Intrinsic decay rates for the energy of a nonlinear viscoelastic equation modeling the vibrations of thin rods with variable density

Abstract

We consider the long-time behavior of a nonlinear PDE with a memory term which can be recast in the abstract form$\frac{d}{dt}\rho(u_{t})+Au_{tt}+\gamma A^{\theta}u_{t}+Au-\int_{0}^{t}g(s)Au(t% -s)=0,$where A is a self-adjoint, positive definite operator acting on a Hilbert space H, ${\rho(s)}$ is a continuous, monotone increasing function, and the relaxation kernel ${g(s)}$ is a continuous, decreasing function in ${L_{1}(\mathbb{R}_{+})}$ with ${g(0)>0}$. Of particular interest is the case when ${A=-\Delta}$ with appropriate boundary conditions and ${\rho(s)=|s|^{\rho}s}$. This model arises in the context of solid mechanics accounting for variable density of the material. While finite energy solutions of the underlying PDE solutions exhibit exponential decay rates when strong damping is active (${\gamma>0,\theta=1}$), this uniform decay is no longer valid (by spectral analysis arguments) for dynamics subjected to frictional damping only, say, ${\theta=0}$ and ${g=0}$. In the absence of mechanical damping (${\gamma=0}$), the linearized version of the model reduces to a Volterra equation generated by bounded generators and, hence, it is exponentially stable for exponentially decaying kernels. The aim of the paper is to study intrinsic decays for the energy of the nonlinear model accounting for large classes of relaxation kernels described by the inequality ${g^{\prime}+H(g)\leq 0}$ with H convex and subject to the assumptions specified in (1.13) (a general framework introduced first in [1] in the context of linear second-order evolution equations with memory). In the context of frictional damping, such a framework was introduced earlier in [15], where it was shown that the decay rates of second-order evolution equations with frictional damping can be described by solutions of an ODE driven by a suitable convex function H which captures the behavior at the origin of the dissipation. The present paper extends this analysis to nonlinear equations with viscoelasticity. It is shown that the decay rates of the energy are intrinsically described by the solution of the dissipative ODE${S_{t}+c_{1}H(c_{2}S)=0}$with given intrinsic constants ${c_{1},c_{2}>0}$. The results obtained are sharp and they improve (by introducing a novel methodology) previous results in the literature (see [20, 19, 21, 6]) with respect to (i) the criticality of the nonlinear exponent ρ and (ii) the generality of the relaxation kernel.

Blow-up of arbitrarily positive initial energy solutions for a viscoelastic wave system with nonlinear damping and source terms

“…For the single viscoelastic wave equation, we refer the reader to [1,2] (the case g = 0) and [3][4][5][6][7] (the case g = 0), where blowup solutions with initial negative energy, positive energy and arbitrarily positive energy are [1][2][3][4][5][6][7], respectively. Moreover, for general energy decay estimates on global solutions of a nonlinear abstract viscoelastic equation with variable density and the oscillation criteria and numerical solution of damped wave models, we refer the reader to [8][9][10].…”

Existence of fast homoclinic solutions for a class of second-order damped vibration systems

“…For the existence of homoclinic solutions for damped vibration problem (1.4), please see the literature [20][21][22][23][24][25], and the references cited therein. For other kinds of damped vibration problem, please see the literature [26] and [27]. Besides, by applying the mountain pass theorem and symmetric mountain pass theorem in critical point theory, Zhang and Li [28] investigated the existence and multiplicity of fast homoclinic solutions for a class of nonlinear second-order nonautonomous systems and obtained some results, which generalized and improved problem (1.4).…”

Exponential stability of a mildly dissipative viscoelastic plate equation with variable density

“…The term | u t | ρ , with ρ > 0, can be seen as a model for material density ϱ = ϱ ( u t ) dependent on the velocity. Two concrete examples of such materials are described in Cavalcanti et al…”