Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $ G $-framework

<abstract><p>Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $ G $-Brownain motion ($ G $-NSFDEs in short). This paper focuses on the $ G $-NSFDEs, and the coefficients of these considered $ G $-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $ G $-NSFDE. It is then obtained the trivial solution of the $ G $-NSFDE is mean square stable with general decay rate (including the trivial solution of the $ G $-NSFDE is mean square exponentially stable and the trivial solution of the $ G $-NSFDE is mean square polynomially stable) by $ G $-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.</p></abstract>

Stability Result for a Weakly Nonlinearly Damped Porous System with Distributed Delay

In this paper, we consider a one-dimensional porous system damped with a single weakly nonlinear feedback and distributed delay term. Without imposing any restrictive growth assumption near the origin on the damping term, we establish an explicit and general decay rate, using a multiplier method and some properties of convex functions in case of the same speed of propagation in the two equations of the system. The result is new and opens more research areas into porous-elastic system.

Asymptotic behavior for a viscoelastic Kirchhoff equation with distributed delay and Balakrishnan–Taylor damping

A nonlinear viscoelastic Kirchhoff-type equation with Balakrishnan–Taylor damping and distributed delay is studied. By the energy method we establish the general decay rate under suitable hypothesis.

A New General Decay Rate of Wave Equation with Memory-Type Boundary Control

Of interest is a wave equation with memory-type boundary oscillations, in which the forced oscillations of the rod is given by a memory term at the boundary. We establish a new general decay rate to the system. And it possesses the character of damped oscillations and tends to a finite value for a large time. By assuming the resolvent kernel that is more general than those in previous papers, we establish a more general energy decay result. Hence the result improves earlier results in the literature.

Global existence and blow-up results for a nonlinear model for a dynamic suspension bridge

<p style='text-indent:20px;'>The paper deals with global existence and blow-up results for a class of fourth-order wave equations with nonlinear damping term and superlinear source term with the coefficient depends on space and time variable. In the case the weak solution is global, we give information on the decay rate of the solution. In the case the weak solution blows up in finite time, estimate the lower bound and upper bound of the lifespan of the blow-up solution, and also estimate the blow-up rate. Finally, if our problem contains an external vertical load term, a sufficient condition is also established to obtain the global existence and general decay rate of weak solutions.</p>

Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term

<p style='text-indent:20px;'>In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and generalizes earlier results in the literature.</p>

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise

<p style='text-indent:20px;'>This paper focuses on the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L<inline-formula><tex-math id="M3">\begin{document}$ \acute{e} $\end{document}</tex-math></inline-formula>vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, <inline-formula><tex-math id="M4">\begin{document}$ H_\infty $\end{document}</tex-math></inline-formula> stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.</p>

General and optimal decay for a quasilinear parabolic viscoelastic system

<p style='text-indent:20px;'>In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying <inline-formula><tex-math id="M1">\begin{document}$ g'(t) \leq - \xi(t) H(g(t)) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M2">\begin{document}$ H $\end{document}</tex-math></inline-formula> is an increasing, convex function and <inline-formula><tex-math id="M3">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when <inline-formula><tex-math id="M4">\begin{document}$ H(t) = t^p, \ t\geq 0, \forall p&gt;1 $\end{document}</tex-math></inline-formula>, instead the parameter <inline-formula><tex-math id="M5">\begin{document}$ p \in [1, \frac{3}{2}[ $\end{document}</tex-math></inline-formula>.</p>