Existence and general decay estimate for a nonlinear plate problem

The study of plate problems has been widely investigated by mathematicians and other scientists. Plate problems have a lot of applications in different areas of science and engineering such as material engineering, mechanical engineering, nuclear physics and optics. In order to describe the structural behaviour and the stability of large structures in our societies, plate models have been extensively used. For instance, the Kirchhoff theory of plates [1] establishes a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. The stability of Kirchhoff plates in the presence of a linear or nonlinear source has been studied by many authors. See, for instance, the results obtained in Komornik [2], Lagnese [3] and Lasiecka [4, 5]. Al-Gharabli and Messaoudi [6] studied the following nonlinear plate problem: and established decay of solutions. Lu [7] investigated the nonautonomous plate-type evolutionary problem and proved the existence of a uniform attractor. Ji and Lasiecka [8] considered a semilinear Kirchhoff plate with a nonlinear dissipation acting via moments and proved that the plate is uniformly stabilizable with uniform energy decay rates with respect to the parameter γ which represents rotational force. Moreover, they showed that as \(\gamma\longrightarrow 0\), the solutions of the Kirchhoff plate equation converge to the solutions of the semilinear Euler-Bernoulli plate, which is also uniformly stable in finite energy norm.

Recently, plate models have also been of great importance in studying the structural behaviour and instability of suspension bridges. The first attempt to model a suspension bridge through a plate is due to Ferrero and Gazzola [9], where the following hyperbolic problem was introduced: where \({\Omega=(0,\pi) \times(-\ell,\ell) \subset\mathbb {R}^{2}}\) is a planar rectangular plate, σ is the Poisson ratio, η is the damping coefficient, h is the nonlinear restoring force of the hangers and f is an external force. The well-posedness and long-time behaviour of this problem were proved in [9] under suitable assumptions on h. A quasilinear stationary variant of this equation was as well suggested in [10]. Wang [11] considered the following fourth-order equation: with the same boundary conditions and initial data as in [10]. He proved the existence and uniqueness of local solution and a finite time blow up result. Messaoudi and co-authors [12–16] have carried out extensive analysis of the suspension bridge plate model (1.4), where existence, decay and global attractor results have been established. For more related results, see Gazzola and Wang [17], Berchio et al. [18] and the book [19] on mathematical models for suspension bridges by Gazzola.

In this paper, we consider the following fourth-order plate equation: with the same boundary and initial conditions as in (1.4), where g is a nonlinear function to be specified later, β is the damping coefficient and u represents the downward displacement of a vibrating suspension bridge under the effect of weak frictional damping.

In this section, we state some preliminary material that will be helpful in achieving our result. We assume that the functions β, g and h satisfy the following assumptions:

As in [9], let us introduce the space together with the inner product For the completeness of the space \({H_{*}}^{2}(\Omega)\), we have the following results by Ferrero and Gazzola [9].

Now, we reformulate problem (1.6) into a semigroup setting. Let \(u_{t} = v\), then problem (1.6) becomes where We introduce the Hilbert space equipped with the inner product where Next, we consider the following stationary boundary conditions: The domain of the operator A is defined as We have the following existence and uniqueness result for problem (3.3).

In this section, we illustrate our result with some examples. As in [21], let \({g_{0}\in C^{2} ( [0,+\infty) )}\) be a strictly increasing function such that \({g_{0}(0)=0}\), and for some positive constants \(c_{1}\), \(c_{2}\) and ϵ, the function g satisfies Define the function Then M is a \(C^{2}\)-strictly convex function on \((0,\epsilon]\) when \({g_{0}}\) is nonlinear and thus satisfies assumption (A₃). We give some examples of \({g_{0}}\) such that g satisfies (5.1) near 0.

1. (1)
   Let \({g_{0}(s)= ks}\), where k is a positive constant, then \({M(s)= ks}\) satisfies (A₃), and we get

   $${E(t)\leq ke^{-k_{1} \int_{0}^{t}\beta(s)\, ds},\quad \forall t\geq0.} $$
    
2. (2)
   Let \({g_{0}(s) = \frac{1}{s}e^{-\frac{1}{s^{2}}}}\), then \({M(s)= e^{-\frac{2}{s}}}\) satisfies (A₃) near 0 and

   $${E(t)\leq k \biggl(\ln \biggl(k_{1} \int_{0}^{t} \beta(s)\, ds+ k_{2} \biggr) \biggr)^{-1},\quad \forall t\geq0.} $$
    
3. (3)
   Let \({g_{0}(s) = e^{-\frac{1}{s}}}\), then \({M(s)= \sqrt{\frac{s}{2}} e^{-\sqrt{\frac{2}{s}}}}\) satisfies (A₃) near 0, and we obtain

   $${E(t)\leq k \biggl(\ln \biggl(k_{1} \int_{0}^{t} \beta(s)\, ds + k_{2} \biggr) \biggr)^{-2},\quad \forall t\geq0.} $$