates. Usefulness of the present method is demonstrated by treating off-center control rods in a one-dimensional slab model of an FBR.
In this paper we consider a nonlinear wave equation with nonlinear boundary damping and source terms, subject to a localized viscoelastic damping and localized frictional damping. We obtain the ...global existence of solution via potential well method. Furthermore, by introducing suitable Lyapunov functionals, using multiplier method and constructing convex differential inequality, we compare the effect of locally viscoelastic term and frictional damping and establish an explicit and general decay rate result. This stability result allows the localized viscoelastic damping and localized frictional damping acting simultaneously or complementarily in the domain, which gives the competing relationship of the viscoelastic and frictional damping.
We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The ...results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.
In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) ...obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295–303) to the nonlinear damping case under weaker assumption on the relaxation function
g
(
t
)
. Then, we give an exponential decay result without the relation between
g
′
(
t
)
and
g
(
t
)
for the linear damping case, provided that
‖
g
‖
L
1
(
0
,
∞
)
is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.
In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove ...the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function g(·), nonlinear function f(·), the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions I(u0)≥0 and E(0)=d.
In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function
g
and with certain ...initial data, the global existence of solutions and a general decay for the energy have been established.
The paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is
u
tt−
Δu=0
...in
(0,∞)×Ω,
u=0
on
0,∞)×Γ
0,
∂u
∂ν
=−|u
t|
m−2u
t+|u|
p−2u
on
0,∞)×Γ
1,
u(0,x)=u
0(x),
u
t(0,x)=u
1(x)
on
Ω,
where
Ω⊂
R
n
(n⩾1)
is a regular and bounded domain,
∂Ω=Γ
0∪Γ
1
,
m>1, 2⩽
p<
r, where
r=2(
n−1)/(
n−2) when
n⩾3,
r=∞ when
n=1,2, and the initial data are in the energy space. We prove local existence of the solutions in the energy space when
m>
r/(
r+1−
p) or
n=1,2, and global existence when
p⩽
m or the initial data are inside the potential well associated to the stationary problem.
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