In this paper we study an initial–Neumann boundary value problem for the following non-autonomous stochastic partial differential equation of parabolic type in a cylinder domain D×[0,T] , driven by an infinite-dimensional fractional noise: (1) du(x,t)=(div(k(x,t)∇u(x,t))+f(u(x,t)))dt+h(u(x,t))WH(x,dt),(x,t)∈D×(0,T],u(x,0)=φ(x),x∈D,∂u(x,t)∂n(k)=0,(x,t)∈∂D×(0,T]. Here D⊂Rd is a bounded domain with the boundary ∂D of class C2+β with some β∈(0,1) , WH is an L2(D) -valued fractional Brownian motion with the Hurst index H∈(12,1) , k=ki,j:D‾×[0,T]→Rd×d is a matrix-valued field, n(k)(x):=k(x,t)n(x) denotes the conormal vector-field, and the last relation in (1) refers to the conormal derivative of u relative to k, that is ∂u(x,t)∂n(k)= ∑i,j=1dki,j(x,t)ni(x)∂∂xju(x,t), n(x)∈Rd is an outer normal vector to ∂D .

Equations similar to (1) were studied extensively in literature, so we will mention only several most relevant articles here. The articles [5] and [7] considered heat equations with additive and multiplicative fractional noise, respectively. The articles [1, 10, 18] are devoted to general non-linear evolution equations with fractional noise; however, the equations are considered in some functional spaces, and the assumptions on the coefficients imposed there do not cover general nonlinear equations of the form (1). The problem (1) was considered in [14] and then in [16], where the notions of variational and mild solutions were introduced. The article [14] established the existence of a variational solution to this equation, but the uniqueness was shown under the assumption that the function h is affine. In [16], it was shown that a variational solution to (1) is a mild solution too, however, the uniqueness, both in the variational and in the mild sense, was established also under the assumption that h is affine, moreover, a rather restrictive assumption H>d+1d+2 \frac{d+1}{d+2}$]]> on the Hurst exponent was imposed.

Our goal is to extend the uniqueness results of [16] to the case of arbitrary H∈(12,1) and non-affine h. Specifically, we prove the uniqueness of a mild solution, assuming that h and its derivative h′ are Lipschitz continuous. Since the existence of a variational solution is known from [14], and [16] established that each variational solution is a mild solution, we get existence and uniqueness of a variational solution too, thus finally answering a question posed in [14]. We also show that the solution to (1) has finite moments of any order.

It is worth to mention that a similar uniqueness result holds in the case where the function h in front of WH depends on t sufficiently regularly, say, Hölder continuous with exponent greater than 1/2 . However, since our main reference for existence results are the articles [14, 16], in which h is assumed to be independent of t, we will follow this assumption.

The paper is organized as follows. In Section 2, we formulate the main hypotheses, and give the definition of a mild solution and basic facts on an L2(D) -valued fractional Brownian process and stochastic integration with respect to it. Section 3 contains auxiliary results concerning the parabolic Green’s function. In Section 4, we give a priori upper bounds for mild solutions. Finally, the main result on existence and uniqueness of a mild solution is proved in Section 5.

This section is devoted to the precise statement of the problem (1). We introduce necessary notation, give the definition of a mild solution, and formulate the assumptions for its existence and uniqueness.

In this section we collect several upper bounds for Green’s function G, needed for the proof of the main result.

Denote ΦtC(x):=t−d/2exp{−Cx2/t},t>0,x∈Rd. 0,\hspace{0.2778em}x\in {\mathbb{R}^{d}}.\]]]>

It is known from [2, 3] that under assumptions (A1) and (A2) G is a continuous function, twice continuously differentiable in x, once continuously differentiable in t. Moreover, G satisfies the heat kernel estimates (6) ∂xμ∂tνG(x,t;y,s)≤C(t−s)−(μ1+2ν)/2Φt−sC(x−y) for μ=(μ1,…,μd) , μ1,…,μd,ν∈N∪0 , and μ1+2ν≤2 with μ1=∑j=1dμj . In particular, for μ1=ν=0 , we have (7) G(x,t;y,s)≤CΦt−sC(x−y). The inequality (7) is sometimes called the Gaussian property of G ([15, 16]).

Several important properties of the parabolic Green’s function G follow from the fact that it is, for every (x,t)∈D×[0,T] , a classical solution to the linear boundary value problem ∂sG(x,t;y,s)=−div(k(y,s)∇yG(x,t;y,s)),(y,s)∈D×(0,T],∂G(x,t;y,s)∂n(k)=0,(y,s)∈∂D×(0,T], dual to (4). In particular, along with (6) we have also (8) ∂yμ∂sνG(x,t;y,s)≤C(t−s)−(μ1+2ν)/2Φt−sC(x−y) for μ1+2ν≤2 , and, moreover, the following convolution formula holds: (9) G(x,t;y,s)=∫DG(x,t;z,σ)G(z,σ;y,s)dzfor allσ∈(s,t), see [2, Appendix, p. 232–233] for the details.

Furthermore, according to Eqs. (3.4)–(3.5) from [16], G satisfies the following inequalities for all x,y∈D and δ∈(dd+2,1) . (i)

For all 0<r<v<t<T and some t∗∈(r,v) , (10) G(x,t;y,v)−G(x,t;y,r)≤C(t−v)−δ(v−r)δΦt−t∗C(x−y).

Fix H∈(1/2,1) and α∈(1−H,1/2) . Let u be a mild solution to (1), defined by (5). Note that the random variable ξα,H,T:=1+ ∑j=1∞λj1/2BjHα,0,T is finite a. s., see [11].

The goal of this section is to prove the following result.

We split the proof of Proposition 1 into two lemmas. In Lemma 2 we establish an upper bound for sups∈[0,t]supx∈Du(x,s) . In Lemma 3 we obtain similar estimate for uα,1,t .

In the calculations below we shall often refer to the following simple formulas: for all a>0 0$]]>, b>0 0$]]>, and 0<v<t , (14) ∫vt(t−s)a−1(s−v)b−1ds=C(t−v)a+b−1, (15) ∫0v(t−s)−a−b(v−s)b−1ds≤C(t−v)−a, where C=B(a,b) , the beta function. The formula (14) follows directly from the definition of the beta function by the substitution z=s−vt−v . The inequality (15) is obtained by the substitution z=v−st−v as follows: ∫0v(t−s)−a−b(v−s)b−1ds=(t−v)−a∫0vt−vzb−1(1+z)a+bdz≤(t−v)−a∫0∞zb−1(1+z)a+bdz=B(a,b)(t−v)−a.

Denote for brevity us=sups∈[0,t]supx∈Du(x,s).