General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities

This paper investigates the existence, energy decay, and finite-time blow-up of solutions for a class of fourth-order viscoelastic evolution equations involving variable exponent nonlinearities, logarithmic terms, and strong damping effects. Such models arise naturally in the mathematical description of heterogeneous viscoelastic media and nonlinear plate equations with memory. By applying the Nehari manifold method and suitable energy estimates, we establish the global existence of weak solutions under appropriate assumptions on the initial data and the variable exponents. Moreover, using a perturbed energy method combined with a carefully constructed Lyapunov functional, we prove that the solutions exhibit general energy decay rates depending on the properties of the relaxation kernel and the spatially variable nonlinearities. Finally, we derive sufficient conditions ensuring finite-time blow-up for solutions with negative initial energy, thereby highlighting the competing effects of strong damping, viscoelastic memory, and logarithmic nonlinearities.

In this paper, we investigate the general decay rate of the solutions for a class of fractional Laplace viscoelastic equations in the whole space , given by with and satisfying . We prove the existence of a solution formula then use the energy method in the Fourier space to establish a general decay result which depends on and . We also present three illustrative examples by the end.

<p>In this article, we aim to investigate the decay characteristics of a system consisting of two viscoelastic wave equations with Dirichlet boundary conditions, where the dispersion term and nonlinear weak damping term are taken into account. Under appropriate conditions, we establish both general and optimal decay results. This work generalizes and improves earlier results in the literature.</p>

In this paper, we consider a viscoelastic plate equation in the presence of a strong time-varying delay, namely; $$\begin{aligned} u_{tt}+\Delta ^2 u-\int _0^t g(t-s)\Delta ^2 u(s)ds-\mu _1\Delta u_t-\mu _2\Delta u_t(\cdot ,t-\tau (t))=0. \end{aligned}$$ We proved a general and optimal decay result for the associated energy functional of the system. The following are novel achievements of the present work: A more general relaxation function g with minimal conditions is used, this allowed us to obtain exponential and polynomial decay rates as special cases. Furthermore, our results hold with lesser restriction on the values of $$\mu _1$$ and $$\mu _2$$ as against previous results which assume that $$|\mu _2|<\sqrt{1-d}\mu _1$$ . This work improves and generalizes previous results in the literature.

In this article, we investigate a system of two viscoelastic equations with Dirichlet boundary conditions. Under some suitable assumptions on the function g_{i}(cdot ), f_{i}(cdot ,cdot ) (i=1,2) and the initial data, we obtain general and optimal decay results. Moreover, for certain initial data, we establish a finite time blow-up result. This work generalizes and improves earlier results in the literature. The conditions of the relaxation functions g_{1}(t) and g_{2}(t) in our work are weak and seldom appear in previous literature, which is an important breakthrough.

General decay for a viscoelastic equation of variable coefficients with a time-varying delay in the boundary feedback and acoustic boundary conditions

General decay for a differential inclusion of Kirchhoff type with a memory condition at the boundary

In this paper, we consider a viscoleastic equation with a nonlinear feedback localized on a part of the boundary and a relaxation function satisfying g′(t) ≤−ξ(t)G(g(t)). We establish an explicit and general decay rate results, using the multiplier method and some properties of the convex functions. Our results are obtained without imposing any restrictive growth assumption on the damping term. This work generalizes and improves earlier results in the literature, in particular those of Messaoudi (Topological Methods in Nonlinear Analysis 51(2):413–427, 2018), Messaoudi and Mustafa (Nonlinear Analysis: Theory Methods & Applications 72(9–10):3602–3611, 2010), Mustafa (Mathematical Methods in the Applied Sciences 41(1): 192–204, 2018) and Wu (Zeitschrift fur angewandte Mathematik und Physik 63(1):65–106, 2012).

In this paper, we are concerned with a system of nonlinear viscoelastic wave equations with initial and Dirichlet boundary conditions in R n (n=1,2,3). Under suitable assumptions, we establish a general decay result by multiplier techniques, which extends some existing results for a single equation to the case of a coupled system.MSC:35L05, 35L55, 35L70.

In a bounded domain, we consider a viscoelastic equation with a nonlinear feedback localized on a part of the boundary and certain initial data. We establish an explicit and general decay rate result, using some properties of the convex functions. Our new results substantially improve several earlier related results in the literature. Keywords: General decay, nonlinear source, viscoelastic, wave equation, relaxation function.

In this paper, we consider the dynamical von Karman equations for viscoelastic plates with nonlinear boundary dissipation. We show the existence of solutions using the Galerkin method and then prove the asymptotic behaviour of the corresponding solutions by choosing suitable Lyapunov functional.

Vanishing solutions of anisotropic parabolic equations with variable nonlinearity

Analytical and computational results for the decay of solutions of a damped wave equation with variable-exponent nonlinearities

We consider the following nonlinear parabolic equation: ut-div(|∇u|p(x)-2∇u)=f(x,t), where f:Ω×(0,T)→R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.

In this article, we deal with a strongly damped von Karman equation with variable exponent source and memory effects. We investigate blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time.