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\(L^p\)-estimates and regularity for SPDEs with monotone semilinearity

Abstract

Semilinear stochastic partial differential equations on bounded domains \({\mathscr {D}}\) are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen–Cahn and Ginzburg–Landau equations. The first main result of this article are \(L^p\)-estimates for such equations. The \(L^p\)-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space \(H^2({\mathscr {D}}')\) and \(\ell ^2\)-integrable with values in \(H^3({\mathscr {D}}')\), for any compact \({\mathscr {D}}' \subset {\mathscr {D}}\). Using results from \(L^p\)-theory of SPDEs obtained by Kim (Stoch Proc Appl 112:261–283, 2004) we get analogous results in weighted Sobolev spaces on the whole \({\mathscr {D}}\). Finally it is shown that the solution is Hölder continuous in time of order \(\frac{1}{2} - \frac{2}{q}\) as a process with values in a weighted \(L^q\)-space, where q arises from the integrability assumptions imposed on the initial condition and forcing terms.

Introduction

The aim of this article is to obtain \(L^p\)-estimates and regularity of solutions to the semilinear stochastic partial differential equation (SPDE)

$$\begin{aligned} \begin{aligned} du_t&=(L_tu_t+f_t(u_t,\nabla u_t)+f_t^0)dt+\sum _{k\in {\mathbb {N}}}(M_t^ku_t+g_t^k)dW_t^k \,\,\,\ \text {on}\,\,\, [0,T]\times {\mathscr {D}}\\ u_t&=0 \,\,\, \text {on}\,\,\, \partial {\mathscr {D}}, \quad u_0 =\phi \,\,\, \text {on}\,\,\, {\mathscr {D}}, \end{aligned} \end{aligned}$$
(1)

where

$$\begin{aligned} L_t u:= \sum _{j=1}^d\partial _j\Big (\sum _{i=1}^da_t^{ij}\partial _iu\Big )+\sum _{i=1}^db_t^i\partial _iu+c_tu \,\,\,\,\,\,\text {and}\,\,\,\, \,\, M_t^k u :=\sum _{i=1}^d\sigma _t^{ik}\partial _iu+\mu _t^ku.\nonumber \\ \end{aligned}$$
(2)

Here \({\mathscr {D}}\) is a bounded domain in \({\mathbb {R}}^d\) and \(W^k\) are independent Wiener processes. The coefficients a and \(\sigma \) are assumed to satisfy stochastic parabolicity condition (and thus our equation is non-degenerate). Moreover all the coefficients a, bc, \(\sigma \) and \(\mu \) are assumed to be measurable and bounded, \(f = f_t(\omega , x, r, z)\) is measurable, continuous in (rz), monotone in r except perhaps around the origin, Lipschitz continuous in z, bounded in x and of polynomial growth in r (of arbitrary order). The forcing terms \(f^0\) and g are assumed to satisfy appropriate integrability conditions. A typical example of equation fitting this setting is the stochastic Ginzburg–Landau equation. In this case

$$\begin{aligned} f(r) = -|r|^{\alpha -2}r , \,\,\,\alpha \ge 2. \end{aligned}$$

To obtain higher interior regularity we will have to impose further regularity assumptions on the coefficients. To obtain regularity up to the boundary (in weighted Sobolev spaces) we will also need to impose regularity assumptions on the domain. The assumptions will be formulated precisely in further sections.

The main aim of this article is to obtain regularity results for the solutions to the SPDE (1). This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. For a semilinear equation it is natural to consider the term \(f:=f(u,\nabla u) + f^0\) as a free term in an appropriate linear SPDE and to use established methods and theory to obtain regularity for this linear SPDE. Due to uniqueness of solutions to (1), see Lemma 1, we then get the same regularity for the semilinear equation (1). However, for the theory of regularity of linear SPDEs to apply, we need to show that the new free term f satisfies appropriate integrability conditions. This would typically mean at least \(L^2\)-integrability. Since the semilinear term in (1) is allowed to have arbitrary polynomial growth, it is clear that we need to obtain \(L^p\)-estimates for solution to (1) with \(p\ge 2\) sufficiently large. Note that if one attempts to do this using Sobolev embedding theorem then one immediately runs into restrictions on the combination of dimension of \({\mathscr {D}}\) and the growth of the semilinear term.

The main novelty of this article is in allowing arbitrary dimension of \({\mathscr {D}}\) and growth of the semilinear term, see Theorem 1. This is achieved by using the monotonicity property of the semilinear term and a cutting argument to obtain the required \(L^p\)-estimate. Once these have been established we then obtain new spatial regularity results for the SPDE (1), these are both interior regularity and up-to-the-boundary regularity in weighed Sobolev spaces, see Theorems 2 and 5. Finally we have a new time regularity result (in weighted space again), see Theorem 6. These effectively say that under appropriate assumptions the SPDE (1) has two additional derivatives. It seems however that our method does not allow one to obtain arbitrarily high regularity (even for equation with smooth data and coefficients), see Remark 5 for explanation. Nevertheless, raising the regularity twice is enough to find the rate of convergence of various numerical approximations using the techniques from e.g. Gyöngy and Millet [9].

Regularity of solutions to linear PDEs has been studied intensively, see e.g. Evans [4], Gilbarg and Trudinger [8] for elliptic PDEs, Ladyženskaja et al. [20] for parabolic PDEs and references therein. Regularity results for linear elliptic and parabolic PDEs in Hölder spaces can be found in Krylov [16]. Regularity of solutions to SPDEs has been an area of active interest for quite some time and here we point out some of the main results. Regularity of solutions to linear SPDEs on the whole space has been proved in Rozovskii [23]. On domains with a boundary the situation is much more involved and one cannot expect the same regularity up to the boundary as in the interior of the domain, see e.g. Examples 1.1 and 1.2 in Krylov [18]. After this observation two approaches to dealing with boundaries emerge: one is to quantify the loss of regularity near the boundary using weighted Sobolev spaces. These allow oscillations and explosion of the spatial derivatives of the solution near the boundary. The other approach is to side-step the problems created by the boundary by restricting the class of equations under consideration by imposing additional restriction on the noise term near the boundary (effectively disallowing stochastic forcing near the boundary), see Flandoli [5]. Weighted Sobolev spaces have also been employed, in the context of \(L^p\)-theory for linear SPDEs, by Kim [14]. Unsurprisingly, there are fewer results for nonlinear SPDEs. Kim and Kim use the \(L^p\)-theory in [12] and [13] to obtain regularity for quasilinear SPDEs where the coefficients are uniformly bounded. Current results in Gerencsér [7] show that for a class of SPDEs, including (1), there exists some Hölder exponent such that the solution is Hölder continuous in space up to the boundary with this exponent. For interior regularity of a class of quasilinear equations associated with the “p-Laplace” operator see Breit [1]. For SPDEs with drift given by the subgradient of a quasi-convex function and with sufficiently regular noise Gess [6] proves higher regularity and existence of (analytically) strong solutions. All the aforementioned work on regularity of nonlinear SPDEs has been done using the variational approach. For results obtained in the semigroup framework we refer the reader to the work of Jentzen and Röckner [11] and references therein. Regularity results for quasilinear PDEs of parabolic type can be found in [20]. However, the results are obtained under the restrictions on the combination of dimension of \({\mathscr {D}}\) and the growth of the nonlinear term. Thus, to the best of our knowledge, our results are new even for deterministic semilinear PDEs with monotone semilinear term.

The article is organised as follows: Sect. 2 is devoted to the proof of Theorem 1 which gives us the desired \(L^p\)-estimates for the solution to semilinear SPDE (1). In Sect. 3, we first prove interior regularity for the associated linear SPDE, see Theorem 3. We then use the results on interior regularity of the linear SPDE to prove Theorem 2. In Sect. 4, we prove regularity results up to the boundary and time regularity in weighted Sobolev spaces using \(L^p\)-theory from Kim [14]. The main results and required assumptions are stated at the beginning of each section.

\(L^p\)-estimates for the semilinear equation

Let \(T>0\) be given, \((\Omega ,{\mathscr {F}},({\mathscr {F}}_t)_{t\in [0,T]},{\mathbb {P}})\) be a stochastic basis, \({\mathscr {P}}\) be the predictable \(\sigma \)-algebra and \(W:=(W_t)_{t\in [0,T]}\) be an infinite dimensional Wiener martingale with respect to \(({\mathscr {F}}_t)_{t\in [0,T]}\), i.e. the coordinate processes \((W_t^k)_{t\in [0,T]},\,\, k\in {\mathbb {N}}\) are independent \({\mathscr {F}}_t\)-adapted Wiener processes such that \(W_t^k-W_s^k\) is independent of \({\mathscr {F}}_s\) for \(s\le t\). Further, let \({\mathscr {D}}\) be a bounded domain in \({\mathbb {R}}^d\) with Lipschitz boundary. We use standard notation for Lebesgue–Bochner and Sobolev spaces. In general, if X is a normed linear space then we will use \(|\cdot |_X\) to denote the norm in this space. There are exceptions: if \(x\in {\mathbb {R}}^d\) then |x| denotes the Euclidean norm. For Lebesgue and Sobolev spaces over the entire domain \({\mathscr {D}}\) we will omit the dependence on \({\mathscr {D}}\). So e.g. if \(h\in L^p({\mathscr {D}})\) then we will write \(|h|_{L^p}\) for \(|h|_{L^p({\mathscr {D}})}\). If \(h\in L^p((0,T); L^p({\mathscr {D}}))\) then we use \(\Vert h\Vert _{L^p}\) to denote the norm. Throughout this article C denotes a generic constant that may change from line to line.

Let \(n\in \{0\}\cup {\mathbb {N}}\) and fix constants \(K>0\), \(\kappa > 0\), \(\alpha \ge 2\) and \(p \ge \alpha \). We assume the following:

A-1 For any \(i,j=1,\ldots ,d\), the coefficients \(a^{ij}, b^i\) and c are real-valued, \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and are bounded by K. The coefficients \(\sigma ^i=(\sigma ^{ik})_{k=1}^\infty \), \(\mu =(\mu ^k)_{k=1}^\infty \) are \(\ell ^2\)-valued, \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and almost surely

$$\begin{aligned} \sum _{i=1}^d\sum _{k\in {\mathbb {N}}} |\sigma _t^{ik}(x)|^2+\sum _{k\in {\mathbb {N}}}|\mu _t^k(x)|^2 \le K \quad \forall t \in [0,T], x\in {\mathscr {D}}. \end{aligned}$$

A-2 Almost surely

$$\begin{aligned} \sum _{i,j=1}^d \Big (a_t^{ij}(x)-\frac{1}{2}\sum _{k\in {\mathbb {N}}}\sigma _t^{ik}(x)\sigma _t^{jk}(x)\Big )\xi _i \xi _j \ge \kappa |\xi |^2 \quad \forall t \in [0,T], x\in {\mathscr {D}}, \xi \in {\mathbb {R}}^d . \end{aligned}$$

A-3 The function \(f = f_t(\omega , x, r, z)\) is \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}}) \times {\mathscr {B}}({\mathbb {R}})\times {\mathscr {B}}({\mathbb {R}}^d)\)-measurable, it is continuous in (rz) almost surely for all t and x. Furthermore, almost surely

$$\begin{aligned} \begin{aligned} (r-r')(f_t(x,r,z)-f_t(x,r',z))&\le K|r-r'|^2, \\ |f_t(x,r,z)-f_t(x,r,z')|&\le K|z-z'|, \\ |f_t(x,r,z)|&\le K(1+|r|)^{\alpha -1} \end{aligned} \end{aligned}$$

for all \(t, x, r,r', z, z' \).

A-4\(\phi \in L^p(\Omega , {\mathscr {F}}_0; L^p({\mathscr {D}}))\), \(f^0 \in L^p(\Omega \times (0,T), {\mathscr {P}};L^p({\mathscr {D}}))\) and \(g \in L^p(\Omega \times (0,T), {\mathscr {P}};L^p({\mathscr {D}};\ell ^2))\).

Remark 1

Without loss of generality, we may assume that almost surely for all t, x and z the function \(r\mapsto f_t(x,r,z)\) is decreasing. If not, then (1) can be rewritten by replacing \(f_t(x,r,z)\) with \({\bar{f}}_t(x,r,z):=f_t(x,r,z)-Kr\) and \(c_t(x)\) with \({\bar{c}}_t(x):=c_t(x)+K\), where using Assumption A - 3, \({\bar{f}}\) is decreasing in r.

Further, we may assume that almost surely for all t and x, \(f_t(x,0,0)=0\). Otherwise, we can replace \(f_t(x,r,z)\) in (1) by \({\tilde{f}}_t(x,r,z):=f_t(x,r,z)-f_t(x,0,0)\) and \(f_t^0\) by \({\tilde{f}}_t^0(x):=f_t^0(x)+f_t(x,0,0)\).

Definition 1

(\(L^2\)-Solution) An adapted, continuous \(L^2({\mathscr {D}})\)-valued process is said to be a solution of stochastic partial differential equation (1) if

  1. (i)

    \(dt\times {\mathbb {P}}\) almost everywhere \(u \in L^\alpha ({\mathscr {D}}) \cap H_0^1({\mathscr {D}})\) and

    $$\begin{aligned} {\mathbb {E}}\int _0^T(|u_t|_{L^\alpha }^\alpha +|u_t|_{H_0^1}^2)\, dt < \infty , \end{aligned}$$
  2. (ii)

    almost surely for every \(t\in [0,T]\) and \(\xi \in C_0^\infty ({\mathscr {D}})\),

    $$\begin{aligned} (u_t,\xi )= & {} (u_0,\xi ) + \int _0^t \langle L_s(u_s)+f_s(u_s,\nabla u_s)+f_s^0,\xi \rangle ds \\&+ \sum _{k\in {\mathbb {N}}} \int _0^t (\xi ,M_s^k(u_s)+g_s^k)dW_s^k. \end{aligned}$$

The following theorem is the main result of this section.

Theorem 1

If Assumptions A-1 to A-4 hold, then there exists a unique solution u to (1) and

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t \le T}|u_t|_{L^p}^p&+{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|\nabla u_s|^2|u_s|^{p-2}dxds \\&\le {C}{\mathbb {E}}\Big (|\phi |_{L^p}^p+\Vert f^0 \Vert _{L^p}^p+\Vert |g|_{\ell ^2}\Vert ^p_{L^p}\Big ), \end{aligned} \end{aligned}$$
(3)

where \(C=C(d,p,K,\kappa ,T)\).

The rest of Sect. 2 is devoted to proving Theorem 1 but we give a brief outline of the proof here.

  1. 1.

    We replace the semilinear term f by truncations \(f^m\), depending on some \(m\in {\mathbb {N}}\), chosen in such a way that that the monotonicity is preserved and \(f^m\) are bounded. By standard theory of stochastic evolution equations we obtain \(u^m\) which are solutions to the SPDE with f replaced with \(f^m\).

  2. 2.

    We now wish to get the estimate (3) for these \(u^m\) (uniformly in m). If we were allowed to apply Itô’s formula directly to \(r \mapsto |r|^p\) and the process \(u^m_t(x)\) and to integrate over \({\mathscr {D}}\) then (3) for \(u^m\) would follow from A-1, A-2 and A-3.

  3. 3.

    Since, of course, this is not allowed we instead consider an appropriate bounded smooth approximation \(\phi _n\) to \(r\mapsto |r|^p\) and use the Itô formula from Krylov [17]. We then establish an estimate similar to (3) but for \(\phi _n(u^m)\) instead of \(|u^m|^p\) and with the right-hand-side still depending on m but independent of n. See Lemma 2. This allows us to take the limit \(n\rightarrow \infty \) and to use the monotonicity of \(r\mapsto f^m_t(x,r,z)\) to obtain (3) for \(u^m\). See Lemma 3.

  4. 4.

    The final step is then to use compactness argument to obtain u as a weak limit of \((u^m)_{m\in {\mathbb {N}}}\), see Lemma 4, and the usual monotonicity argument to show that u satisfies (1). Fatou’s lemma will then yield (3) for u.

Before proceeding with the proof of Theorem 1, we observe the following:

Remark 2

Assumptions A-1 and A-2 imply, after some computations using Hölder’s and Young’s inequalities, the existence of a constant \(K'\) depending on Kd and \(\kappa \) only such that almost surely for all \(t\in [0,T]\) and \(w,w' \in H_0^1({\mathscr {D}})\),

$$\begin{aligned} 2 \langle L_tw+f_t^0,w\rangle + \sum _{k\in {\mathbb {N}}}|M_t^k w + g_t^k|^2_{L^2}+\kappa |w|^2_{H_0^1} \le K'\Big [|f_t^0|_{L^2}^2+\big ||g_t|_{\ell ^2}\big |^2_{L^2}+|w|^2_{L^2} \Big ] \end{aligned}$$

and

$$\begin{aligned} 2 \langle L_tw-L_tw',w-w'\rangle + \sum _{k\in {\mathbb {N}}}|M_t^kw-M_t^kw'|^2_{L^2}+\kappa |w-w'|^2_{H_0^1}\le K'|w-w'|^2_{L^2} . \end{aligned}$$

Lemma 1

(Uniqueness) The solution to (1) is unique in the sense that if u and \({\bar{u}}\) both satisfy (1) then

$$\begin{aligned} {\mathbb {P}}\Big (\sup _{t\le T}|u_t - {\bar{u}}_t|_{L^2}=0\Big ) = 1. \end{aligned}$$

Proof

Let u and \({\bar{u}}\) be two solutions of (1) in the sense of Definition 1. Then,

$$\begin{aligned} \begin{aligned} u_t - {{\bar{u}}}_t =\int _0^t \left( L_s(u_s)-L_s({\bar{u}}_s)+f_s(u_s, \nabla u_s)-f_s({\bar{u}}_s, \nabla {\bar{u}}_s)\right) \,ds \\ +\sum _{k\in {\mathbb {N}}} \int _0^t \left( M_s^k(u_s)-M_s^k({\bar{u}}_s)\right) \,dW_s^k \end{aligned} \end{aligned}$$
(4)

almost surely for all \( t \in [0,T]\). Using Remark 1, Assumption A-3 and Young’s inequality, we get

$$\begin{aligned} \begin{aligned}&\langle f_t(u_t, \nabla u_t)-f_t({\bar{u}}_t, \nabla {\bar{u}}_t),u_t-{\bar{u}}_t\rangle \\&\quad =\langle f_t(u_t, \nabla u_t)-f_t({\bar{u}}_t, \nabla u_t)+ f_t({\bar{u}}_t, \nabla u_t)-f_t({\bar{u}}_t, \nabla {\bar{u}}_t),u_t-{\bar{u}}_t\rangle \\&\quad \le \frac{\kappa }{2} |\nabla (u_t-{\bar{u}}_t)|^2_{L^2}+N|u_t-{\bar{u}}_t|^2_{L^2} . \end{aligned} \end{aligned}$$
(5)

Using the product rule and applying Itô’s formula for the the square of the norm to (4), see Gyöngy and Šiška [10] or Pardoux [22, Chapitre 2, Theoreme 5.2], we obtain

$$\begin{aligned} \begin{aligned} d&\Big (e^{-K''t}|u_t-{\bar{u}}_t|_{L^2}^2 \Big ) = e^{-K''t}\big [d|u_t-{\bar{u}}_t|_{L^2}^2-K''|u_t-{\bar{u}}_t|_{L^2}^2\,dt\big ]\\&\quad = e^{-K''t}\bigg [\Big (2\langle L_t(u_t)-L_t({\bar{u}}_t)+f_t(u_t, \nabla u_t)-f_t({\bar{u}}_t, \nabla {\bar{u}}_t),u_t-{\bar{u}}_t\rangle \\&\quad +\sum _{k\in {\mathbb {N}}}|M_t^k(u_t) - M_t^k({\bar{u}}_t)|_{L^2}^2 -K''|u_t-{\bar{u}}_t|_{L^2}^2\Big )\,dt \\&\quad + \sum _{k\in {\mathbb {N}}} 2\big (u_t-{\bar{u}}_t,M_t^k(u_t)-M_t^k({\bar{u}}_t)\big ) dW_t^k \bigg ] \end{aligned} \end{aligned}$$
(6)

almost surely for all \( t \in [0,T]\). Substituting (5) in (6) and using Remark 2, we get

$$\begin{aligned} e^{-K''t}|u_t-{\bar{u}}_t|_{L^2}^2 \le 2\sum _{k\in {\mathbb {N}}}\int _0^t e^{-K''s}\big ( u_s-{\bar{u}}_s,M_s^k(u_s)-M_s^k({\bar{u}}_s)\big ) dW_s^k \end{aligned}$$

implying that right hand side is a non-negative local martingale (and thus a super-martingale) starting from 0 and hence for all \( t \in [0,T]\),

$$\begin{aligned} {\mathbb {E}}[e^{-K't}|u_{t}-{\bar{u}}_{t}|_{L^2}^2]\le 0. \end{aligned}$$

Thus for all \( t \in [0,T]\), we get \({\mathbb {P}}(|u_t - {{\bar{u}}}_t|_{L^2}^2=0)=1\) which, along with the continuity of \(u-{{\bar{u}}}\) in \({L^2({\mathscr {D}})}\), concludes the proof. \(\square \)

Having proved uniqueness we start preparing the proof of Theorem 1. For \(m\in {\mathbb {N}}\), consider the truncated function

$$\begin{aligned} f_t^{m}(x,r,z)=\left\{ \begin{array}{ll} f_t(x,-m,z)&{}\quad \text {if}\,\,r<-m \\ f_t(x,r,z)&{}\quad \text {if}\,\,-m\le r \le m\\ f_t(x,m,z)&{}\quad \text {if}\,\,r>m, \end{array} \right. \end{aligned}$$

and the equation

$$\begin{aligned} \begin{aligned} du_t^{m}&=(L_t u_t^{m}+f_t^{m}(u_t^{m},\nabla u_t^{m})+f_t^0)dt+\sum _{k\in {\mathbb {N}}}(M_t^ku_t^{m}+g_t^k)dW_t^k, \\ u_t^m&=0 \,\,\, \text {on}\,\,\, \partial {\mathscr {D}}, \quad u_0^m =\phi \,\,\, \text {on}\,\,\, {\mathscr {D}}. \end{aligned} \end{aligned}$$
(7)

For each \(m\in {\mathbb {N}}\), using Assumption A-3, \(f_t^{m}(x,r,z )\) is bounded and hence (7) can be viewed as a SPDE on the Gelfand triple \(H_0^1({\mathscr {D}}) \hookrightarrow ~ L^2({\mathscr {D}}) \hookrightarrow ~ H^{-1}({\mathscr {D}})\) and all the conditions for existence and uniqueness of solution in [19] are satisfied. Thus (7) has a unique \(L^2\)-solution in the sense of [19, Definition 2.2].

We now prove an estimate similar to (3) for the solutions of (7). We will do this by applying the Itô formula from Krylov [17] similarly to Dareiotis and Gerencsér [3]. To that end we need to consider the functions

$$\begin{aligned} \phi _n(r)=\left\{ \begin{array}{lll} |r|^p &{} \quad \text {if} &{} |r|< n \\ n^{p-2}\frac{p(p-1)}{2}(|r|-n)^2+pn^{p-1}(|r|-n)+n^p &{} \quad \text {if} &{} |r|\ge n. \end{array} \right. \end{aligned}$$

We now collect some key properties of these functions. We see that \(\phi _n\) are twice continuously differentiable and

$$\begin{aligned} |\phi _n(x)|\le {C}|x|^2,\,\,|\phi _n'(x)|\le {C}|x|,\,\,|\phi _n''(x)|\le {C} \end{aligned}$$

where C depends on p and \(n \in {\mathbb {N}}\) only. Further, for any \(r\in {\mathbb {R}}\),

$$\begin{aligned} \phi _n(r)\rightarrow |r|^p, \,\,\phi _n'(r)\rightarrow p|r|^{p-2}r, \,\,\phi _n''(r)\rightarrow p(p-1)|r|^{p-2} \end{aligned}$$
(8)

as \(n \rightarrow \infty \) and

$$\begin{aligned} \phi _n(r)\le {C}|r|^p, \,\, \phi _n'(r)\le {C}|r|^{p-1}, \,\,\phi _n''(r)\le {C}|r|^{p-2}, \,\, \end{aligned}$$
(9)

where C depends on p only.

Remark 3

For any \(r\in {\mathbb {R}}\) we have

  1. (a)

    \( |r\phi _n'(r)|\le p\phi _n(r) \),

  2. (b)

    \(|r^2\phi _n''(r)|\le p(p-1)\phi _n(r)\),

  3. (c)

    \(|\phi _n'(r)|^2\le 4p\phi _n''(r)\phi _n(r)\),

  4. (d)

    \(|\phi _n''(r)|^\frac{p}{p-2}\le [p(p-1)]^\frac{p}{p-2}\phi _n(r) \).

These inequalities along with Young’s inequality imply, for any \(\epsilon > 0\),

  1. (i)

    \( |u_s^m\phi _n'(u_s^m)|\le {C}\phi _n(u_s^m) \),

  2. (ii)

    \(|u_s^m|^2\phi _n''(u_s^m)\le {C}\phi _n(u_s^m)\),

  3. (iii)

    \(\sum _{i=1}^d \partial _iu_s^m \phi _n'(u_s^m)\le \epsilon \phi _n''(u_s^m)|\nabla u_s^m|^2+ {C} \phi _n(u_s^m)\),

  4. (iv)

    \(|f_s^0\phi _n'(u_s^m)|\le {C}|f_s^0|[\phi _n''(u_s^m)]^\frac{1}{2}[\phi _n(u_s^m)]^\frac{1}{2}\le {C}|f_s^0|^p+{C}\phi _n(u_s^m)\),

  5. (v)

    \(|f_s^m(u_s^m, \nabla u_s^m)\phi _n'(u_s^m)|\le {C}|f_s^m(u_s^m, \nabla u_s^m)|[\phi _n''(u_s^m)]^\frac{1}{2}[\phi _n(u_s^m)]^\frac{1}{2} \le {C}|f_s^m(u_s^m, \nabla u_s^m)|^p+{C}\phi _n(u_s^m)\le {C}|f_s(-m, \nabla u_s^m)|^p+{C}\phi _n(u_s^m)\),

  6. (vi)

    \(|g_s|_{\ell ^2}^2\phi _n''(u_s^m) \le {C}\phi _n(u_s^m) + {C} |g_s|_{\ell ^2}^p\),

where the last inequality is obtained using Hölder’s inequality and C depends only on dp and \(\epsilon \).

Using Theorem 3.1 from [17], we get that almost surely

for any \(t\in [0,T]\) and \(n\in {\mathbb {N}}\). Thus using Assumptions A-1, A-2 and Young’s inequality for any \(\epsilon >0\), we obtain almost surely

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}}&\phi _n(u_t^m)dx \le \int _{\mathscr {D}}\phi _n(u_0^m)dx+ {\mathscr {M}}_t^{n,m} \\&+ \int _0^t\int _{\mathscr {D}} \Big (\sum _{i=1}^d b_s^i \partial _iu_s^m + c_s u_s^m + f_s^m(u_s^m, \nabla u_s^m)+f^0_s\Big )\phi _n'(u_s^m)dxds \\&- \int _0^t\int _{\mathscr {D}}\kappa |\nabla u_s^m|^2 \phi _n''(u_s^m)dxds\\&+ \int _0^t\int _{\mathscr {D}} \Big ( \epsilon |\nabla u_s^m|^2 + {C} |u_s|^2+ {C} |g_s|_{\ell ^2}^2\Big )\phi _n''(u_s^m)\,dxds, \end{aligned} \end{aligned}$$
(10)

for any \(t\in [0,T]\) and \(n\in {\mathbb {N}}\). Here the generic constant C depends only on dK and \(\epsilon \) and

$$\begin{aligned} {\mathscr {M}}_t^{n,m}:=\sum _{k\in {\mathbb {N}}} \int _0^t\int _{\mathscr {D}}\Big (\sum _{i=1}^d\sigma _s^{ik}\partial _iu_s^m+\mu _s^ku_s^m+g_s^k\Big )\phi _n'(u_s^m) dxdW_s^k \end{aligned}$$

is a martingale.

Further, using Burkholder–Davis–Gundy’s inequality, Remark 3(c) and Hölder’s inequality, we see that

$$\begin{aligned} \begin{aligned}&{\mathbb {E}} \sup _{0\le t \le T} |{\mathscr {M}}_t^{n,m}| \\&\quad \le {C}{\mathbb {E}}\Bigg (\int _0^T\sum _k\bigg ( \int _{\mathscr {D}}\Big |\sum _{i=1}^d\sigma _s^{ik}\partial _iu_s^m{+}\mu _s^ku_s^m+g_s^k\Big |\Big (\phi _n''(u_s^m)\phi _n(u_s^m)\Big )^\frac{1}{2}dx\bigg )^2ds\Bigg )^\frac{1}{2}\\&\quad \le {C}{\mathbb {E}}\Bigg (\int _0^T\sum _k\bigg ( \int _{\mathscr {D}}\Big |\sum _{i=1}^d\sigma _s^{ik}\partial _iu_s^m{+}\mu _s^ku_s^m{+}g_s^k\Big |^2\phi _n''(u_s^m)dx\int _{\mathscr {D}}\phi _n(u_s^m)dx\bigg )ds\Bigg )^\frac{1}{2} \end{aligned} \end{aligned}$$

which, using the same steps as before, in particular Remark 3 points (ii) and (iv), gives

$$\begin{aligned}&{\mathbb {E}} \sup _{0\le t \le T} |{\mathscr {M}}_t^{n,m}| \nonumber \\&\quad \le {C}{\mathbb {E}}\Bigg (\int _0^T\bigg ( \int _{\mathscr {D}}\Big (|\nabla u_s^m|^2+|u_s^m|^2+|g_s|_{\ell ^2}^2\Big )\phi _n''(u_s^m)dx\int _{\mathscr {D}}\phi _n(u_s^m)dx\bigg )ds\Bigg )^\frac{1}{2}\nonumber \\&\quad \le {C}{\mathbb {E}}\Bigg (\sup _{0\le t\le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx\int _0^T \int _{\mathscr {D}}\Big [|\nabla u_s^m|^2\phi _n''(u_s^m)+\phi _n(u_s^m) +|g_s|_{\ell ^2}^p\Big ]dxds\Bigg )^\frac{1}{2} \nonumber \\&\quad \le \frac{1}{2} {\mathbb {E}}\sup _{0\le t\le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx {+} {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}}\Big [|\nabla u_s^m|^2\phi _n''(u_s^m){+}\phi _n(u_s^m){+}|g_s|_{\ell ^2}^p\Big ]dxds\nonumber \\ \end{aligned}$$
(11)

Lemma 2

If \(u^m\) is the solution to (7), then

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}|u_t^{m}|_{L^p}^p&+ {\mathbb {E}}\int _0^t\int _{\mathscr {D}}|\nabla u_s^m|^2|u_s^m|^{p-2}dxds \\&\le {C}{\mathbb {E}}\Big (|\phi |_{L^p}^p+ C_m+\Vert f^0 \Vert _{L^p}^p+\Vert |g|_{\ell ^2}\Vert ^p_{L^p}\Big ), \end{aligned} \end{aligned}$$
(12)

where \({C}={C}(d,K,\kappa , p)\) and \(C_m:={\mathbb {E}}\int _0^T\int _{{\mathscr {D}}}(1+|m|)^{\alpha (p-1)}dxds\) are constants.

Proof

From  (10) and Remark 3(iv),(v) and Assumption A-3, we get

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\int _{\mathscr {D}}\phi _n&(u_t^m)dx+\frac{\kappa }{2} {\mathbb {E}}\int _0^t\int _{\mathscr {D}}|\nabla u_s^m|^2\phi _n''(u_s^m)dxds \le {C}{\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_0^m)dx + C_m\\&+ {\mathbb {E}}\int _0^t\int _{\mathscr {D}}|f_s^0|^pdxds + {C}{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|g_s|_{\ell ^2}^p dxds+ {C}\int _0^t{\mathbb {E}} \int _{\mathscr {D}}\phi _n(u_s^m)dxds\\&\le {C}{\mathbb {E}}{\mathcal {K}}_t^m+{C}\int _0^t{\mathbb {E}} \int _{\mathscr {D}}\phi _n(u_s^m)dxds, \end{aligned} \end{aligned}$$

where \({C}={C}(d,p,K,\epsilon )\) and

$$\begin{aligned} {\mathcal {K}}_t^m:=\int _{\mathscr {D}}|\phi |^pdx+ C_m +\int _0^t\int _{\mathscr {D}}|f_s^0|^pdxds+\int _0^t\int _{\mathscr {D}}|g_s|_{\ell ^2}^p\,dxds. \end{aligned}$$

Applying Gronwall’s lemma, we obtain for any \(t\in [0,T]\)

$$\begin{aligned} {\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_t^m)dx+{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|\nabla u_s^m|^2\phi _n''(u_s^m)dxds \le {C}{\mathbb {E}}{\mathcal {K}}_t^m \end{aligned}$$
(13)

where \({C}={C}(d,p,K,\kappa ,T)\).

Further, taking the supremum over \(t\in [0,T]\) in (10), using the same estimates as given above and then taking expectation, we get using (11)

$$\begin{aligned} \begin{aligned} {\mathbb {E}}&\sup _{0\le t \le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx \\&\le {C}{\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_0^m)dx+ {\mathbb {E}}\sup _{0\le t\le T}\int _0^t\int _{\mathscr {D}}f_s^m(u_s^m, \nabla u_s^m)\phi _n'(u_s^m)dxds \\&\quad +{C}{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|f_s^0|^pdxds+{C}{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|g_s|_{\ell ^2}^p\,dxds+{C}\int _0^T{\mathbb {E}} \int _{\mathscr {D}}\phi _n(u_s^m)dxds\\&\quad + \frac{1}{2} {\mathbb {E}}\sup _{0\le t\le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx + {C}{\mathbb {E}}\int _0^T \int _{\mathscr {D}}\Big [|\nabla u_s^m|^2\phi _n''(u_s^m)+\phi _n(u_s^m)\Big ]dxds \\&\le {C}{\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_0^m)dx+ {C}C_m+{C}{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|f_s^0|^pdxds \\&\quad + {C}{\mathbb {E}}\int _0^T\int _{{\mathscr {D}}} \big [|g_s|_{\ell ^2}^p + \phi _n(u_s^m)\big ]\,dxds\\&\quad + \frac{1}{2} {\mathbb {E}}\sup _{0\le t\le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx +{C}{\mathbb {E}}\int _0^T \int _{\mathscr {D}}|\nabla u_s^m|^2\phi _n''(u_s^m)dxds\\&\le {C}{\mathbb {E}}{\mathcal {K}}_T^m+ \frac{1}{2} {\mathbb {E}}\sup _{0\le t\le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx <\infty \end{aligned} \end{aligned}$$

where C does not depend on n and m. Thus, we have

$$\begin{aligned} {\mathbb {E}}\sup _{0\le t \le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx+{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|\nabla u_s^m|^2\phi _n''(u_s^m)dxds\le {C}{\mathbb {E}}{\mathcal {K}}_T^m < \infty , \end{aligned}$$

where \({C}={C}(d,p,K,\kappa ,T)\). Now we let \(n\rightarrow \infty \) and apply Fatou’s lemma to complete the proof. \(\square \)

We can now use Lemma 2 and the monotonicity of \(r\mapsto f^m_t(x,r,z)\) to obtain an estimate for \(u^m_t\), where the right-hand-side no longer depends on m. Let

$$\begin{aligned} {\mathscr {K}}_t:=\int _{\mathscr {D}}|\phi |^p dx + \int _0^t\int _{\mathscr {D}}\big [|f_s^0|^p + |g_s|_{\ell ^2}^p\big ]\,dxds. \end{aligned}$$

Lemma 3

If \(u^m\) is the solution to (7) then there is \({C}={C}(d,p,K,\kappa ,T)\) such that

$$\begin{aligned} {\mathbb {E}}\sup _{0\le t \le T}|u_t^m|_{L^p}^p +{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|\nabla u_s^m|^2|u_s^m|^{p-2}\,dxds\le {C}{\mathbb {E}}{\mathscr {K}}_T. \end{aligned}$$
(14)

Proof

From  (10) and Remark 3(iv), we get

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_t^m)&dx+\frac{\kappa }{2}{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|\nabla u_s^m|^2\phi _n''(u_s^m)dxds \\ \le&{C}{\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_0^m)dx+ {\mathbb {E}}\int _0^t\int _{\mathscr {D}}\big [f_s^m(u_s^m, \nabla u_s^m)\phi _n'(u_s^m) + |f_s^0|^p\big ]\,dxds \\&+{C}{\mathbb {E}}\int _0^t\int _{\mathscr {D}}\big [|g_s|_{\ell ^2}^p + \phi _n(u_s^m)\big ]\,dxds, \end{aligned} \end{aligned}$$

where \({C}={C}(d,p,K,\kappa )\).

Taking limit \(n\rightarrow \infty \) and using Lebesgue’s dominated convergence theorem in view of (12), (8) and (9), we get

$$\begin{aligned} \begin{aligned} {\mathbb {E}}&\int _{\mathscr {D}}|u_t^m|^pdx+p(p-1)\frac{\kappa }{2}{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|\nabla u_s^m|^2|u_s^m|^{p-2}dxds \\&\le {C}{\mathbb {E}}{\mathscr {K}}_t+ p{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|u_s^m|^{p-2}f_s^m(u_s^m, \nabla u_s^m)u_s^mdxds+{C} {\mathbb {E}} \int _0^t \int _{\mathscr {D}}|u_s^m|^p dxds. \end{aligned} \end{aligned}$$
(15)

Using the fact \(rf_t^m(r,0)\le 0\) for any \(r\in {\mathbb {R}},m\in {\mathbb {N}}, t\in [0,T]\), Young’s inequality and Assumption A-3, we get

$$\begin{aligned} \begin{aligned}&p{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|u_s^m|^{p-2}f_s^m(u_s^m, \nabla u_s^m)u_s^mdxds \\&\quad = p{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|u_s^m|^{p-2} \big [f_s^m(u_s^m, \nabla u_s^m)-f_s^m(u_s^m,0)+f_s^m(u_s^m,0)\big ]u_s^mdxds \\&\quad \le {\mathbb {E}}\int _0^t\int _{\mathscr {D}}|u_s^m|^{p-2} \big [\frac{\kappa }{4} |f_s^m(u_s^m, \nabla u_s^m)-f_s^m(u_s^m,0)|^2 + {C}|u_s^m|^2\big ]dxds \\&\quad \le \frac{\kappa }{4} {\mathbb {E}}\int _0^t\int _{\mathscr {D}}|u_s^m|^{p-2} | \nabla u_s^m|^2 dxds + {C} {\mathbb {E}}\int _0^t\int _{\mathscr {D}}|u_s^m|^pdxds \end{aligned} \end{aligned}$$

Substituting this in (15) and then applying Gronwall’s lemma, we obtain for any \(t\in [0,T]\)

$$\begin{aligned} {\mathbb {E}}\int _{\mathscr {D}}|u_t^m|^pdx+{\mathbb {E}}\int _0^t\int _{\mathscr {D}}|\nabla u_s^m|^2|u_s^m|^{p-2}dxds\le {C}{\mathbb {E}}{\mathscr {K}}_t \end{aligned}$$

where \({C}={C}(d,p,K,\kappa ,T)\).

Further, taking the supremum over \(t\in [0,T]\) in (10), using the same estimates as given above and then taking expectation, we get using (11)

$$\begin{aligned} \begin{aligned} {\mathbb {E}}&\sup _{0\le t \le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx \\&\le {C}{\mathbb {E}}\int _{\mathscr {D}}\phi _n(u_0^m)dx+ {\mathbb {E}}\sup _{0\le t\le T}\int _0^t\int _{\mathscr {D}}f_s^m(u_s^m, \nabla u_s^m)\phi _n'(u_s^m)dxds \\&\quad + {C}{\mathbb {E}}\int _0^T\int _{\mathscr {D}}\big [|f_s^0|^p + |g_s|_{\ell ^2}^p + \phi _n(u_s^m)\big ]\,dxds\\&\quad + \frac{1}{2} {\mathbb {E}}\sup _{0\le t\le T}\int _{\mathscr {D}}\phi _n(u_t^m)dx + {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}}|\nabla u_s^m|^2\phi _n''(u_s^m)dxds, \end{aligned} \end{aligned}$$

where C does not depend on n and m. Taking limit \(n\rightarrow \infty \) using Lebesgue’s dominated convergence theorem and using (13) along with the steps as above, we get

$$\begin{aligned} {\mathbb {E}}\sup _{0\le t \le T}\int _{\mathscr {D}}|u_t^m|^pdx\le {C}{\mathbb {E}}{\mathscr {K}}_T+ \frac{1}{2} {\mathbb {E}}\sup _{0\le t\le T}\int _{\mathscr {D}}|u_t^m|^pdx \end{aligned}$$

and hence the lemma. \(\square \)

To complete the proof of Theorem 1 we need to take the limit, as \(m\rightarrow \infty \) in (14) and to show that (1) has a solution. To that end we obtain the following result.

Lemma 4

There is a subsequence of (m) denoted by \((m')\) and an adapted process u such that \(u \in L^\alpha (\Omega \times (0,T),{\mathscr {P}};L^\alpha ({\mathscr {D}})) \cap L^2(\Omega \times (0,T),{\mathscr {P}};H_0^1({\mathscr {D}}))\) and almost surely \(u\in C([0,T]; L^2({\mathscr {D}}))\). Moreover, there exists \(f'\in L^\frac{\alpha }{\alpha -1}\big (\Omega \times (0,T),{\mathscr {P}};L^\frac{\alpha }{\alpha -1}({\mathscr {D}})\big )\) such that

$$\begin{aligned} \begin{aligned} u^{m'}\rightharpoonup u\quad \text {in}\quad {L^\alpha (\Omega \times (0,T),{\mathscr {P}};L^\alpha ({\mathscr {D}})) \cap L^2(\Omega \times (0,T),{\mathscr {P}};H_0^1({\mathscr {D}}))}, \end{aligned} \\ f^{m'}(u^{m'}, \nabla u^{m'})\rightharpoonup f' \quad \text {in}\quad L^\frac{\alpha }{\alpha -1}\big (\Omega \times (0,T),{\mathscr {P}};L^\frac{\alpha }{\alpha -1}({\mathscr {D}})\big ), \\ \begin{aligned} L(u^{m'})\rightharpoonup L({u})\quad \text {in}\quad L^2\big (\Omega \times (0,T),{\mathscr {P}};H^{-1}({\mathscr {D}})\big ),\\ M(u^{m'})\rightharpoonup M({u}) \quad \text {in}\quad L^2\big (\Omega \times (0,T),{\mathscr {P}};\ell ^2(L^2({\mathscr {D}}))\big ). \end{aligned} \end{aligned}$$

Finally for all \(t \in [0,T]\),

$$\begin{aligned} u_t=u_0+\int _0^t (L_s u_s+f'_s+f_s^0)ds +\sum _{k\in {\mathbb {N}}}\int _0^t (M_s^k u_s+g_s^k)dW_s^k\,\,a.s. \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} |u_t|_{L^2}^2 =&|\psi |_{L^2}^2 + 2\int _0^t \langle L_s u_s+f_s^0, u_s \rangle \,ds + 2 \int _0^t \langle f'_s, u_s \rangle \,ds \\&+ 2 \sum _{k\in {\mathbb {N}}} \int _0^t (M_s^k u_s+g_s^k, u_s) \,dW^k_s + \sum _{k\in {\mathbb {N}}} \int _0^t |M_s^k u_s+g_s^k|_{L^2}^2\, ds . \end{aligned} \end{aligned}$$

Proof

By Lemma 3, we have \(u^{m}\in L^\alpha (\Omega \times (0,T),{\mathscr {P}};L^\alpha ({\mathscr {D}}))\cap L^2(\Omega \times (0,T),{\mathscr {P}};H_0^1({\mathscr {D}}))\). Moreover, using Assumption A-3 and (14), we have

$$\begin{aligned} \begin{aligned} {\mathbb {E}} \int _0^T \int _{{\mathscr {D}}}|f^m_t(u^m_t(x), \nabla u^m_t(x))|^\frac{\alpha }{\alpha -1}&\,dxdt \le K {\mathbb {E}} \int _0^T \int _{{\mathscr {D}}}(1+|u_t^m(x)|)^\alpha dxdt \\&\le {C}+{C} {\mathbb {E}}\sup _{0\le t \le T}\int _{\mathscr {D}}|u_t^m(x)|^\alpha dx <\infty . \end{aligned} \end{aligned}$$
(16)

Thus, \(f^m(u^m,\nabla u^m)\in L^\frac{\alpha }{\alpha -1}\big (\Omega \times (0,T),{\mathscr {P}};L^\frac{\alpha }{\alpha -1}({\mathscr {D}})\big )\) such that (14) and (16) holds for each \(m \in {\mathbb {N}}\) with a constant independent of m. Since these Banach spaces are reflexive, there exists a subsequence \({(m')}\) (see, e.g. Theorem 3.18 in [2]) such that

$$\begin{aligned} u^{m'}\rightharpoonup v&\text {in}&L^\alpha (\Omega \times (0,T),{\mathscr {P}};L^\alpha ({\mathscr {D}})) ,\\ u^{m'}\rightharpoonup {\bar{v}}&\text {in}&L^2(\Omega \times (0,T),{\mathscr {P}};H_0^1({\mathscr {D}}))\,\,\text {and}\\ f^{m'}(u^{m'},\nabla u^{m'})\rightharpoonup f'&\text {in}&L^\frac{\alpha }{\alpha -1}\big (\Omega \times (0,T),{\mathscr {P}};L^\frac{\alpha }{\alpha -1}({\mathscr {D}})\big ) . \end{aligned}$$

Moreover, the operators L and M are bounded and linear and hence map a weakly convergent sequence to a weakly convergent sequence. Thus, we have

$$\begin{aligned} L(u^{m'})\rightharpoonup L({\bar{v}})&\text {in}&L^2\big (\Omega \times (0,T),{\mathscr {P}};H^{-1}({\mathscr {D}})\big )\,\,\text {and}\\ M(u^{m'})\rightharpoonup M({\bar{v}})&\text {in}&L^2\big (\Omega \times (0,T),{\mathscr {P}};\ell ^2(L^2({\mathscr {D}}))\big ) . \end{aligned}$$

Note that for any adapted and bounded real valued process \(\eta _t\) and \(\xi \in C_0^\infty ({\mathscr {D}})\), we have

$$\begin{aligned} {\mathbb {E}}\int _0^T\eta _t \langle v_t-{\bar{v}}_t,\xi \rangle dt={\mathbb {E}}\int _0^T\eta _t \langle v_t-u_t^{m'},\xi \rangle dt+{\mathbb {E}}\int _0^T\eta _t \langle u_t^{m'}-{\bar{v}}_t,\xi \rangle dt \rightarrow 0 \end{aligned}$$

as \(m' \rightarrow \infty \). Since \(C_0^\infty ({\mathscr {D}})\) is dense in \(L^\alpha ({\mathscr {D}})\) and \(H_0^1({\mathscr {D}})\), we have the processes v and \({\bar{v}}\) are equal \(dt\times {\mathbb {P}}\)  almost everywhere. Further, the Bochner integral and the stochastic integral are bounded linear operators and hence are continuous with respect to weak topologies. Again, we have

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\int _0^T\eta _t(u_t^{m'},\xi )dt \\&\quad ={\mathbb {E}}\int _0^T\eta _t\Big ((u_0^{m'},\xi )+\int _0^t\langle L_su_s^{m'}+f_s^{m'}+f_s^0,\xi \rangle ds \\&\quad +\sum _{k\in {\mathbb {N}}}\int _0^t(\xi ,M_s^ku_s^{m'}+g_s^k)dW_s^k \Big )dt. \end{aligned} \end{aligned}$$

On taking limit \(m' \rightarrow \infty \), we get

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\int _0^T\eta _t({v}_t,\xi )dt \\&\quad ={\mathbb {E}}\int _0^T\eta _t\Big ((u_0,\xi )+\int _0^t\langle L_s{v}_s+f'_s+f_s^0,\xi \rangle ds \\&\quad +\sum _{k\in {\mathbb {N}}}\int _0^t(\xi ,M_s^k{v}_s+g_s^k)dW_s^k \Big )dt \end{aligned} \end{aligned}$$

for any adapted and bounded real valued process \(\eta _t\) and \(\xi \in C_0^\infty ({\mathscr {D}})\). Since \(C_0^\infty ({\mathscr {D}})\) is dense in \(L^\alpha ({\mathscr {D}})\) and \(H_0^1({\mathscr {D}})\), we have

$$\begin{aligned} {v}_t=u_0+\int _0^t (L_s{v}_s+f'_s+f_s^0)ds +\sum _{k\in {\mathbb {N}}}\int _0^t (M_s^k{v}_s+g_s^k)dW_s^k \end{aligned}$$

\(dt\times {\mathbb {P}}\)  almost everywhere. Using Itô formula for processes taking values in intersection of Banach spaces from Gyöngy and Šiška [10], there exists an \(L^2({\mathscr {D}})\)-valued continuous modification u of v which satisfies above equality almost surely for all \(t\in [0,T]\). \(\square \)

Remark 4

For \(\psi \in L^\alpha (\Omega \times (0,T),{\mathscr {P}};L^\alpha ({\mathscr {D}}))\cap L^2(\Omega \times (0,T),{\mathscr {P}};H_0^1({\mathscr {D}})), \) we have

$$\begin{aligned} f^{m'}(\psi ,\nabla \psi )\rightarrow f(\psi , \nabla \psi ) \end{aligned}$$

in \(L^\frac{\alpha }{\alpha -1}(\Omega \times (0,T),{\mathscr {P}};L^\frac{\alpha }{\alpha -1}({\mathscr {D}}))\). Indeed, by definition of \(f^{m'}\), as \(m'\rightarrow \infty \)

$$\begin{aligned} f_s^{m'}(\psi _s(x),\nabla \psi _s(x))\rightarrow f_s(\psi _s(x),\nabla \psi _s(x))\,\,\, \forall \omega , s, x . \end{aligned}$$

Moreover \(|f_s^{m'}(r,z)|\le |f_s(r,z)|\) and due to Assumption A-3,

$$\begin{aligned} {\mathbb {E}}\int _0^T|f_s(\psi _s, \nabla \psi _s(x))|_{L^\frac{\alpha }{\alpha -1}}^\frac{\alpha }{\alpha -1}ds \le {C}{\mathbb {E}}\int _0^T\int _{\mathscr {D}}\Big (1+|\psi _s(x)|^\alpha \Big ) dxds <\infty . \end{aligned}$$

Therefore we may use Lebesgue Dominated Convergence Theorem to obtain

$$\begin{aligned} \begin{aligned} \lim _{m' \rightarrow \infty }&{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|f_s^{m'}(\psi _s(x),\nabla \psi _s(x))-f_s(\psi _s(x),\nabla \psi _s(x))|^\frac{\alpha }{\alpha -1}dxds \\&={\mathbb {E}}\int _0^T\int _{\mathscr {D}}\lim _{m' \rightarrow \infty }|f_s^{m'}(\psi _s(x),\nabla \psi _s(x))-f_s(\psi _s(x),\nabla \psi _s(x))|^\frac{\alpha }{\alpha -1}dxds=0. \end{aligned} \end{aligned}$$

Proof of Theorem 1

In order to show the weak limit u obtained in Lemma 4 is indeed the unique solution of SPDE (1), it remains to show that \(f'=f(u,\nabla u)\) which can be shown using the monotonicity argument as below.

Define for each \(w\in L^\alpha ({\mathscr {D}})\cap H_0^1({\mathscr {D}}), s\in (0,T)\) and \(k\in {\mathbb {N}}\), the operators

$$\begin{aligned} A_sw:=L_sw+f_s^0 \quad \text {and} \quad B_s^kw:=M_s^kw+g_s^k. \end{aligned}$$

Then for any \(w,w' \in L^\alpha ({\mathscr {D}})\cap H_0^1({\mathscr {D}})\), we have using Remark 2

$$\begin{aligned} 2\langle A_sw-A_sw',w-w' \rangle {+} \sum _{k\in {\mathbb {N}}} |B_s^kw{-}B_s^kw'|_{L^2}^2 \le {-}\kappa |w-w'|^2_{H_0^1}+K'|w-w'|^2_{L^2}.\nonumber \\ \end{aligned}$$
(17)

Consider \(\psi \in L^\alpha (\Omega \times (0,T),{\mathscr {P}};L^\alpha ({\mathscr {D}})) \cap L^2(\Omega \times (0,T),{\mathscr {P}};H_0^1({\mathscr {D}}))\). Then using Assumption A-3, Remark 1 and definition of \(f^m\), we have

$$\begin{aligned} \langle f^{m'}_s(u_s^{m'}, \nabla u_s^{m'})-f^{m'}_s(\psi _s,\nabla u_s^{m'}),u_s^{m'}-\psi _s\rangle \le 0 \end{aligned}$$
(18)

almost surely for all \(s\in [0,T]\). Moreover using Young’s inequality and Assumption A-3, we have almost surely for all \(s\in [0,T]\)

$$\begin{aligned} 2\langle f^{m'}_s(\psi _s,\nabla u_s^{m'}){-}f^{m'}_s(\psi _s,\nabla \psi _s),u_s^{m'}{-}\psi _s\rangle \le \kappa |\nabla (u_s^{m'}{-}\psi _s)|^2_{L^2}+{C}|u_s^{m'}-\psi _s|^2_{L^2}.\nonumber \\ \end{aligned}$$
(19)

Define \(K'':=K'+{C}\), where \(K'\) and C are as in (17) and (19) above. Then using the product rule and Itô’s formula, we obtain

$$\begin{aligned} \begin{aligned} {\mathbb {E}} \big (e^{-K''t}&|u_t|_{L^2}^2\big ) - {\mathbb {E}}(|u_0|_{L^2}^2) \\&= {\mathbb {E}}\Big [\int _0^te^{-K''s}\Big (2\langle A_su_s+f'_s,u_s\rangle +\sum _{k\in {\mathbb {N}}} |B^k_su_s|_{L^2}^2-K''|u_s|_{L^2}^2\Big )ds\Big ] \end{aligned}\nonumber \\ \end{aligned}$$
(20)

and

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\big (e^{-K''t}|u_t^{m'}|_{L^2}^2\big ) -{\mathbb {E}}(|u_0^{m'}|_{L^2}^2) = {\mathbb {E}}\Big [&\int _0^te^{-K''s}\Big (2\langle A_su_s^{m'}+f_s^{m'}(u_s^{m'},\nabla u_s^{m'} ),u_s^{m'}\rangle \\&+\sum _{k\in {\mathbb {N}}} |B_s^ku_s^{m'}|_{L^2}^2-K''|u_s^{m'}|_{L^2}^2\Big )ds\Big ] \end{aligned} \end{aligned}$$
(21)

for all \(t\in [0,T]\).

We now need to re-arrange the right-hand side of (21) so that we can use the monotonicity assumptions. We have

$$\begin{aligned}&{\mathbb {E}}\Big [\int _0^te^{-K''s}\Big (2\langle A_su_s^{m'}+f_s^{m'}(u_s^{m'}, \nabla u_s^{m'}),u_s^{m'}\rangle +\sum _{k\in {\mathbb {N}}} |B_s^ku_s^{m'}|_{L^2}^2-K''|u_s^{m'}|_{L^2}^2\Big )ds\Big ] \nonumber \\&={\mathbb {E}}\Big [\int _0^te^{-K''s}\Big (2\langle A_su_s^{m'}-A_s\psi _s,u_s^{m'}\rangle +2\langle A_s\psi _s,u_s^{m'}\rangle +2 \langle A_su_s^{m'}-A_s\psi _s,\psi _s\rangle \nonumber \\&\quad + 2\langle f^{m'}_s(u_s^{m'}, \nabla u_s^{m'}){-}f^{m'}_s(\psi _s,\nabla \psi _s),u_s^{m'}{-}\psi _s\rangle +2\langle f^{m'}_s(\psi _s,\nabla \psi _s),u_s^{m'} \rangle \nonumber \\&\quad +2 \langle f_s^{m'}(u_s^{m'},\nabla u_s^{m'}){-}f_s^{m'}(\psi _s,\nabla \psi _s),\psi _s\rangle {+}\sum _{k\in {\mathbb {N}}} \big |B_s^ku_s^{m'}{-}B_s^k\psi _s\big |_{L^2}^2{-}\sum _{k\in {\mathbb {N}}}|B_s^k\psi _s|_{L^2}^2 \nonumber \\&\quad +2\sum _{k\in {\mathbb {N}}}\big (B_s^ku_s^{m'},B_s^k\psi _s\big )-K''\left[ |u_s^{m'}-\psi _s|_{L^2}^2-|\psi _s|_{L^2}^2 +2(u_s^{m'},\psi _s)\right] \Big )ds\Big ] .\nonumber \\ \end{aligned}$$
(22)

Using (18) and (19), we have

$$\begin{aligned}&2\langle f^{m'}_s(u_s^{m'}, \nabla u_s^{m'})-f^{m'}_s(\psi _s,\nabla \psi _s),u_s^{m'}-\psi _s\rangle \\&\quad =2\langle f^{m'}_s(u_s^{m'}, \nabla u_s^{m'}){-}f^{m'}_s(\psi _s,\nabla u_s^{m'})\\&\quad +f^{m'}_s(\psi _s,\nabla u_s^{m'}){-}f^{m'}_s(\psi _s,\nabla \psi _s),u_s^{m'}-\psi _s\rangle \\&\quad \le \kappa |\nabla (u_s^{m'}-\psi _s)|^2_{L^2}+{C}|u_s^{m'}-\psi _s|^2_{L^2} \end{aligned}$$

and hence using (17) in (22) together with (21), we obtain for all \(t\in [0,T]\)

$$\begin{aligned} \begin{aligned} {\mathbb {E}}&\big (e^{-K''t}|u_t^{m'}|_{L^2}^2\big )-{\mathbb {E}}(|u_0^{m'}|_{L^2}^2) \\&\le {\mathbb {E}}\Big [\int _0^te^{-K''s}\Big ( 2\langle A_s\psi _s,u_s^{m'}\rangle +2 \langle A_su_s^{m'}-A_s\psi _s,\psi _s\rangle \\&\quad +2\langle f_s^{m'}(\psi _s,\nabla \psi _s),u_s^{m'}\rangle +2 \langle f_s^{m'}(u_s^{m'},\nabla u_s^{m'})-f_s^{m'}(\psi _s,\nabla \psi _s),\psi _s\rangle \\&\quad -\sum _{k\in {\mathbb {N}}}|B_s^k\psi _s|_{L^2}^2 +2\sum _{k\in {\mathbb {N}}}\big (B_s^ku_s^{m'},B_s^k\psi _s\big )+K''\big [|\psi _s|_{L^2}^2 -2(u_s^{m'},\psi _s)\big ] \Big )ds\Big ]. \end{aligned} \end{aligned}$$

Now, integrating over t from 0 to T, letting \(m' \rightarrow \infty \) and using the weak lower semicontinuity of the norm, we obtain

$$\begin{aligned} {\mathbb {E}}\Big [\int _0^T&\big (e^{-K''t}|u_t|_{L^2}^2-|u_0|_{L^2}^2\big )dt\Big ] \nonumber \\\le & {} \liminf _{{m'}\rightarrow \infty }{\mathbb {E}}\Big [\int _0^T\big (e^{-K''t}|u_t^{m'}|_{L^2}^2-|u_0^{m'}|_{L^2}^2\big )dt\Big ] \nonumber \\\le & {} {\mathbb {E}}\Big [\int _0^T\int _0^te^{-K''s}\Big ( 2\langle A_s\psi _s,u_s\rangle +2 \langle A_su_s-A_s\psi _s,\psi _s\rangle \nonumber \\&+\,2\langle f_s(\psi _s, \nabla \psi _s),u_s\rangle +2 \langle f'_s-f_s(\psi _s,\nabla \psi _s),\psi _s\rangle -\sum _{k\in {\mathbb {N}}}|B_s^k\psi _s|_{L^2}^2 \nonumber \\&+\,2\sum _{k\in {\mathbb {N}}}(B^k_su_s,B_s^k(\psi _s))+K''\left[ |\psi _s|_{L^2}^2 -2(u_s,\psi _s)\right] \Big )dsdt\Big ]\nonumber \\ \end{aligned}$$
(23)

where we have used Remark 4 in last inequality. Again, integrating from 0 to T in (20) and combining this with (23), we get

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\Big [\int _0^T&\int _0^t e^{-K''s}\Big ( 2\langle A_su_s-A_s\psi _s,u_s-\psi _s \rangle +2\langle f'_s-f_s(\psi _s,\nabla \psi _s),u_s-\psi _s \rangle \\&\quad +\sum _{k\in {\mathbb {N}}}|B_s^k\psi _s-B^k_su_s|_{L^2}^2 -K''|u_s-\psi _s|_{L^2}^2 \Big )dsdt\Big ]\le 0 \end{aligned} \end{aligned}$$

which on using (17) gives

$$\begin{aligned} {\mathbb {E}}\Big [\int _0^T\int _0^t e^{-K''s}\Big ( 2\langle f'_s-f_s(\psi _s, \nabla \psi _s),u_s-\psi _s \rangle \Big )dsdt\Big ]\le 0. \end{aligned}$$
(24)

Let \(\eta \in L^\infty ((0,T)\times \Omega ;{\mathbb {R}})\), \(\phi \in C_0^\infty ({\mathscr {D}})\), \(\epsilon \in (0,1)\) and let \(\psi =u-\epsilon \eta \phi \). Then from (24) one obtains that

$$\begin{aligned} {\mathbb {E}} \Big [ \int _0^T \int _0^t 2\epsilon e^{-K''s} \langle f'_s-f_s(u_s-\epsilon \eta _s \phi ,\nabla u_s-\epsilon \eta _s \nabla \phi ),\eta _s \phi \rangle dsdt\Big ]\le 0. \end{aligned}$$

Dividing by \(\epsilon \), letting \(\epsilon \rightarrow 0\), using Lebesgue dominated convergence theorem and Assumption A-3 leads to

$$\begin{aligned} {\mathbb {E}}\Big [\int _0^T \int _0^t 2 e^{-K''s} \eta _s \langle f'_s-f_s(u_s,\nabla u_s),\phi \rangle dsdt\Big ]\le 0. \end{aligned}$$

Since this holds for any \(\eta \in L^\infty ((0,T)\times \Omega , {\mathscr {P}} ;{\mathbb {R}})\) and \(\phi \in C_0^\infty ({\mathscr {D}})\), one gets that \(f(u,\nabla u)= ~f'\) which concludes the proof.

Further, taking \(m \rightarrow \infty \) in (14) and using the weak lower semicontinuity of the norm, we obtain the following estimates for the solution of (1)

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t \le T}|u_t|_{L^p}^p +&{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|\nabla u_s|^2|u_s|^{p-2}dxds \\&\le \liminf _{m\rightarrow \infty } \Big [{\mathbb {E}}\sup _{0\le t \le T}|u_t^m|_{L^p}^p+{\mathbb {E}}\int _0^T\int _{\mathscr {D}}|\nabla u_s^m|^2|u_s^m|^{p-2}dxds\Big ]\\&\le {C}{\mathbb {E}}\Big (|\phi |_{L^p}^p+\Vert f^0 \Vert _{L^p}^p+\Vert |g|_{\ell ^2}\Vert ^p_{L^p}\Big ). \end{aligned} \end{aligned}$$

\(\square \)

Interior regularity

In this section, we present the results on interior regularity of the solution to SPDE (1). The main result is stated in Theorem 2. The idea is to prove the result for the linear SPDE first and then use it along with the \(L^p\)-estimates obtained in Sect. 2 to prove Theorem 2. We do not claim the result for the linear case to be new, however we could not find such result in literature in sufficient generality.

To raise the regularity of the solution one needs the given data to be sufficiently smooth. Thus, we assume the following condition on the coefficients before stating the main result of this section.

A-5 For any \(i,j=1,\ldots ,d\), the coefficients \(a^{ij}, b^i\) and c and their spatial derivatives up to order n are real-valued, \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and are bounded by K. The coefficients \(\sigma ^i=(\sigma ^{ik})_{k=1}^\infty \), \(\mu =(\mu ^k)_{k=1}^\infty \) and their spatial derivatives up to order n are \(\ell ^2\)-valued, \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and almost surely

$$\begin{aligned} \sum _{i=1}^d\sum _{k\in {\mathbb {N}}} \sum _{|\gamma |\le n}|D^\gamma \sigma _t^{ik}(x)|^2+\sum _{k\in {\mathbb {N}}}\sum _{|\gamma |\le n}|D^\gamma \mu _t^k(x)|^2 \le K \end{aligned}$$

for all t and x.

For A, B subsets of \({\mathbb {R}}^d\) let \({{\,\mathrm{dist}\,}}(A, B)\) denote the distance between A and B. Further, for \(\ell = 1,2\) define

$$\begin{aligned} \begin{aligned} {\mathcal {I}}^\ell :={\mathbb {E}}\Big [ \sum _{|\gamma |\le \ell }|D^\gamma \phi |^2_{L^2}+&\sum _{|\gamma |\le \ell -1}\Vert D^\gamma f^0 \Vert _{L^2}^2 +\sum _{|\gamma |\le \ell }\Vert |D^\gamma g|_{\ell ^2}\Vert ^2_{L^2} \\&+|\phi |_{L^{2\alpha -2}}^{2\alpha -2}+\Vert f^0 \Vert _{L^{2\alpha -2}}^{2\alpha -2}+\Vert |g|_{\ell ^2}\Vert ^{2\alpha -2}_{L^{2\alpha -2}} \Big ] . \end{aligned} \end{aligned}$$

Theorem 2

Let Assumptions A-2 to A-4 hold and u be the solution to (1). Fix some open \({\mathscr {D''}}\Subset {\mathscr {D'}}\Subset {\mathscr {D}}\) such that \({{\,\mathrm{dist}\,}}({\mathscr {D'}}, \partial {\mathscr {D}})<1\) and \({{\,\mathrm{dist}\,}}({\mathscr {D''}}, \partial {\mathscr {D}}')<1\).

  1. (i)

    If Assumption A-5 holds with \(n=1\), and if \(\phi \in L^2(\Omega ,{\mathscr {F}}_0;H^1({\mathscr {D}}))\) and \(g \in ~ L^2( \Omega \times (0,T),{\mathscr {P}};H^1({\mathscr {D}};\ell ^2))\), then

    $$\begin{aligned} u \in C([0,T], H^1({\mathscr {D}}'))\,\,\,\text {a.s. and} \,\,\, u\in L^2(\Omega \times (0,T),{\mathscr {P}};H^{2}({\mathscr {D}}')). \end{aligned}$$

    Moreover, there is \(C=C(d,T,K,\kappa )\) such that

    $$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}|\partial _i u_t|_{L^2({\mathscr {D'}})}^2 +&{\mathbb {E}}\int _0^T|\partial _i u_t|^2_{H^1({\mathscr {D'}})} dt \le C{{\,\mathrm{dist}\,}}({\mathscr {D'}},\partial {\mathscr {D}})^{-2} {\mathcal {I}}^1 \end{aligned} \end{aligned}$$
    (25)

    for all \(i=1,\ldots ,d\).

  2. (ii)

    Further, in case the semilinear term f does not depend on z, if Assumption A-1 holds with \(n=2\), if \(\phi \in L^2(\Omega ,{\mathscr {F}}_0;H^2({\mathscr {D}}))\), \(f^0 \in L^2( \Omega \times (0,T),{\mathscr {P}};H^1({\mathscr {D}}))\) and \(g \in L^2( \Omega \times (0,T),{\mathscr {P}};H^2({\mathscr {D}};\ell ^2))\) and if almost surely

    $$\begin{aligned} |\partial _rf_t(x,r)|\le K(1+|r|)^{\alpha -2}\,\,\,\text {and}\,\,\, |\partial _i f_t(x,r)| \le K(1+|r|)^{\alpha - 1} \end{aligned}$$
    (26)

    for all \(i=1,\dots , d\), \(t\in [0,T], x\in {\mathscr {D}}\) and all \(r\in {\mathbb {R}}\), then we have

    $$\begin{aligned} u\in C([0,T], H^2({\mathscr {D}}'')) \,\,\,\text {a.s. and} \,\,\, u\in L^2(\Omega \times (0,T),{\mathscr {P}};H^{3}({\mathscr {D}}'')). \end{aligned}$$

    Furthermore, there is \(C=C(d,T,K,\kappa )\) such that

    $$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}|\partial _i \partial _j u_t|_{L^2({\mathscr {D''}})}^2 +{\mathbb {E}}\int _0^T|&\partial _i \partial _j u_t|^2_{H^1({\mathscr {D''}})} dt \le C {{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')^{-2} {\mathcal {I}}^2 \\&+ C {{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')^{-2} {{\,\mathrm{dist}\,}}({\mathscr {D}}',\partial {\mathscr {D}})^{-2}{\mathcal {I}}^1 \end{aligned} \end{aligned}$$
    (27)

    for all \(i,j=1,\ldots ,d\).

One can obtain regularity results up to the boundary in appropriate weighted Sobolev spaces using results from Krylov [18] along with the \(L^p\)-estimates obtained in Theorem 1. However, obtaining the similar results for the linear equations using \(L^p\)-theory is more useful. We will discuss this in Sect. 4.

As mentioned before, we will first get the results for linear equations. So, we consider the following linear stochastic evolution equation:

$$\begin{aligned} \begin{aligned} dv_t&=(L_tv_t+f_t)dt+\sum _{k\in {\mathbb {N}}}(M_t^kv_t+g_t^k)dW_t^k \,\,\,\ \text {on}\,\,\, [0,T]\times {\mathscr {D}}, \end{aligned} \end{aligned}$$
(28)

where the operators L and \(M^k\) are defined in (2). As can be seen in what follows, one can raise the regularity to any order for the linear equation by assuming the given data to be sufficiently smooth. Thus we make the following assumption on initial data and the free terms and then state the result in Theorem 3.

Let \(n\ge 0\) be an integer.

A-6 Assume that \(v_0 \in L^2(\Omega ,{\mathscr {F}}_0;H^n({\mathscr {D}}))\), \(g \in ~L^2( \Omega \times (0,T),{\mathscr {P}};H^n({\mathscr {D}};\ell ^2))\) and \(f\in L^2( \Omega \times (0,T),{\mathscr {P}};H^{n-1}({\mathscr {D}}))\).

Theorem 3

Assume that v is a continuous \(L^2({\mathscr {D}})\)-valued adapted process such that \(v \in ~ L^2(\Omega \times (0,T), {\mathscr {P}}; H^1({\mathscr {D}}))\), and it satisfies (28). If Assumptions A- 2, A- 5 and A- 6 hold, then for all open \({\mathscr {D'}}\Subset {\mathscr {D}}\),

$$\begin{aligned} v \in C([0,T], H^n({\mathscr {D}}')) \,\,\,\text {a.s.} \,\,\, \text {and}\,\,\, v \in L^2(\Omega \times (0,T),{\mathscr {P}};H^{n+1}({\mathscr {D}}')) \end{aligned}$$

We will prove Theorem 3 via Lemmas 5 and 6. In Lemma 5, we first prove the special case \(n=1\).

Lemma 5

Assume that \(v \in C([0,T]; L^2({\mathscr {D}}))\) a.s., v is adapted and satisfies (28) and moreover \(v \in L^2(\Omega \times (0,T), {\mathscr {P}}; H^1({\mathscr {D}}))\). If Assumptions A-2, A-5 and A-6 hold with \(n=1\), then there is \(C=C(d,T,K,\kappa )\) such that

$$\begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}|\partial _i v_t|_{L^2({\mathscr {D'}})}^2+ & {} {\mathbb {E}}\int _0^T|\partial _i v_t|^2_{H^1({\mathscr {D'}})} dt \le {C{{\,\mathrm{dist}\,}}({\mathscr {D'}},\partial {\mathscr {D}})^{-2}} \Bigg [{\mathbb {E}} \int _{\mathscr {D}}|\nabla v_0|^2 dx \nonumber \\&+\, {\mathbb {E}}\int _0^T\int _{\mathscr {D}} \Big [|\nabla v_t|^2 + |f_t|^2 + |v_t|^2 + \sum _{k\in {\mathbb {N}}}|\nabla g_t^k|^2 \Big ]dxdt\Bigg ]\nonumber \\ \end{aligned}$$
(29)

for all \(i=1,\ldots ,d\) and all open \({\mathscr {D'}}\Subset {\mathscr {D}}\) such that \({{\,\mathrm{dist}\,}}({\mathscr {D'}},\partial {\mathscr {D}})<1\).

Proof

Let \(\zeta = {{\,\mathrm{dist}\,}}({\mathscr {D'}},\partial {\mathscr {D}})\). We consider a cut-off function \(\eta \in C_0^\infty ({\mathscr {D}})\) which is 1 on \({\mathscr {D}}'\) and such that \(\eta \le 1\) and \(|\partial _i\eta |\le C \zeta ^{-1}\) for \(i=1,2,\ldots , d\). Define the lth-difference quotient, \(l\in \{1,2,\ldots ,d \}\), by

$$\begin{aligned} \delta _l^hu(x):=\frac{1}{h}\big (T_l^hu-u\big )(x),\qquad x\in {\mathbb {R}}^d \end{aligned}$$

where \(T_l^hu(x)=u(x+he_l)\) is the shift operator and the step-size h satisfies \(2|h|< {{\,\mathrm{dist}\,}}({{\,\mathrm{supp}\,}}\eta ,\partial {{\mathscr {D}}})\). From (28), we get

$$\begin{aligned} d(\eta \delta _l^h v_t)=\eta \delta _l^h(L_tv_t+f_t)dt+\eta \sum _{k\in {\mathbb {N}}}\delta _l^h(M_t^kv_t+g_t^k)dW_t^k. \end{aligned}$$

Applying Itô’s formula for the square of \(L^2\)-norm, we get

$$\begin{aligned} \begin{aligned} d|\eta \delta _l^h v_t|_{L^2({\mathscr {D}})}^2 =2 \langle \eta \delta _l^h(L_tv_t+f_t),\eta \delta _l^h v_t \rangle dt&+ 2 \sum _{k\in {\mathbb {N}}}(\eta \delta _l^h(M_t^kv_t+g_t^k), \eta \delta _l^h v_t)dW_t^k \\&+ \sum _{k\in {\mathbb {N}}}|\eta \delta _l^h(M_t^kv_t+g_t^k)|_{L^2({\mathscr {D}})}^2 dt. \end{aligned} \end{aligned}$$

It follows from the definition of \(\delta _l^h\) and linearity of \(\partial _j\), that the two operators commute. Thus, using integration by parts and the formula

$$\begin{aligned} \delta _l^h(vw)(x)=\delta _l^hv(x)T_l^hw(x)+v(x)\delta _l^hw(x) \end{aligned}$$

we get,

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}}&\eta ^2 |\delta _l^h v_t|^2 dx = \int _{\mathscr {D}}\eta ^2 |\delta _l^h v_0|^2 dx +2 \int _0^t\int _{\mathscr {D}}\eta ^2 \delta _l^h(L_sv_s+f_s)\delta _l^h v_s dxds \\&+ {\mathscr {M}}_t^h + \sum _{k\in {\mathbb {N}}}\int _0^t\int _{\mathscr {D}}\eta ^2 |\delta _l^h(M_s^kv_s+g_s^k)|^2 dxds \\ =&I_0-2 \int _0^t\int _{\mathscr {D}}\eta ^2\sum _{i,j=1}^d a_s^{ij} \, \partial _i(\delta _l^h v_s)\, \partial _j(\delta _l^h v_s)+I_1+I_2+I_3+ {\mathscr {M}}_t^h +I_4 \end{aligned} \end{aligned}$$
(30)

where,

$$\begin{aligned} \begin{aligned} I_0 :=&\int _{\mathscr {D}}\eta ^2 |\delta _l^h v_0|^2 dx , \\ I_1 :=&-2 \int _0^t\int _{\mathscr {D}}\eta ^2 \sum _{i,j=1}^d\delta _l^h a_s^{ij}\,\partial _i (T_l^hv_s)\partial _j(\delta _l^h v_s) dxds , \\ I_2 :=&-4\int _0^t\int _{\mathscr {D}}\eta \sum _{i,j=1}^d\big [\delta _l^h a_s^{ij}\, \partial _i (T_l^hv_s)+a_s^{ij} \, \partial _i(\delta _l^h v_s)\big ] \partial _j\eta \delta _l^h v_s dxds \\ I_3 :=&2 \int _0^t\int _{\mathscr {D}}\eta ^2 \Big [\sum _{i=1}^d\{\delta _l^h b_s^i \, \partial _i (T_l^hv_s)+ b_s^i \, \delta _l^h(\partial _i v_s)\} \\&+\delta _l^h c_s \, T_l^hv_s+c_s \, \delta _l^h v_s+\delta _l^h f_s \Big ]\delta _l^h v_sdxds ,\\ I_4 :=&\sum _{k\in {\mathbb {N}}}\int _0^t\int _{\mathscr {D}} \eta ^2 \Big |\sum _{i=1}^d\delta _l^h \sigma _s^{ik} \, \partial _i(T_l^hv_s)+ \delta _l^h\mu _s^k \, T_l^hv_s \\&+\sum _{i=1}^d \sigma _s^{ik} \, \partial _i(\delta _l^h v_s)+\mu _s^k \, \delta _l^hv_s+\delta _l^hg_s^k \Big |^2 dxds \end{aligned} \end{aligned}$$

and

$$\begin{aligned} {\mathscr {M}}_t^h:= 2 \sum _{k\in {\mathbb {N}}}\int _0^t\int _{\mathscr {D}}\eta ^2 \delta _l^h(M_s^kv_s+g_s^k) \delta _l^h v_sdxdW_s^k. \end{aligned}$$

Now, we see that

$$\begin{aligned} \begin{aligned} I_4&= \sum _{k\in {\mathbb {N}}} \int _0^t \int _{\mathscr {D}} \eta ^2 \left[ \Big |\sum _{i=1}^d \delta _l^h \sigma _s^{ik} \, \partial _i(T_l^hv_s)+ \delta _l^h \mu _s^k \, T_l^hv_s \Big |^2 \right. \\&\quad +2\Big [\sum _{i=1}^d \delta _l^h \sigma _s^{ik} \, \partial _i(T_l^hv_s) + \delta _l^h\mu _s^k \, T_l^hv_s \Big ]\Big [\sum _{i=1}^d \sigma _s^{ik} \, \partial _i(\delta _l^hv_s)+\mu _s^k \delta _l^hv_s+\delta _l^hg_s^k \Big ] \\&\quad \left. +\Big |\sum _{i=1}^d \sigma _s^{ik} \, \partial _i(\delta _l^h v_s) {+}\mu _s^k \, \delta _l^h v_s {+} \delta _l^h g_s^k \Big |^2 \right] dxds {\le } \sum _{i,j{=}1}^d \sigma _s^{ik} \, \partial _i(\delta _l^h v_s) \, \sigma _s^{jk} \, \partial _j(\delta _l^h v_s){+} \bar{I_4} \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \bar{I_4}:= & {} \sum _{k\in {\mathbb {N}}}\int _0^t\int _{\mathscr {D}}\eta ^2\left[ (d+1)\sum _{i=1}^d|\delta _l^h \sigma _s^{ik}|^2|\partial _i(T_l^hv_s)|^2+ (d+1)|\delta _l^h\mu _s^k \, T_l^hv_s|^2\right. \\&+\, 2 \sum _{i,j=1}^d \delta _l^h \sigma _s^{ik} \, \partial _i(T_l^hv_s) \, \sigma _s^{jk} \, \partial _j(\delta _l^h v_s) +2\sum _{i,j=1}^d \delta _l^h \sigma _s^{ik} \, \partial _i(T_l^h v_s) \, \mu _s^k \, \delta _l^h v_s \\&+ 2\sum _{i,j=1}^d \delta _l^h \sigma _s^{ik} \, \partial _i(T_l^hv_s) \, \delta _l^h g_s^k + 2 \sum _{i=1}^d \sigma _s^{ik} \, \partial _i(\delta _l^h v_s) \,\delta _l^h \mu _s^k \, T_l^h v_s\\&+ 2 \delta _l^h \mu _s^k \, T_l^hv_s \, \mu _s^k \, \delta _l^h v_s + 2\delta _l^h \mu _s^k \, T_l^hv_s \, \delta _l^h g_s^k \\&+|\mu _s^k \, \delta _l^h v_s|^2 +|\delta _l^h g_s^k|^2+2\sum _{i=1}^d \sigma _s^{ik} \, \partial _i(\delta _l^h v_s) \, \mu _s^k \, \delta _l^hv_s\\&\left. + 2\sum _{i=1}^d \sigma _s^{ik} \, \partial _i(\delta _l^h v_s) \, \delta _l^h g_s^k +2 \mu _s^k \, \delta _l^h v_s \, \delta _l^h g_s^k \right] dxds \end{aligned}$$

Substituting this in (30), we get

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}}&\eta ^2 |\delta _l^h v_t|^2 dx \\ \le&I_0 +I_1 -2 \int _0^t\int _{\mathscr {D}}\eta ^2 \sum _{i,j=1}^d \Big [a_s^{ij} -\frac{1}{2}\sum _{k\in {\mathbb {N}}}\sigma _s^{ik}\sigma _s^{jk} \Big ] \partial _i(\delta _l^h v_s) \, \partial _j(\delta _l^h v_s) dxds\\&+I_2 +I_3 + {\mathscr {M}}_t^h + \bar{I_4}.\\ \end{aligned} \end{aligned}$$

which on using Assumptions A-2, A-5 (with \(n=1\)) and Young’s inequality for an \(\epsilon >0\) gives

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}} \eta ^2&|\delta _l^h v_t|^2 dx \le \int _{\mathscr {D}}\eta ^2 |\delta _l^h v_0|^2 dx -2 \kappa \int _0^t\int _{\mathscr {D}}\eta ^2 |\nabla (\delta _l^hv_s)|^2 dxds + {\mathscr {M}}_t^h\\ +&\int _0^t\int _{\mathscr {D}}\sum _{i,j=1}^d {\big [ \epsilon K|\eta \partial _i(T_l^hv_s)|^2 + \epsilon K | \eta \partial _i(\delta _l^h v_s)|^2 + \frac{C}{\epsilon } |\partial _j\eta \delta _l^h v_s|^2\big ]} \, \, dxds \\ +&\int _0^t\int _{\mathscr {D}}\eta ^2\left[ 2\delta _l^h f_s \, \delta _l^h v_s + {\frac{C_{K,d}}{\epsilon }\sum _{i=1}^d|\partial _i(T_l^hv_s)|^2 +\frac{ C_{K,d}}{\epsilon }|T_l^hv_s|^2} \right. \\&\left. + C\sum _{k\in {\mathbb {N}}}|\delta _l^hg_s^k|^2+ { \epsilon C_K \sum _{i=1}^d| \partial _i(\delta _l^h v_s)|^2 + \frac{C_{K}}{\epsilon }|\delta _l^hv_s|^2} \right] dxds. \end{aligned} \end{aligned}$$
(31)

Now extending \(\eta , f,g\) and v to \({\mathbb {R}}^d\) by setting them to 0 on \({\mathbb {R}}^d\setminus {\mathscr {D}}\) and using the fact that \({{\,\mathrm{supp}\,}}\eta \subset {\mathscr {D}}\) and \({{\,\mathrm{supp}\,}}(T_l^{-h}\eta ) \subset {\mathscr {D}}\) for our choice of h, we get

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}}\eta ^2 \, \delta _l^h f_s \, \delta _l^h v_s dx&= \int _{{\mathbb {R}}^d}\eta ^2 \, \delta _l^h f_s \, \delta _l^h v_s dx\\&= \int _{{\mathbb {R}}^d}\eta ^2 \, \frac{1}{h} T_l^hf_s \, \delta _l^h v_s dx-\int _{{\mathbb {R}}^d}\eta ^2 \, \frac{1}{h} f_s \, \delta _l^h v_s dx \\&= \int _{{\mathbb {R}}^d} T_l^{-h}(\eta ^2) \frac{1}{h}f_s \, T_l^{-h}(\delta _l^h v_s) dx-\int _{{\mathbb {R}}^d}\eta ^2 \, \frac{1}{h} f_s \, \delta _l^h v_s dx \\&= \int _{{\mathbb {R}}^d}f_s\frac{1}{h}\big [T_l^{-h}(\eta ^2 \delta _l^h v_s)-(\eta ^2 \delta _l^h v_s)\big ] dx \\&= - \int _{{\mathbb {R}}^d}f_s \, \delta _l^{-h}(\eta ^2 \, \delta _l^h v_s) dx = - \int _{{\mathscr {D}}}f_s \,\delta _l^{-h}(\eta ^2 \, \delta _l^h v_s) dx \\&\le \epsilon \int _{{\mathscr {D}}}|\delta _l^{-h}(\eta ^2 \, \delta _l^h v_s)|^2 dx +{\frac{1}{\epsilon }}\int _{{\mathscr {D}}}|f_s|^2dx \end{aligned} \end{aligned}$$
(32)

where last inequality has been obtained using Young’s inequality.

Since \(\eta ^2 \, \delta _l^h v_s \in H^1({\mathscr {D}})\), using the relation between difference quotients and weak derivatives (see e.g. [4, Ch. 5, Sec. 8, Theorem 3]), we have

$$\begin{aligned} \int _{{\mathscr {D}}}|\delta _l^{-h}(\eta ^2 \, \delta _l^h v_s)|^2 dx = \int _{{\mathscr {D}}_l^h(\eta )}|\delta _l^{-h} (\eta ^2 \, \delta _l^h v_s)|^2 dx \le C\int _{{\mathscr {D}}}| \nabla (\eta ^2 \, \delta _l^h v_s)|^2 dx \end{aligned}$$

for some constant C and \({\mathscr {D}}_l^h(\eta ):={{\,\mathrm{supp}\,}}\eta \cup {{\,\mathrm{supp}\,}}(T_l^h\eta ) \cup {{\,\mathrm{supp}\,}}(T_l^{-h}\eta ) \Subset {\mathscr {D}}\). Substituting this in (32), we get

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}}\eta ^2 \, \delta _l^h f_s \,&\delta _l^h v_s dx \le \epsilon C \int _{{\mathscr {D}}}|\nabla (\eta ^2 \, \delta _l^h v_s)|^2 dx +{\frac{1}{\epsilon }} \int _{{\mathscr {D}}}|f_s|^2dx \\&= \epsilon C \int _{{\mathscr {D}}}|\eta ^2 \, \nabla (\delta _l^h v_s) +2 \eta \, \nabla \eta \, \delta _l^h v_s|^2 dx + {\frac{1}{\epsilon }} \int _{{\mathscr {D}}}|f_s|^2 dx \\&\le {\epsilon C \int _{{\mathscr {D}}} |\eta \, \nabla (\delta _l^h v_s)|^2 dx +\epsilon C \mu ^{-2} \int _{{\mathscr {D}}} |(\eta \, \delta _l^h v_s)|^2 dx +\frac{1}{\epsilon } \int _{{\mathscr {D}}}|f_s|^2dx.} \end{aligned} \end{aligned}$$
(33)

Similarly,

$$\begin{aligned} \begin{aligned} \int _{\mathscr {D}}&\eta ^2 |T_l^h v_s|^2 dx = \int _{{\mathscr {D}}_l^h(\eta )} \eta ^2 |T_l^h v_s|^2 dx = \int _{{\mathscr {D}}_l^h(\eta )} |T_l^{-h}\eta |^2 |v_s|^2 dx \le {C} \int _{{\mathscr {D}}} |v_s|^2 dx \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \sum _{i=1}^d \int _{\mathscr {D}}\eta ^2 |\partial _i (T_l^h v_s)|^2 dx&= \sum _{i=1}^d \int _{{\mathscr {D}}_l^h(\eta )} \eta ^2 |T_l^h(\partial _i v_s)|^2 dx \\&\le C \sum _{i=1}^d \int _{{\mathscr {D}}} |\partial _i v_s|^2 dx = {C} \int _{{\mathscr {D}}} |\nabla v_s|^2 dx. \end{aligned} \end{aligned}$$

Using the assumption \(g \in L^2(\Omega \times (0,T), {\mathscr {P}};H^{1}({\mathscr {D}};\ell ^2))\) and the property of difference quotients mentioned above,

$$\begin{aligned} \begin{aligned} \sum _{k\in {\mathbb {N}}} \int _{\mathscr {D}}\eta ^2 |\delta _l^h \, g_s^k|^2 dx = \sum _{k\in {\mathbb {N}}} \int _{{\mathscr {D}}_l^h(\eta )}\eta ^2 |\delta _l^h g_s^k|^2 dx \le {C}\sum _{k\in {\mathbb {N}}} \int _{{\mathscr {D}}} |\nabla g_s^k|^2 dx. \end{aligned} \end{aligned}$$

Similarly, \(v \in L^2(\Omega \times (0,T), {\mathscr {P}}; H^1({\mathscr {D}}))\) and the property of difference quotients imply

$$\begin{aligned} \begin{aligned}&{\int _{\mathscr {D}} |\delta _l^h v_s|^2 dx \le C \int _{{\mathscr {D}}} |\nabla v_s|^2 dx}. \end{aligned} \end{aligned}$$
(34)

Substituting (33)–(34) in (31), we get

$$\begin{aligned} {\begin{aligned} \int _{\mathscr {D}} \eta ^2 |\delta _l^h v_t|^2&dx \le {C} \int _{\mathscr {D}}|\nabla v_0|^2 dx -2 \kappa \int _0^t\int _{\mathscr {D}}\eta ^2 |\nabla (\delta _l^hv_s)|^2 dxds\\&+ {\mathscr {M}}_t^h + \int _0^t\int _{\mathscr {D}} {\Big [ \frac{ C_{K,d}}{\epsilon }\mu ^{-2}|\nabla v_s|^2 + \epsilon C_{K} |\eta \nabla (\delta _l^hv_s)|^2 +\frac{1}{\epsilon } |f_s|^2 \Big .} \\&{ \Big . + \frac{C_{K,d}}{\epsilon }|v_s|^2 +C \sum _{k\in {\mathbb {N}}}|\nabla g_s^k|^2 \Big ]} dxds.\qquad \qquad \end{aligned}}\end{aligned}$$
(35)

Further, it can be seen that the process \({\mathscr {M}}_t^h\) defined in (30) is a local martingale where a localizing sequence of stopping times converging to T as \(n \rightarrow \infty \) is given by

$$\begin{aligned} \tau _n:=\inf \{t\in [0,T]: |\eta \delta _l^hv_s|_{L^2({\mathscr {D}})}| > n\} \wedge T. \end{aligned}$$
(36)

Thus, replacing t by \(t\wedge \tau _n\) in (35), then taking expectation and choosing \(\epsilon >0\) small enough such that \(2\kappa -\epsilon C_{K}=C_\kappa >0\) and finally using Fatou’s lemma, we get

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\int _{\mathscr {D}} \eta ^2 |\delta _l^h v_t|^2 dx+C_\kappa {\mathbb {E}}\int _0^t\int _{\mathscr {D}}\eta ^2 |\nabla (\delta _l^hv_s)|^2 dxds \le { C} {\mathbb {E}}\int _{\mathscr {D}}|\nabla v_0|^2 dx \\&\quad + {\mathbb {E}}\int _0^t\int _{\mathscr {D}} {\Big [\frac{ C_{K,d}}{\epsilon }\mu ^{-2}|\nabla v_s|^2 + \frac{1}{\epsilon } |f_s|^2 + \frac{C_{K,d}}{\epsilon }|v_s|^2 +C \sum _{k\in {\mathbb {N}}}|\nabla g_s^k|^2 \Big ]}dxds. \end{aligned} \end{aligned}$$
(37)

Using the inequalities of Burkholder–Davis–Gundy, Hölder and Young together with the estimates above we get that

$$\begin{aligned}&{\mathbb {E}}\sup _{0\le t \le T}|{\mathscr {M}}^h_{t\wedge \tau _n}|={\mathbb {E}}\sup _{0\le t \le T}\Big |2 \sum _{k\in {\mathbb {N}}}\int _0^{t\wedge {\tau _n}}\int _{\mathscr {D}}\eta ^2 \delta _l^h(M_s^kv_s+g_s^k) \delta _l^h v_sdxdW_s^k\Big | \nonumber \\&\quad \le 4{\mathbb {E}} \Big (\sum _{k\in {\mathbb {N}}}\int _0^{\tau _n}\Big | 2\int _{\mathscr {D}}\eta ^2 \delta _l^h(M_s^kv_s+g_s^k) \delta _l^h v_sdx\Big |^2ds\Big )^\frac{1}{2} \nonumber \\&\quad \le 8 {\mathbb {E}}\Big (\sum _{k\in {\mathbb {N}}}\int _0^{\tau _n} |\eta \,\delta _l^h(M_s^kv_s+g_s^k)|^2_{L^2({\mathscr {D}})} |\eta \, \delta _l^h v_s|^2_{L^2({\mathscr {D}})}ds\Big )^\frac{1}{2} \nonumber \\&\quad \le \frac{1}{2} {\mathbb {E}} \sup _{0 \le t \le T}|\eta \, \delta _l^h v_t| ^2_{L^2({\mathscr {D}})}+{C}\sum _{k\in {\mathbb {N}}}{\mathbb {E}}\int _0^{\tau _n} |\eta \,\delta _l^h(M_s^kv_s+g_s^k)|^2_{L^2({\mathscr {D}})} ds \nonumber \\&\quad \le \frac{1}{2} {\mathbb {E}} \sup _{0 \le t \le T}|\eta \, \delta _l^h v_t| ^2_{L^2({\mathscr {D}})}+{C \zeta ^{-2}}{\mathbb {E}}\int _0^{\tau _n}\int _{{\mathscr {D}}} \big [|\nabla v_s|^2 {+}|f_s|^2 {+}|v_s|^2 {+} |\nabla g_s|_{\ell ^2}^2 \big ]dxds.\nonumber \\ \end{aligned}$$
(38)

Replacing t by \(t\wedge \tau _n\) in (35), taking the supremum over \(t\in [0,T]\) and using (38) we obtain

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\sup _{0\le t \le T}\int _{\mathscr {D}} \eta ^2 |\delta _l^h v_{t\wedge \tau _n}|^2 dx \\&\quad \le {C \zeta ^{-2}} \left[ {\mathbb {E}} \int _{\mathscr {D}}|\nabla v_0|^2 dx + {\mathbb {E}}\int _0^T\int _{\mathscr {D}} \Big [|\nabla v_s|^2 + |f_s|^2 + |v_s|^2 + |\nabla g_s|_{\ell ^2}^2 \Big ]\,dxds\right] , \end{aligned} \end{aligned}$$

which, on applying Fatou’s lemma, yields

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\sup _{0\le t \le T}\int _{\mathscr {D}} \eta ^2 |\delta _l^h v_t|^2 dx \\&\quad \le {C \zeta ^{-2}} \left[ {\mathbb {E}} \int _{\mathscr {D}}|\nabla v_0|^2 dx + {\mathbb {E}}\int _0^T\int _{\mathscr {D}} \Big [|\nabla v_s|^2 + |f_s|^2 + |v_s|^2 + |\nabla g_s|_{\ell ^2}^2 \Big ]\,dxds\right] , \end{aligned} \end{aligned}$$

where \({C}={C}(K,d,\epsilon )\). Now note that the right hand side of above equation and (37) are independent of h and are finite and hence using e.g. [4, Ch. 5, Sec. 8, Theorem 3]), we get (29). \(\square \)

We now extend the result to the case \(n=2\) as follows. From Lemma 5 we have that v is a continuous \(H^1({\mathscr {D'}})\)-valued adapted process such that \(v \in L^2(\Omega \times (0,T),{\mathscr {P}}; H^2({\mathscr {D'}}))\), and it satisfies (28). If Assumptions A-5 and A-6 hold for \(n=2\), then from (28), we get

$$\begin{aligned} \begin{aligned} d(\partial _l v_t)&=\partial _l(L_tv_t+f_t)dt+\sum _{k\in {\mathbb {N}}}\partial _l(M_t^kv_t+g_t^k)dW_t^k\\&= \big (L_t(\partial _l v_t)+{\bar{f}}_t\big )dt + \sum _{k\in {\mathbb {N}}}\big (M_t^k(\partial _l v_t)+{\bar{g}}_t^k \big )dW_t^k \end{aligned} \end{aligned}$$
(39)

on \([0,T] \times {\mathscr {D}}'\), where

$$\begin{aligned} {\bar{f}}_t:=\sum _{j=1}^d \partial _j\Big (\sum _{i=1}^d \partial _la_t^{ij} \, \partial _iv_t\Big ) +\sum _{i=1}^d \partial _lb_t^i \, \partial _iv_t+ \partial _lc_t \, v_t+ \partial _l f_t \end{aligned}$$

and

$$\begin{aligned} {\bar{g}}_t^k:=\sum _{i=1}^d \partial _l\sigma _t^{ik} \, \partial _i v_t+ \partial _l\mu _t^k \, v_t+ \partial _lg_t^k. \end{aligned}$$

Using Assumptions A-5, A-6 with \(n=2\) we get that \({\bar{f}}\in L^2(\Omega \times (0,T),{\mathscr {P}};L^2({\mathscr {D}}'))\) and \({\bar{g}} \in L^2(\Omega \times (0,T),{\mathscr {P}};H^1({\mathscr {D}}';\ell ^2))\).

Thus replacing \(f, g^k,{\mathscr {D}}\) in (28) by \({\bar{f}},{\bar{g}}^k\) and \({\mathscr {D'}}\) respectively, we see that \(z=\partial _l v\) satisfies (28). Clearly \(z \in C([0,T];L^2({\mathscr {D'}}))\) almost surely and \(z \in L^2(\Omega \times (0,T); H^1({\mathscr {D'}}))\) and hence all the assumptions of Lemma 5 are satisfied for the new linear equation (39). Therefore for all open \({\mathscr {D''}}\Subset {\mathscr {D'}}\) such that \({{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')<1\), we have

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}|\partial _i z_t|_{L^2({\mathscr {D''}})}^2 +{\mathbb {E}}\int _0^T&|\partial _i z_t|^2_{H^1({\mathscr {D''}})} dt \le {C {{\,\mathrm{dist}\,}}({\mathscr {D''}},\partial {\mathscr {D}}')^{-2}}\Bigg [{\mathbb {E}} \int _{\mathscr {D'}}|\nabla z_0|^2 dx \\&+ {\mathbb {E}}\int _0^T\int _{\mathscr {D'}} \Big [|\nabla z_t|^2 + |{\bar{f}}_t|^2 + |z_t|^2 + |\nabla {\bar{g}}_t|_{\ell ^2}^2 \Big ]dxdt\Bigg ] . \end{aligned} \end{aligned}$$

which, substituting back the values of \({\bar{f}},{\bar{g}}^k\) and \(z=\partial _l v\) and then using Assumption A-5 with \(n=2\) and (29), gives

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}&|\partial _i \partial _l v_t|_{L^2({\mathscr {D''}})}^2 +{\mathbb {E}}\int _0^T|\partial _i \partial _l v_t|^2_{H^1({\mathscr {D''}})} dt \\ \le&{C {{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')^{-2}}\Bigg [{\mathbb {E}} \int _{\mathscr {D'}}\sum _{|\gamma |\le 2}|D^\gamma v_0|^2 dx \\&+ {\mathbb {E}}\int _0^T \int _{\mathscr {D'}} \Big [{\sum _{|\gamma |\le 2}}|D^\gamma v_t|^2 +\sum _{|\gamma |\le 1}|D^\gamma f_t|^2 +\sum _{|\gamma |\le 2} |D^\gamma g_t|_{\ell ^2}^2 \Big ]dxdt\Bigg ] \end{aligned} \end{aligned}$$
(40)

for all \(i=1,\ldots ,d\) and open \({\mathscr {D''}}\Subset {\mathscr {D'}}\) where \({C}={C}(d,T,K,\kappa )\). Repeating the above procedure k times, we have the following result.

Lemma 6

Assume that v is a continuous \(L^2({\mathscr {D}})\)-valued adapted process satisfying (28) and such that \(v \in L^2(\Omega \times (0,T),{\mathscr {P}}; H^1({\mathscr {D}})).\) If Assumptions A-2, A-5 and A-6 hold for \(n=k\), then

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\sup _{0\le t\le T}|\partial _{i_k}\ldots \partial _{i_1} v_t|_{L^2({\mathscr {D}}^k)}^2 +{\mathbb {E}}\int _0^T|\partial _{i_k}\ldots \partial _{i_1} v_t|^2_{H^1({\mathscr {D}}^k)} dt \le {C {{\,\mathrm{dist}\,}}({\mathscr {D}}^k,\partial {\mathscr {D}}^{k-1})^{-2}}\\&\quad \Bigg [{\mathbb {E}} \int _{{\mathscr {D}}^{k-1}}\sum _{|\gamma |\le k}|D^\gamma v_0|^2 dx + {\mathbb {E}}\int _0^T\int _{{\mathscr {D}}^{k-1}} \Big [{\sum _{|\gamma |\le k}}|D^\gamma v_t|^2 +\sum _{|\gamma |\le k-1}|D^\gamma f_t|^2 \Big . \\&\quad \Big . +\sum _{|\gamma |\le k}| D^\gamma g_t|_{\ell ^2}^2 \Big ]dxdt\Bigg ] \end{aligned} \end{aligned}$$

for all \(i_k=1,\ldots ,d\) and open \({\mathscr {D}}^k\Subset {\mathscr {D}}^{k-1}\) such that \( {{\,\mathrm{dist}\,}}({\mathscr {D}}^k,\partial {\mathscr {D}}^{k-1})<1\) where \(C=C(d,T,K,\kappa )\).

We immediately see that Theorem 3 follows from Lemma 6. Using Theorems 1 and 3, we can now prove Theorem 2.

Proof of Theorem 2

Let u be the solution to (1) given by Theorem 1. Then considering \(f_t(u_t, \nabla u_t)+f_t^0\) as a new free term \(f_t\), we observe that u satisfies (28) with such free term.

Now under the Assumptions A-3, A-4 and due to Theorem 1, applied with \(p \ge 2\alpha -2\), we get the estimate (3) and hence

$$\begin{aligned} {\mathbb {E}}\int _0^T |f_t|^2 _{L^2({\mathscr {D}})}dt= & {} {\mathbb {E}}\int _0^T \int _{\mathscr {D}}|f(u_t, \nabla u_t)+f_t^0|^2 dxdt \nonumber \\&\le 2 \Big [{\mathbb {E}}\int _0^T \int _{\mathscr {D}} K^2(1+|u_t|)^{2\alpha -2} dxdt + {\mathbb {E}}\int _0^T \int _{\mathscr {D}}|f_t^0|^2 dxdt \Big ]\nonumber \\&\le {C} \Big [1 {+} {\mathbb {E}} \sup _{0\le t\le T} \int _{\mathscr {D}}|u_t|^{2\alpha -2} dx\Big ]{+} 2{\mathbb {E}}\int _0^T \int _{\mathscr {D}}|f_t^0|^2 dxdt <\infty .\nonumber \\ \end{aligned}$$
(41)

Hence we can apply Theorem 3 with \(n=1\) thus proving the first claim in (i). Again using (29) for the new free term \(f_t\) we get for each \(i=1,\ldots ,d\),

$$\begin{aligned} \begin{aligned} {\mathbb {E}}&\sup _{0\le t\le T}|\partial _i u_t|_{L^2({\mathscr {D'}})}^2 + {\mathbb {E}}\int _0^T|\partial _i u_t|^2_{H^1({\mathscr {D'}})} dt \le C{{\,\mathrm{dist}\,}}({\mathscr {D'}},\partial {\mathscr {D}})^{-2} {\mathbb {E}}\Bigg [ \int _{{\mathscr {D}}}|\nabla \phi |^2dx\\&+ \int _0^T \int _{{\mathscr {D}}}\Big [|\nabla u_t|^2+|f_t|^2+|u_t|^2 + \sum _{k \in {\mathbb {N}}}|\nabla g_t^k|^2 \Big ] dxdt\Bigg ] \end{aligned} \end{aligned}$$

which on using (41), then Theorem 1 with \(p=2\alpha -2\) and finally Hölder’s inequality proves (25).

Further if f is a function of \(t,\omega ,x\) and r only such that (26) holds, then taking \(f_t(u_t)+~f_t^0\) as a new free term \(f_t\), similarly as above, we get

$$\begin{aligned} \begin{aligned}&{\mathbb {E}}\int _0^T |\partial _i f_t|_{L^2({\mathscr {D}})}^2 dt = {\mathbb {E}}\int _0^T \int _{\mathscr {D}}|\partial _i u_t \, \partial _r f_t(u_t) + \partial _i f_t(u_t)+\partial _if_t^0|^2 dxdt \\&\quad \le {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} \big [ |\nabla u_t|^2(1+|u_t|)^{2\alpha -4} + (1+|u_t|)^{2\alpha - 2} + |\partial _i f^0_t|^2 \big ] dxdt \\&\quad \le {C} {\mathbb {E}} \int _0^T \int _{\mathscr {D}}\big [1 + |\nabla u_t|^2 +|\nabla u_t|^2|u_t|^{2\alpha -4} + |u_t|^{2\alpha - 2} + |\partial _if_t^0|^2 \big ] dxdt <\infty \end{aligned} \end{aligned}$$
(42)

for any \(i \in \{1,\ldots ,d \}\). Hence \(f(u) + f^0\) is in \(L^2(\Omega \times (0,T), {\mathscr {P}}, H^1({\mathscr {D}}))\). Thus all the conditions of Theorem 3 are satisfied for \(n=2\). This yields the first claim in (ii). Again, using (40) for the new free term \(f_t\), we obtain for each \(i,j=1,\ldots ,d\)

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\sup _{0\le t\le T}&|\partial _i \partial _j u_t|_{L^2({\mathscr {D''}})}^2 +{\mathbb {E}}\int _0^T|\partial _i \partial _j u_t|^2_{H^1({\mathscr {D''}})} dt \\&\,\le C {{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')^{-2}{\mathbb {E}}\Bigg [ \sum _{\gamma \le 2}\int _{{\mathscr {D}}'}|D^\gamma \phi |^2dx + \int _0^T \int _{{\mathscr {D}}'}\Big [\sum _{\gamma \le 2}|D^\gamma u_t|^2\\&+ \sum _{\gamma \le 1}|D^\gamma f_t|^2 + \sum _{\gamma \le 2}|D^\gamma g_t|^2_{\ell ^2} \Big ] dxdt\Bigg ] \\&\,\le C {{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')^{-2}{\mathbb {E}}\Bigg [ \sum _{\gamma \le 2}\int _{{\mathscr {D}}'}|D^\gamma \phi |^2dx + \int _0^T \int _{{\mathscr {D}}'}\Big [\sum _{\gamma \le 1}|D^\gamma u_t|^2\\&+ \sum _{\gamma \le 1}|D^\gamma f_t|^2 + \sum _{\gamma \le 2}|D^\gamma g_t|^2_{\ell ^2} \Big ] dxdt\Bigg ] \\&+ C {{\,\mathrm{dist}\,}}({\mathscr {D}}'',\partial {\mathscr {D}}')^{-2}{\mathbb {E}}\int _0^T \int _{{\mathscr {D}}'}\sum _{\gamma = 2}|D^\gamma u_t|^2 dxdt \end{aligned} \end{aligned}$$

which on using (41), (42), then Theorem 1 with \(p=2\alpha -2\) and (25) proves (27).

\(\square \)

Remark 5

Note that to prove even higher regularity than that given by Theorem 2 one would need to show that

$$\begin{aligned} {\mathbb {E}}\int _0^T |\partial _j \partial _i f_t|_{L^2({\mathscr {D}})}^2 dt < \infty . \end{aligned}$$

Using our approach we would require that

$$\begin{aligned} {\mathbb {E}}\int _0^T \int _{{\mathscr {D}}}|\partial _ju_t \partial _i u_t \partial _r^2 f_t(u_t)|^2 \, dxdt < \infty . \end{aligned}$$

However the \(L^p\)-estimates from Theorem 1 are not sufficient. To overcome this, one may try to formally apply \(\partial _i\) to the SPDE (1) and then to try to get the analogous \(L^p\)-estimates for the equation for the derivative. However, since the semilinear term is no longer monotone, the proof will break down.

Regularity in weighted spaces using \(L^p\)-theory and time regularity

In this section, we raise the regularity of the solution to the SPDE (1) using \(L^p\)-theory from Kim [14]. The reason for using \(L^p\)-theory is that one gets better estimates for the solution of the corresponding linear equation, see Theorem 4, given below, which follows immediately from Kim [14, Theorem 2.9].

We will use this together with the \(L^p\)-estimates we proved in Theorem 1 to obtain regularity results (both space and time) for the solution of the semilinear equation (1), see Theorems 5 and 6 below. In particular we obtain Hölder continuity in time of order \(\frac{1}{2} - \frac{2}{q}\) for the solution to (1) as a process in weighted \(L^q\)-space, where q comes from the integrability assumptions imposed on the data.

First, we introduce some notations, concepts and assumptions from Kim [14]. For \(r_0 >~ 0\) and \(x\in {\mathbb {R}}^d\), let \(B_{r_0}(x):=\{y \in {\mathbb {R}}^d : |x-y|< r_0 \}\).

Definition 2

(Domain of class\(C^1_u\)) The domain \({\mathscr {D}}\subset {\mathbb {R}}^d\) is said to be of class \(C^1_u\) if for any \(x_0 \in \partial {\mathscr {D}}\), there exist \(r_0,K_0,L_0 >0\) and a one-one, onto continuously differentiable map \(\Psi :B_{r_0}(x_0)\rightarrow G\), for a domain \(G\subset {\mathbb {R}}^d\), satisfying the following:

  1. (i)

    \(\Psi (x_0)=0\) and \(\Psi \big (B_{r_0}(x_0)\cap {\mathscr {D}} \big )\subset \{y \in {\mathbb {R}}^d : y^1>0 \} \) ,

  2. (ii)

    \(\Psi \big (B_{r_0}(x_0)\cap \partial {\mathscr {D}} \big )= G \cap \{ y\in {\mathbb {R}}^d: y^1=0\)},

  3. (iii)

    \(|\Psi |_{C^1(B_{r_0}(x_0))}\le K_0\) and \(|\Psi ^{-1}(y_1)-\Psi ^{-1}(y_2)|\le K_0|y_1-y_2|\) for any \(y_1,y_2 \in G\),

  4. (iv)

    \(|\Psi _x(x_1)-\Psi _x(x_2)|\le L_0|x_1-x_2|\) for any \(x_1,x_2 \in B_{r_0}(x_0)\).

Let \({\mathscr {D}}\) be of class \(C^1_u\) and \(\rho (x):={{\,\mathrm{dist}\,}}(x,\partial {\mathscr {D}})\). Then, by [14, Lemma 2.5] and [15, Remark 2.7] (since \({\mathscr {D}}\) is bounded), there exists a bounded real valued function \(\psi \) defined on \(\bar{{\mathscr {D}}}\) satisfying

$$\begin{aligned} \sup _{x\in {\mathscr {D}}}\rho ^{|\gamma |}(x)|D^\gamma \partial _i\psi (x)|<\infty \end{aligned}$$
(43)

for any \(i=1,\ldots , d\) and any multi-index \(\gamma \), such that

$$\begin{aligned} \frac{1}{{C}} \rho \le \psi \le {C} \rho \,\,\,\text {in}\,\,\, {\mathcal {D}}, \end{aligned}$$

for some constant C. In other words, \(\psi \) and \(\rho \) are comparable in \({\mathscr {D}}\), and in estimates they can be used interchangeably (up to multiplication by a constant). Moreover this implies \(\psi \ge 0\).

For \(1 \le q < \infty , \,\, \theta \in {\mathbb {R}}\) and a non-negative integer n, define the weighted Sobolev space \( H^{n,q}_\theta ({\mathscr {D}})\) by

$$\begin{aligned} H^{n,q}_\theta ({\mathscr {D}}):= \{u : \rho ^{|\gamma |+(\theta -d)/q}D^\gamma u \in L^q({\mathscr {D}})\,\,\, \text {for any}\,\,\, |\gamma |\le n\} \end{aligned}$$

where the norm for \(u \in H^{n,q}_\theta ({\mathscr {D}})\) is given by

$$\begin{aligned} |u|^q_{H^{n,q}_\theta }:= \sum _{i=0}^n \sum _{|\gamma |=i}\int _{{\mathscr {D}}}|D^\gamma u(x)|^q \rho ^{\theta -d+iq}(x)dx . \end{aligned}$$

For functions \(u:{\mathbb {R}}^d \rightarrow {\mathbb {R}}^{d'}\), we define the norm analogously and use the same notation. The following result from Lototsky [21] plays an important role in proving our results.

Remark 6

The following are equivalent:

  1. (i)

    \(u \in H^{n,q}_\theta ({\mathscr {D}})\),

  2. (ii)

    \( u \in H^{n-1,q}_\theta ({\mathscr {D}})\) and \(\psi \partial _i u \in H^{n-1,q}_\theta ({\mathscr {D}})\) for all \(i=1,2,\ldots d\) ,

  3. (iii)

    \( u \in H^{n-1,q}_\theta ({\mathscr {D}})\) and \( \partial _i(\psi u) \in H^{n-1,q}_\theta ({\mathscr {D}})\) for all \(i=1,2,\ldots d\) .

Further, let

$$\begin{aligned} {\mathbb {H}}^{n,q}_\theta ({\mathscr {D}}):=L^q(\Omega \times (0,T),{\mathscr {P}}, H^{n,q}_\theta ({\mathscr {D}}) ). \end{aligned}$$

In the rest of the article, we assume that

$$\begin{aligned} q\ge 2 \quad \text {and} \quad d-2+q<\theta <d-1+q \end{aligned}$$
(44)

so that in view of [14, Remark 2.7], the assumption regarding existence of an \({\mathscr {A}}_{p,\theta }\)-type set (see [14, Assumption 2.8]), is satisfied. Finally, we need the following assumption on the coefficients:

A-7 For any \(i,j=1,\ldots ,d\),

  1. (i)

    the real valued coefficients \(a^{ij}\) and their spatial derivatives up to order \(n+1\) are \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and bounded by K,

  2. (ii)

    the real-valued coefficients \(b^i,\,\,c\) and their spatial derivatives up to order n are \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and are bounded by K,

  3. (iii)

    the coefficients \(\sigma ^i=(\sigma ^{ik})_{k=1}^\infty , \,\, \mu =(\mu ^k)_{k=1}^\infty \) and their spatial derivatives up to order \(n+1\) are \(\ell ^2\)-valued \({\mathscr {P}} \times {\mathscr {B}}({\mathscr {D}})\)-measurable and almost surely

    $$\begin{aligned} \sum _{i=1}^d\sum _{k\in {\mathbb {N}}} \sum _{|\gamma |\le n+1}|D^\gamma \sigma _t^{ik}(x)|^2+\sum _{k\in {\mathbb {N}}}\sum _{|\gamma |\le n+1}|D^\gamma \mu _t^k(x)|^2 \le K \end{aligned}$$

    for all t and x,

  4. (iv)

    and for almost every \((t,\omega )\), the coefficients \(a^{ij}(t,x)\) and \(\sigma ^i(t,x)\) are uniformly continuous in \(x \in {\mathscr {D}}\).

Note that, the operator L given by (2) is in divergence form but the results from [14] are for operators in non-divergence form. One knows that (1) can be expressed in non-divergence form if the coefficients \(a^{ij}\) are differentiable. Thus Assumption A-7 implies Assumptions 2.2 and 2.3 in [14]. Hence the following theorem follows from Theorem 2.9 of Kim [14].

Theorem 4

Assume \({\mathscr {D}}\) is of class \(C^1_u\). Further, let Assumptions A-2 and A-7 hold with some \(n \ge 0\). If \(\psi f\in {\mathbb {H}}^{n,q}_\theta ({\mathscr {D}})\), \(g \in {\mathbb {H}}^{n+1,q}_\theta ({\mathscr {D}};\ell ^2)\) and \(\psi ^{\frac{2}{q}-1}\phi \in {\mathbb {H}}^{n+2,q}_\theta ({\mathscr {D}})\), then

$$\begin{aligned} \left\{ \begin{aligned} dv_t&=(L_tv_t+f_t)dt+\sum _{k\in {\mathbb {N}}}(M_t^kv_t+g_t^k)dW_t^k \,\,\,\ \text {on}\,\,\, [0,T]\times {\mathscr {D}},\\ v_t&=0 \,\,\, \text {on}\,\,\, \partial {\mathscr {D}}, \quad v_0 =\phi \,\,\, \text {on}\,\,\, {\mathscr {D}} \end{aligned} \right. \end{aligned}$$
(45)

has a unique solution v such that \(\psi ^{-1}v\in {\mathbb {H}}^{n+2,q}_\theta ({\mathscr {D}})\).

In fact Theorem 2.9 in Kim [14] is proved even for fractional weighted Sobolev spaces and under somewhat weaker assumptions. We do not use fractional spaces here to keep the presentation simpler. As to being able to use weaker assumptions: to obtain results for the semilinear equation (1) we will need to apply our results from Sect. 2, in particular Theorem 1 and thus we cannot substantially weaken our assumptions here. Finally, we can state the main results on regularity for the solution to semilinear SPDE (1).

Theorem 5

Assume \({\mathscr {D}}\) is of class \(C^1_u\) and u is the solution to (1). Further, let Assumptions A-2 to A-4 hold with \(p\ge \max (q\alpha -q,2)\) and Assumption A-7 holds with \(n=0\). If for some q satisfying (44), \(\psi ^{\frac{2}{q}-1}\phi \in {\mathbb {H}}^{2,q}_\theta ({\mathscr {D}}),g~ \in ~ {\mathbb {H}}^{1,q}_\theta ({\mathscr {D}};\ell ^2)\) and \(f^0\in {\mathbb {H}}_\theta ^{0,q}({\mathscr {D}})\), then \( \psi ^{-1}u\in {\mathbb {H}}^{2,q}_\theta ({\mathscr {D}}).\)

Moreover, in the case Assumption A-7 holds with \(n=1\) and almost surely

$$\begin{aligned} \begin{aligned}&|\partial _i f_t(x,r,z)| \le K(1+|r|)^{\alpha - 1},\,\,\, |\partial _rf_t(x,r,z)|\le K(1+|r|)^{\alpha -2} \\&\text {and}\,\,\, |\partial _z f_t(x,r,z)| \le K(1+|r|)^{\alpha - 1} \end{aligned} \end{aligned}$$
(46)

for all \(i=1,\dots , d\), \(t\in [0,T], x\in {\mathscr {D}}, r\in {\mathbb {R}}\) and all \(z\in {\mathbb {R}}^d\), if for some q satisfying (44), \(\psi ^{\frac{2}{q}-1}\phi \in {\mathbb {H}}^{3,q}_\theta ({\mathscr {D}})\), \(g~ \in ~ {\mathbb {H}}^{2,q}_\theta ({\mathscr {D}};\ell ^2)\) and \(f^0\in {\mathbb {H}}_\theta ^{1,q}({\mathscr {D}})\), then \( \psi ^{-1}u\in {\mathbb {H}}^{3,\frac{q}{2}}_\theta ({\mathscr {D}})\).

Remark 7

Note that if \(\psi ^{-1}u\in {\mathbb {H}}^{2,q}_\theta ({\mathscr {D}})\), then by using Remark 6, we get

$$\begin{aligned} \psi ^{-1}u\in {\mathbb {H}}^{1,q}_\theta ({\mathscr {D}}) \,\,\, \text {and} \,\,\, \partial _i u \in {\mathbb {H}}^{1,q}_\theta ({\mathscr {D}}) \,\,\, \forall i=1,2,\ldots d. \end{aligned}$$

Invoking Remark 6 again, we have

$$\begin{aligned} \psi ^{-1}u\in {\mathbb {H}}^{0,q}_\theta ({\mathscr {D}}),\,\,\, \partial _i u \in {\mathbb {H}}^{0,q}_\theta ({\mathscr {D}}) \,\,\, \text {and} \,\,\, \psi \partial _i \partial _j u \in {\mathbb {H}}^{0,q}_\theta ({\mathscr {D}}) \,\,\, \forall i,j=1,2,\ldots d.\nonumber \\ \end{aligned}$$
(47)

Finally, we present the result on time regularity of the solution of (1).

Theorem 6

Under the assumptions of Theorems 1 and 5,

$$\begin{aligned} u \in {C^\gamma \big ([0,T]; H^{0,q}_{\theta +q} ({\mathscr {D}}) \big )} \quad a.s. \end{aligned}$$

i.e. the solution u to SPDE (1), as a \(H^{0,q}_{\theta +q} ({\mathscr {D}})\)-valued process, is Hölder continuous of order \(\gamma \) for every \(\gamma < \frac{1}{2}-\frac{2}{q}\,\) for every q satisfying (44).

Note that one would like u to be Hölder continuous with exponent \(\gamma \) as a process with values in a weighted Sobolev space with the same weight exponent \(\theta \) as in the results for spatial regularity (Theorem 5). However we need to use (47) in our arguments when proving Theorem 6 which leads to requiring the weight exponent to be \(\theta + q\).

Before proving these theorems, we first prove the following lemma:

Lemma 7

Let \({\tilde{\theta }} > d\) and \({\tilde{q}} \ge 1\). Further, let assumptions of Theorem 1 hold with \(p~\ge ~ \max ({\tilde{q}}\alpha - {\tilde{q}},2)\) and \(f^0\in {\mathbb {H}}_{{\tilde{\theta }}}^{0,{\tilde{q}}}({\mathscr {D}})\). If u is the solution to (1) and \(f_t:=f_t(u_t,\nabla u_t)+~f_t^0\), then \(f \in {\mathbb {H}}_{{\tilde{\theta }}}^{0,{\tilde{q}}}({\mathscr {D}})\) and thus \(\psi f \in {\mathbb {H}}_{{\tilde{\theta }}}^{0,{\tilde{q}}}({\mathscr {D}})\).

Proof

First we note that \({\tilde{\theta }} > d\) and \({\mathscr {D}}\) is bounded, therefore \(\sup _{x\in {\mathscr {D}}}\rho ^{{\tilde{\theta }}-d}(x)<\infty \). Using this along with Assumption A-3 implies

$$\begin{aligned} \begin{aligned} {\mathbb {E}}\int _0^T \int _{\mathscr {D}}&|f_t|^{{\tilde{q}}} \rho ^{{\tilde{\theta }}-d}dxdt ={\mathbb {E}}\int _0^T \int _{\mathscr {D}}|f_t(u_t, \nabla u_t)+f_t^0|^{{\tilde{q}}} \rho ^{{\tilde{\theta }}-d}dxdt \\&\le {C} \Big [{\mathbb {E}}\int _0^T \int _{\mathscr {D}} (1+|u_t|)^{{{\tilde{q}}}\alpha -{{\tilde{q}}}} dxdt + {\mathbb {E}}\int _0^T \int _{\mathscr {D}}|f_t^0|^{{{\tilde{q}}}} \rho ^{{\tilde{\theta }}-d}dxdt \Big ]\\&\le {C} \Big [1 + {\mathbb {E}} \sup _{0\le t\le T} |u_t|^{{{\tilde{q}}}\alpha -{{\tilde{q}}}}_{L^{{{\tilde{q}}}\alpha -{{\tilde{q}}}}}\Big ]+ {C}{\mathbb {E}}\int _0^T \int _{\mathscr {D}}|f_t^0|^{{\tilde{q}}} \rho ^{{\tilde{\theta }}-d}dxdt \end{aligned} \end{aligned}$$
(48)

which is finite in view of Theorem 1 and the fact \(f^0\in {\mathbb {H}}_{{\tilde{\theta }}}^{0,{{\tilde{q}}}}({\mathscr {D}})\). Now note that \(\psi \) is bounded on \(\bar{{\mathscr {D}}}\) and hence

$$\begin{aligned} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\psi f_t|^q \rho ^{\theta -d}\,dxdt \le {C}{\mathbb {E}}\int _0^T \int _{\mathscr {D}} |f_t|^q \rho ^{\theta -d}\,dxdt<\infty . \end{aligned}$$

\(\square \)

Proof of Theorem 5

Let u be the solution to (1) given by Theorem 1. Then considering \(f_t(u_t,\nabla u_t)+f_t^0\) as a new free term \(f_t\), the solution u satisfies (45). We wish to apply Theorem 4 with \(n=0\) and in order to do so we need to show that \(\psi f\in {\mathbb {H}}^{0,q}_\theta ({\mathscr {D}})\). Indeed this follows immediately by using Lemma 7 with \({\tilde{\theta }}=\theta \) and \({{\tilde{q}}}=q\). Hence applying Theorem 4 with \(n=0\) we obtain \( \psi ^{-1}u\in {\mathbb {H}}^{2,q}_\theta ({\mathscr {D}})\). This completes the proof of the first statement of the theorem.

We now consider the case when Assumption A-7 holds with \(n=1\). Again we will apply Theorem 4 (but now with \(n=1\) and \(\frac{q}{2}\) in place of q) and so we need to show that \(\psi f\in {\mathbb {H}}^{1,{\bar{q}}}_\theta ({\mathscr {D}})\) with \({\bar{q}}:=\frac{q}{2}\). Taking \({\tilde{\theta }}=\theta \) and \({{\tilde{q}}}={\bar{q}}\) in Lemma 7, we get \(\psi f\in {\mathbb {H}}^{0,{\bar{q}}}_\theta ({\mathscr {D}})\). Thus we consider

$$\begin{aligned} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _i \big (\psi f_t\big )|^{{\bar{q}}} \rho ^{\theta -d+{{\bar{q}}}}dxdt = I_1 + I_2 , \end{aligned}$$

where

$$\begin{aligned} I_1 :={\mathbb {E}}\int _0^T \int _{\mathscr {D}} |f_t|^{{\bar{q}}} |\partial _i \psi |^{{\bar{q}}} \rho ^{\theta -d+{{\bar{q}}}}dxdt \,\,\, \text {and} \,\,\, I_2:={\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _i f_t|^{{\bar{q}}} \psi ^{{\bar{q}}} \rho ^{\theta -d+{{\bar{q}}}}dxdt . \end{aligned}$$

Clearly \(I_1<\infty \) using (43), the fact \(\rho \) is bounded on \({\mathscr {D}}\) and Lemma 7 (with \({\tilde{\theta }}=\theta \) and \({{\tilde{q}}}={\bar{q}}\)). Further observe that

$$\begin{aligned} \begin{aligned} \partial _i f_t&= \partial _i (f_t(u_t, \nabla u_t) + f^0_t) \\&= \partial _i f_t(u_t, \nabla u_t)+\partial _i u_t \, \partial _r f_t(u_t, \nabla u_t)+\partial _i (\nabla u_t) \, \nabla _z f_t(u_t, \nabla u_t) +\partial _i f_t^0, \end{aligned} \end{aligned}$$

where \(\nabla _z f_t\) is the gradient with respect to z of \(f_t=f_t(x,r,z)\). Thus, we have

$$\begin{aligned} I_2 \le {C} (I_3+I_4+I_5+I_6) \end{aligned}$$
(49)

where

$$\begin{aligned} I_3:= & {} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _i f_t(u_t, \nabla u_t)|^{{{\bar{q}}}} \psi ^{{{\bar{q}}}} \rho ^{\theta -d+{{\bar{q}}}}\, dxdt , \\ I_4:= & {} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _i u_t \, \partial _r f_t(u_t, \nabla u_t)|^{{{\bar{q}}}} \psi ^{{{\bar{q}}}} \rho ^{\theta -d+{{\bar{q}}}}\, dxdt , \\ I_5:= & {} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _i (\nabla u_t) \, \nabla _z f_t(u_t, \nabla u_t)|^{{{\bar{q}}}}\psi ^{{{\bar{q}}}} \rho ^{\theta -d+{{\bar{q}}}}\, dxdt , \end{aligned}$$

and

$$\begin{aligned} I_6 := {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _i f_t^0|^{{{\bar{q}}}} \psi ^{{{\bar{q}}}} \rho ^{\theta -d+{{\bar{q}}}}\, dxdt . \end{aligned}$$

Now, using the fact that \(\psi \) and \(\rho \) are bounded on \({\mathscr {D}}\) and the assumption on growth of derivatives of the semilinear term, see (46), we observe that

$$\begin{aligned} I_3 \le {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} \big ( 1{+}|\partial _i f_t(u_t, \nabla u_t)| \big )^q dxdt \le {C} \Big [ 1{+} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} (1+|u_t|)^{q\alpha -q} dxdt \Big ] . \end{aligned}$$

This is finite in view of Theorem 1, see the estimate (48) for details. Further, using Young’s inequality and the fact that \(\psi \) and \(\rho \) are bounded on \({\mathscr {D}}\) along with growth assumption (46), we get

$$\begin{aligned} \begin{aligned} I_4&\le {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} \Big [ |\partial _i u_t|^q + |\partial _r f_t(u_t, \nabla u_t)|^q \Big ] \rho ^{\theta -d} dxdt \\&\le {C} \Big [|\partial _i u|_{{\mathbb {H}}^{0,q}_\theta }^q + {\mathbb {E}}\int _0^T \int _{\mathscr {D}} (1+|u_t|)^{q\alpha -2q} dxdt\Big ] . \end{aligned} \end{aligned}$$

We see that this is finite using Remark 7 and Theorem 1 again. Furthermore, using Young’s inequality, growth assumption (46) and the fact that \(\psi \) and \(\rho \) are comparable, we obtain

$$\begin{aligned} \begin{aligned} I_5&\le {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} \Big [ |\partial _i (\nabla u_t)|^q + |\nabla _z f_t(u_t, \nabla u_t)|^q \Big ] \psi ^q \rho ^{\theta -d} dxdt \\&\le {C} \Big [|\psi \partial _i (\nabla u)|_{{\mathbb {H}}^{0,q}_\theta }^q + {\mathbb {E}}\int _0^T \int _{\mathscr {D}} (1+|u_t|)^{q\alpha -q} dxdt\Big ]. \end{aligned} \end{aligned}$$

Thus, applying Remark 7 and Theorem 1 as before, we obtain \(I_5<\infty \). Finally, the fact that \(\psi \) and \(\rho \) are comparable and bounded on \({\mathscr {D}}\) implies

$$\begin{aligned} I_6 \le {C} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} \big ( 1{+}|\partial _i f_t^0| \big )^q \rho ^{\theta -d{+}q} dxdt \le {C} \Big [ 1{+} {\mathbb {E}}\int _0^T \int _{\mathscr {D}} |\partial _if_t^0|^q \rho ^{\theta -d+q}dxdt \Big ] \end{aligned}$$

which is finite since \(f^0\in {\mathbb {H}}_\theta ^{1,q}({\mathscr {D}})\). Thus \( \psi f \in {\mathbb {H}}_\theta ^{1,{\bar{q}}}({\mathscr {D}})\) and we can apply Theorem 4 with \(n=1\) and \({\bar{q}}\) in place of q to complete the proof. \(\square \)

Proof of Theorem 6

We will prove the result using Kolmogorov continuity theorem. To ease the notation we let \(f_t:=f_t(u_t, \nabla u_t)+f_t^0\). Then from (1) we see that

$$\begin{aligned} {\mathbb {E}}|u_t-u_s|_{H^{0,q}_{\theta +q}}^q \le 2^{q-1}(I_1(s,t) + I_2(s,t)), \end{aligned}$$
(50)

where

$$\begin{aligned} I_1(s,t):={\mathbb {E}}\Big |\int _s^t (L_ru_r{+}f_r)dr\Big |^q_{H^{0,q}_{\theta {+}q}} \,\,\,\text {and}\,\,\, I_2(s,t):= \Big |\sum _{k\in {\mathbb {N}}}\int _s^t (M^k_ru_r{+}g^k_r)dW^k_r\Big |^q_{H^{0,q}_{\theta +q}} . \end{aligned}$$

We note that \(f^0\in {\mathbb {H}}_\theta ^{0,q}({\mathscr {D}})\) implies \(f^0\in {\mathbb {H}}_{\theta +q}^{0,q}({\mathscr {D}})\) because \(\rho \) is bounded on \({\mathscr {D}}\). Now using Hölder’s inequality, we get

$$\begin{aligned} \begin{aligned} I_1(s,t)&\le (t-s)^{q-1}{\mathbb {E}}\int _s^t |L_ru_r+f_r|^q_{H^{0,q}_{\theta +q}}dr \\&\le {C}(t-s)^{q-1}\Big [{\mathbb {E}}\int _s^t |L_ru_r|^q_{H^{0,q}_{\theta +q}}dr+{\mathbb {E}}\int _s^t|f_r|^q_{H^{0,q}_{\theta +q}}dr\Big ] . \end{aligned} \end{aligned}$$
(51)

Using Assumption A-7 with \(n=0\), we get

$$\begin{aligned} \begin{aligned}&|L_ru_r|^q_{H^{0,q}_{\theta +q}} = \int _{{\mathscr {D}}} \Big |\sum _{j=1}^d\partial _j\Big (\sum _{i=1}^da_t^{ij}\partial _iu_r\Big ) +\sum _{i=1}^db_t^i\partial _iu_r+c_tu_r \Big |^q \rho ^{\theta +q-d}dx \\&\quad \le {C} \int _{{\mathscr {D}}}\Big ( \sum _{i,j=1}^d| \partial _i\partial _ju_r|^q +\sum _{i=1}^d |\partial _iu_r|^q + |u_r|^q \Big )\rho ^{\theta +q-d} dx \\&\quad \le {C} \Bigg (\sum _{i,j=1}^d|\psi \partial _i\partial _ju_r|^q_{H^{0,q}_{\theta }}+|\psi |_{C(\mathscr {{{\bar{D}}}})}^q \sum _{i=1}^d |\partial _iu_r|^q_{H^{0,q}_{\theta }}+|\psi |_{C(\mathscr {{{\bar{D}}}})}^{2q}|\psi ^{-1}u_r|^q_{H^{0,q}_{\theta }} \Bigg ). \end{aligned} \end{aligned}$$

Substituting this in (51) and using the fact that \(\psi \) is bounded on \(\bar{{\mathscr {D}}}\), we obtain

$$\begin{aligned}&I_1(s,t) \nonumber \\&\quad \le {C}(t-s)^{q-1}\Bigg (\sum _{i,j=1}^d|\psi \partial _i\partial _ju|^q_{{\mathbb {H}}^{0,q}_{\theta }}+ \sum _{i=1}^d |\partial _iu|^q_{{\mathbb {H}}^{0,q}_{\theta }}+|\psi ^{-1}u|^q_{{\mathbb {H}}^{0,q}_{\theta }} +|f|^q_{{\mathbb {H}}^{0,q}_{\theta +q}} \Bigg ) \nonumber \\&\quad \le {C}(t-s)^{q-1},\nonumber \\ \end{aligned}$$
(52)

where last statement follows using Remark 7 and Lemma 7 with \({\tilde{\theta }}=\theta +q\) and \({\tilde{q}}=q\).

Furthermore using Burkholder–Davis–Gundy’s inequality, Assumption A-7 with \(n=0\), Hölder’s inequality and the fact that \(\rho \) is bounded on \({\mathscr {D}}\), we see that

$$\begin{aligned} \begin{aligned} I_2(s,t)&= {\mathbb {E}} \int _{\mathscr {D}}\Big |\sum _{k\in {\mathbb {N}}} \int _s^t (M^k_ru_r+g^k_r)dW^k_r \Big |^q \rho ^{\theta +q-d}dx\\&\le \int _{{\mathscr {D}}}{\mathbb {E}}\Big [\int _s^t \sum _{k\in {\mathbb {N}}}|M^k_ru_r+g^k_r|^2dr\Big ]^\frac{q}{2} \rho ^{\theta +q-d} dx \\&= \int _{{\mathscr {D}}}{\mathbb {E}}\Big [\int _s^t \sum _{k\in {\mathbb {N}}}\Big |\sum _{i=1}^d\sigma _r^{ik}\partial _iu_r+\mu _r^ku_r+g^k_r\Big |^2dr\Big ]^\frac{q}{2} \rho ^{\theta +q-d} dx \\&\le {C} \int _{{\mathscr {D}}}{\mathbb {E}}\Big [\int _s^t \Big (\sum _{i=1}^d|\partial _iu_r|^2+|u_r|^2+\sum _{k\in {\mathbb {N}}}|g^k_r|^2\Big )dr\Big ]^\frac{q}{2} \rho ^{\theta +q-d} dx \\&\le {C} \int _{\mathscr {D}}(t-s)^{\frac{q}{2}-1}{\mathbb {E}} \Big [\int _s^t \Big ( \sum _{i=1}^d|\partial _iu_r|^q+|u_r|^q+|g_r|_{\ell ^2}^q\Big )dr\Big ] \rho ^{\theta +q-d} dx \\&\le {C}(t-s)^{\frac{q}{2}-1} \Bigg (\sum _{i=1}^d |\partial _iu|^q_{{\mathbb {H}}^{0,q}_{\theta }}+|\psi ^{-1}u|^q_{{\mathbb {H}}^{0,q}_{\theta }} +|g|^q_{{\mathbb {H}}^{0,q}_{\theta }} \Bigg ) \le {C}(t-s)^{\frac{q}{2}-1} . \end{aligned} \end{aligned}$$
(53)

Here, the last inequality is obtained using Remark 7 as before and the assumption that \(g\in {\mathbb {H}}^{1,q}_{\theta }({\mathscr {D}};\ell ^2)\). Using (52) and (53) in (50), we obtain

$$\begin{aligned} {\mathbb {E}}|u_t-u_s|_{H^{0,q}_\theta }^q \le {C} |t-s|^{\frac{q}{2}-1} \end{aligned}$$

which on using Kolmogorov continuity theorem concludes the result. \(\square \)

Corollary 1

Under the assumptions of Theorems 1, 2 (parts (i) and (ii)) and 5 we have

$$\begin{aligned} u \in C^\alpha \big ([0,T];H^1({\mathscr {D}}')\big ) \quad a.s. \end{aligned}$$

for every \(\alpha < \frac{1}{4}-\frac{1}{q}\) with q satisfying (44) and \({\mathscr {D}}' \Subset {\mathscr {D}}\).

Proof

Note that for any open \({\mathscr {D}}' \Subset {\mathscr {D}}\), there exists a constant \(M>0\) such that the distance function \(\rho \) satisfies \(|\rho (x)|\ge M\) for all \(x \in {\mathscr {D}}'\). Therefore using Theorem 6, we get that almost surely

$$\begin{aligned} \begin{aligned} |u_t{-}u_s|_{L^q({\mathscr {D}}')}&=\Big (\int _{{\mathscr {D}}'}|u_t{-}u_s|^q dx\Big )^\frac{1}{q}{\le } \Big (\sup _{x\in {\mathscr {D}}'} \frac{1}{\rho ^{\theta {+}q-d}}\int _{{\mathscr {D}}}|u_t-u_s|^q \rho ^{\theta {+}q-d} dx\Big )^\frac{1}{q} \\&\le \frac{1}{\big (M^{\theta +q-d}\big )^\frac{1}{q}}|t-s|^{\frac{1}{2}-\frac{2}{q}-\epsilon }|u|_{C^{\frac{1}{2}-\frac{2}{q}-\epsilon } \big ([0,T]; H^{0,q}_{\theta +q} ({\mathscr {D}}) \big )} \end{aligned} \end{aligned}$$
(54)

for any \(\epsilon >0\) and all \(s,t\in [0,T]\). Further, since \(q\ge 2\), using Hölder’s inequality we have that there exists a random variable C such that

$$\begin{aligned} |u_t-u_s|_{L^2({\mathscr {D}}')} \le C |t-s|^{\frac{1}{2}-\frac{2}{q}-\epsilon } \end{aligned}$$

which implies that almost surely \(u \in C^{\frac{1}{2}-\frac{2}{q}-\epsilon }\big ([0,T];L^2({\mathscr {D}}')\big ) \) for any \(\epsilon >0\). Furthermore using Theorem 2, we have that almost surely \(u \in C\big ([0,T];H^2({\mathscr {D}}')\big )\). Now using Gagliardo–Nirenberg inequality, we have that almost surely for any \(s,t \in [0,T]\)

$$\begin{aligned} \begin{aligned} |u_t-u_s|_{H^1({{\mathscr {D}}'})}&\le C |u_t-u_s|^\frac{1}{2}_{L^2({\mathscr {D}}')}|u_t-u_s|^\frac{1}{2}_{H^2({\mathscr {D}}')}\\&\le C \Big (|t-s|^{\frac{1}{2}-\frac{2}{q}-\epsilon }|u|_{C^{\frac{1}{2}-\frac{2}{q}-\epsilon } \big ([0,T]; L^2({\mathscr {D}}') \big )}\Big )^\frac{1}{2}\Big (2 |u|_{C\big ([0,T];H^2({\mathscr {D}}')\big )}\Big )^\frac{1}{2} \\&\le C |t-s|^{\frac{1}{4}-\frac{1}{q}-\frac{\epsilon }{2}} \end{aligned} \end{aligned}$$

for some random variable C which concludes the result since \(\epsilon >0\) is arbitrary. \(\square \)

References

  1. 1.

    Breit, D.: Regularity theory for nonlinear systems of SPDEs. Manuscripta Math. 146(3–4), 329–349 (2015)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2010)

    Google Scholar 

  3. 3.

    Dareiotis, K., Gerencsér, M.: On the boundedness of solutions of SPDEs. Stoch. Partial Differ. Equ. Anal. Comput. 3(1), 84–102 (2015)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Evans, L.C.: Partial Differential Equations. American Mathematical Society, Providence, RI (1998)

    MATH  Google Scholar 

  5. 5.

    Flandoli, F.: Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of solutions. Stochastics 29(3), 331–357 (1990)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Gess, B.: Strong solutions for stochastic partial differential equations of gradient type. J. Funct. Anal. 263(8), 2355–2383 (2012)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Gerencsér, M.: Boundary regularity of stochastic PDEs. Ann. Probab. 47(2), 804–834 (2019)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)

    MATH  Google Scholar 

  9. 9.

    Gyöngy, I., Millet, A.: Rate of convergence of space time approximations for Stochastic evolution equations. Potential Anal. 30, 29–64 (2009)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Gyöngy, I., Šiška, D.: Itô formula for processes taking values in intersection of finitely many banach spaces. Stoch. Partial Differ. Equ. Anal. Comput. 5, 428–455 (2017)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Jentzen, A., Röckner, M.: Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. J. Differ. Equ. 252(1), 114–136 (2012)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Kim, I., Kim, K.-H.: Some \(L_p\) and Hölder estimates for divergence type nonlinear SPDEs on \(C^1\)-domains. Potential Anal. 41, 583–612 (2014)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Kim, I., Kim, K.-H.: A regularity theory for quasi-linear stochastic partial differential equations in weighted Sobolev spaces. Stoch. Process. Appl. 128(2), 622–643 (2018)

    MATH  Google Scholar 

  14. 14.

    Kim, K.-H.: On stochastic partial differential equations with variable coefficients in \(C^1\) domains. Stoch. Proc. Appl. 112, 261–283 (2004)

    MATH  Google Scholar 

  15. 15.

    Kim, K.-H., Krylov, N.V.: On the Sobolev space theory of parabolic and elliptic equations in \(C^1\) domains. SIAM J. Math. Anal. 36(2), 618–642 (2004)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. American Mathematical Society, Providence, RI (1996)

    MATH  Google Scholar 

  17. 17.

    Krylov, N.V.: A relatively short proof of Itô’s formula for SPDEs and its applications. Stoch. Partial Differ. Equ. Anal. Comput. 1(1), 152–174 (2013)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Krylov, N.V.: A \(W_2^n\)-theory of the Dirichlet problem for SPDEs in general smooth domains. Probab. Theory Relat. Fields 98, 389–421 (1994)

    MATH  Google Scholar 

  19. 19.

    Krylov, N.V., Rozovskii, B.L.: Stochastic evolution equations. J. Sov. Math. 14, 1233–1277 (1981)

    MATH  Google Scholar 

  20. 20.

    Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-linear Equations of Parabolic Type, Volume 23 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1968)

  21. 21.

    Lototsky, S.V.: Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations. Methods Appl. Anal. 7(1), 195–204 (2000)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Pardoux, E.: Equations aux dérivées partielles stochastiques non lineaires monotones. Thése Doct. Sci. Math. Univ. Paris Sud, Etude des solutions forte de type Ito (1975)

  23. 23.

    Rozovskii, B.L.: Stochastic Evolution Systems. Linear Theory and Applications to Nonlinear Filtering. Kluwer, Dordrecht (1990)

    Google Scholar 

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The authors are grateful to the anonymous referees for their valuable suggestions which helped to significantly improve the paper.

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Neelima, Šiška, D. \(L^p\)-estimates and regularity for SPDEs with monotone semilinearity. Stoch PDE: Anal Comp 8, 422–459 (2020). https://doi.org/10.1007/s40072-019-00150-w

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Keywords

  • Stochastic partial differential equations
  • Regularity
  • Weighted Sobolev spaces

Mathematics Subject Classification

  • 60H15
  • 35R60