Stability of semilinear stochastic evolution equations with monotone nonlinearity

Abstract

Abstract. In this paper, we consider the exponentially asymptotic stability of the mild solutions of semilinear stochastic evolution equations of monotone type. An Itô-type inequality is our main tool to study the stability in the p -th moment and almost sure sample-path stability of the mild solutions. We also give some examples to illustrate the applications of the theorems.Mathematics subject classification (1991): 60H15, 34G20.

Stochastic functional evolution equations with monotone nonlinearity: Existence and stability of the mild solutions

“…The stability theory of semilinear stochastic heat equations has been well-studied by several authors among them we may point out [12,13,25] in the case of Lipschitz nonlinearity and [14] in the case of monotone one. In this section, we study the stochastic functional parabolic initial-boundary value problems of monotone-type and by virtue of the results derived in the previous sections, we justify the existence and stability of the mild solutions.…”

“…Finally, we give the Itô-type inequality [14] which is our main tool in this paper to prove the uniqueness, asymptotic stability and p-th mean boundedness of the mild solutions. Let {W t : t 0} be the cylindrical Brownian motion with respect to (Ω, F , F t , P ).…”

“…In this paper, we will generalize the results of [14,16] to semilinear stochastic functional evolution equations. Our novelty is that the nonlinear part of the equation is demicontinuous and satisfies a special kind of monotone condition instead of Lipschitz one.…”

“…To overcome this difficulty, Ichikawa [12,13] and K. Liu [20] introduced approximating systems with strong solutions and then used a limiting argument to transfer the stability properties to the mild solutions. On the other hand, Taniguchi [25,26] and Jahanipur [14][15][16] solved the problem by giving an L p -estimate for the difference of two mild solutions and proved the stability of the mild solutions of semilinear stochastic (delay) evolution equations. However, the results in [14,16] are more general than the previous works in that the nonlinear part of the equation is of monotone-type and satisfies the weak condition of demicontinuity.…”

“…Since there are not any standard results on the existence of the mild solutions for the class of evolution equations to which the present paper is about, we devote Section 4 to accomplish this task. The proofs of the existence and uniqueness are based on the Itô-type inequality [14] and a version of Burkholder inequality which is established in Section 3. In Section 5, we study the asymptotic behaviour of the mild solutions and finally in Section 6, the results are applied to general stochastic functional initial-boundary value problems.…”