This paper establishes Razumikhin-type theorems on general decay stability for stochastic functional differential equations. This improves existing stochastic Razumikhin-type theorems and can make us examine the stability with general decay rate in the sense of the pth moment and almost sure. These stabilities may be specialized as the exponential stability and the polynomial stability. When the almost sure stability is examined, the conditions of this paper may defy the linear growth condition for the drift term, which implies that the theorems of this paper can work for some cases to which the existing results cannot be applied. This paper also examines some sufficient criteria under which this stability is robust. To illustrate applications of our results clearly, this paper also gives two examples and examines the exponential stability and the polynomial stability, respectively.By condition (H2), letting u = /( +¯ ) and applying the inequality (40) yield (here we assume that ,¯ = 0, otherwise this computation is easier) |g(t, )| 2 1 (t) ( +¯ )| (0)| 2 +¯ ( +¯ ) 0 − | ( )| 2 d ( ) .
In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation _ xðtÞ ¼ f ðxðtÞ, tÞ, in which f satisfies the one-sided polynomial growth condition, we introduce two Brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation dxðtÞ ¼ f ðxðtÞ, tÞdt þ qxðtÞdw 1 ðtÞ þ jxðtÞj xðtÞdw 2 ðtÞ. This article shows that appropriate may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time. Then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable.
The BGK model of the Boltzmann equation plays an important role in the kinetic theory of rarefied gases. Some existence and uniqueness results of solutions to its Cauchy problem were established for large initial data under various circumstances [see, for example, Perthame, B., “Global existence to the BGK model of Boltzmann equation,” J. Differ. Equations 82, 191–205 (1989)]. In this paper, by establishing some weighted Lp estimates of the hydrodynamical quantities of a gas, we prove the existence theorem of the Lp solutions to the Cauchy problem and establish the propagation properties of the Lp moments for this kind of solutions.
SUMMARYIn this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential systeṁ, where (t) is the variable delay and f satisfies the one-sided polynomial growth condition. Since f may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t) = f (x(t), x(t − (t)), t)dt +qx(t)dw 1 (t)+ |x(t)| x(t)dw 2 (t) by introducing two independent Brownian motions w 1 (t) and w 2 (t). This paper shows that the Brownian motion w 2 (t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of under the condition = 0. Moreover, for sufficiently large q, the Brownian motion w 1 (t) may exponentially stabilize this system.
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