Existence and uniqueness of the solution for stochastic super-fast diffusion equations with multiplicative noise

Abstract

In this paper we are concerned with the stochastic partial differential equations of superfast diffusion processes describing behavior of plasmaWe define a strong solution adequate to the properties of the natural logarithm and we prove the corresponding existence and uniqueness result.

State-constrained porous media control systems with application to stabilization

“…Predictability is obvious by definition and the inequality in item 4. in Definition 15 are guaranteed by Fatou's lemma. The fact that (X , Y) extends (X n , Y n ) is a mere consequence of the uniqueness in (17). The continuity of X , Y (see last item in Proposition 16) and the convergence lim…”

“…There is a vast literature on the subject and a complete overview would exceed the framework of this paper. We shall recall the pioneer works from [20], [27], the case of an unbounded domain from [11], the critical cases from [7], [17] and some properties of the solution from [9], [10], [23]. For further details, we invite our readers to refer to [8] for a monograph on the subject.…”

Stochastic porous media equations with divergence Itô noise

“…This type of solution is inspired from [18] and [22] and was already used several times in the study of the stochastic porous media equations. See [8], [9], [14].…”

“…We can now pass to the limit in (14) and get that (X (t) , e j ) −1 = (x, e j ) −1 − In order to finish the proof we only need to show that η = Ψ (X) a.e. on Ω × (0, T ) × O.…”