The main goal of this work is to investigate the following nonlinear plate equation

$$u_{tt}+{\mathrm{\Delta }}^{2}u+\alpha (t)g(u_t)=u|u{|}^{\beta },$$

which models suspension bridges. Firstly, we prove the local existence using Faedo-Galerkin method and Banach fixed point theorem. Secondly, we prove the global existence by using the well-depth method. Finally, we establish explicit and general decay results for the energy of solutions of the problem. Our decay results depend on the functions $\alpha$ and $g$ and obtained without any restriction growth assumption on $g$ at the origin. The multiplier method, properties of the convex functions, Jensen's inequality and the generalized Young inequality are used to establish the stability results.

In this paper, we study the asymptotic behavior of solutions of the dissipative coupled system where we have interactions between a Kirchhoff plate and a Euler-Bernoulli plate. We investigate the interaction between the internal strong damping acting in the Kirchhoff equation and internal weak damping of variable-exponent type acting in the Euler-Bernoulli equation. By using the potential well, the energy method (multiplier method) combined with the logarithmic Sobolev inequality, we prove the global existence and derive the stability results. We show that the solutions of this system decay to zero sometimes exponentially and other times polynomially. We find explicit decay rates that depend on the weak damping of the variable-exponent type. This outcome extends earlier results in the literature.

This paper investigated, for the first time, a coupled Kirchhoff-type wave system incorporating logarithmic damping and logarithmic source terms (external forces). We established both energy decay and finite-time blow-up results, emphasizing the novel interaction between the two logarithmic components acting as dissipative and driving mechanisms. In the stable region, we constructed a suitable Lyapunov functional and employed refined logarithmic estimates to derive a polynomial decay rate of the total energy. Conversely, for initial data belonging to the unstable set, we proved that the corresponding solutions blow up in finite time using a concavity argument. In addition, we provided some numerical examples to illustrate the stability and blow-up theoretical results. This study presents the first comprehensive analysis of a Kirchhoff-type system with logarithmic damping, revealing how the combined effects of the nonlocal Kirchhoff tension and logarithmic nonlinearities govern the transition between global stabilization and blow-up behavior.

In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the variable exponents. Second, we show that the solutions with positive-initial energy blow-up in a finite time. Finally, we establish the global existence as well as the energy decay results of the solutions, using the stable-set and the multiplier methods, under appropriate conditions on the variable exponents and the initial data.

Image deblurring models with a mean curvature functional has been widely used to preserve edges and remove the staircase effect in the resulting images. However, the Euler-Lagrange equations of a mean curvature model can be used to solve fourth-order non-linear integro-differential equations. Furthermore, the discretization of fourth-order non-linear integro-differential equations produces an ill-conditioned system so that the numerical schemes like Krylov subspace methods (conjugate gradient etc.) have slow convergence. In this paper, we propose an augmented Lagrangian method for a mean curvature-based primal form of the image deblurring problem. A new circulant preconditioned matrix is introduced to overcome the problem of slow convergence when employing a conjugate gradient method inside of the augmented Lagrangian method. By using the proposed new preconditioner fast convergence has been observed in the numerical results. Moreover, a comparison with the existing numerical methods further reveal the effectiveness of the preconditioned augmented Lagrangian method.

In this study, a nonlinear damped wave equation within a bounded domain was considered. We began by demonstrating the global existence of solutions through the application of the well-depth method. Following this, a general decay rate for the solutions was established using the multiplier method alongside key properties of convex functions. Notably, these results were derived without the imposition of restrictive growth assumptions on the frictional damping, making this work an improvement and extension of previous findings in the field.