Existence results for linear evolution equations of parabolic type

We study both strict and mild solutions to parabolic evolution equations of the form $dX+AXdt=F(t)dt+G(t)dW(t)$ in Banach spaces. First, we explore the deterministic case. The maximal regularity of solutions has been shown. Second, we investigate the stochastic case. We prove existence of strict solutions and show their space-time regularity. Finally, we apply our abstract results to a stochastic heat equation.

1. Introduction. We consider the Cauchy problem for a linear evolution equation with additive noise (1.1) dX + AXdt = F (t)dt + G(t)dW (t), 0 < t ≤ T, X(0) = ξ in a UMD Banach space E of type 2 with norm · . Here, W denotes a cylindrical Wiener process on a separable Hilbert space H, and is defined on a filtered, complete probability space (Ω, F, {F t } t≥0 , P). Operators G(t), 0 ≤ t ≤ T, are γ -radonifying operators from H to E, whereas F is an E -valued measurable function on [0, T ]. Initial value ξ is an E -valued F 0 -measurable random variable. And, A is a sectorial operator in E, i.e. it is a densely defined, closed linear operator satisfying the condition: The resolvent satisfies the estimate with some constant M ̟ > 0 depending only on the angle ̟.
In the stochastic case (i.e. G ≡ 0), weak solutions in L 2 spaces have been shown in [3,4] by using the semigroup methods, in [13] by using the variational methods, and in [22] by using the martingale methods. After that, some researchers have studied that kind of solutions in weighted Sobolev spaces or weighted Hölder spaces (see [6,7,9]) by using the semigroup or the variational methods.
However, existence of strict solutions to (1.1) is only shown in a very restrictive case. In [3,4], Da Prato et al. showed that when A is a bounded linear operator, (1.1) that is considered in Hilbert spaces possesses strict solutions (under other conditions on coefficients and initial value).
The work in [4] inspired us to study existence of strict solutions to (1.1) when the linear operator A is unbounded. In the present paper, we want to consider the equation in Banach spaces (for the deterministic case) and in UMD Banach spaces of type 2 (for the stochastic case), where both F and G have temporal and spatial regularity. In the deterministic case, our results improve those in [20] and generalize the maximal regularity theorem in [26]. In the stochastic case, we show existence and regularity of strict solutions, provided that A is a (unbounded) sectorial operator.
For the study, we use the semigroup methods. In particular, we very often use an identity: where B(·, ·) is the Beta function and 0 < α, β < 1 are some constants. Notice that when α + β = 1, we have This identity has been used as a key point in the so-called factorization method introduced by Da Prato et al. (see [3,4]). For applications, our results can be applied to a class of stochastic partial differential equations such as heat equations, reaction diffusion equations, FitzHugh-Nagumo models, or Hodgkin-Huxley models (see, e.g., [22,4]). In the last section of the present paper, we consider a special case, namely σ 1 = −a(x)u(t, x) + b(t, x) and σ 2 = σ(t, x) (see (5.1)), of the nonlinear stochastic heat equation: We should mention that weak solutions to (1.2) have been studied in [10,11,15,22] and references therein. By using our abstract results, strict solutions to (5.1) can be obtained (see Theorems 5.1 and 5.2). The paper is organized as follows. Section 2 is preliminary. Section 3 studies the deterministic case of (1.1). The stochastic case is investigated in Section 4. Finally, Section 5 gives an application to heat equations.

Preliminary.
2.1. UMD Banach spaces of type 2. Let us recall the notion of UMD Banach spaces of type 2.
is a martingale) on a complete probability space (Ω ′ , F ′ , P ′ ) and any ǫ : (ii) A Banach space E is said to be of type 2 if there exists c 2 (E) > 0 such that for any Rademacher sequence {ǫ i } i on a complete probability space (Ω ′ , F ′ , P ′ ) and any finite sequence {x k } n k=1 of E, (Recall that a Rademacher sequence is a sequence of independent symmetric random variables each one taking on the set {1, −1}.) Remark 2.2. All Hilbert spaces and L p spaces (2 ≤ p < ∞) are UMD spaces of type 2. When 1 < p < ∞, L p spaces are UMD spaces.
From now on, if not specified we always assume that E is a UMD Banach sapce of type 2 and H is a separable Hilbert space.
2.2. γ -radonifying operators. Let us review the notion of γ -radonifying operators. For more details on the subject, see [21]. Definition 2.3 (γ -radonifying operators). Let {e n } ∞ n=1 be an orthonormal basis of H. Let {γ n } ∞ n=1 be a sequence of independent standard Gaussian random variables on a probability space (Ω ′ , F ′ , P ′ ). A γ -radonifying operator from H to E is an operator, denoted by φ for example, in L(H; E) such that the Gaussian series ∞ n=1 γ n φe n converges in L 2 (Ω ′ , E).
Denote by γ(H; E) the set of all γ -radonifying operators from H to E. Define a norm in γ(H; E) by It is known that the norm is independent of the orthonormal basis {e n } ∞ n=1 and the Gaussian sequence {γ n } ∞ n=1 . Furthermore, (γ(H; E), · γ(H;E) ) is complete.
Let (S, Σ) be a measurable space. A function ϕ : S → E is said to be strongly measurable if it is the pointwise limit of a sequence of simple functions. A function φ : S → L(H; E) is said to be H -strongly measurable if φ(·)h : S → E is strongly measurable for all h ∈ H.
The following result is very often used in this paper.

Stochastic integrals.
Definition 2.6. A family W = {W (t)} t≥0 of bounded linear operators from H to L 2 (Ω) is called a cylindrical Wiener process on H if is defined as a limit of integrals of adapted step processes. By a localization argument stochastic integrals can be extended to the class N ([0, T ]) of all H -strongly measurable and adapted processes φ : [0, T ] × Ω → γ(H; E) which are in L 2 ((0, T ); γ(H; E)) a.s. (see [21]).
Theorem 2.7. There exists c(E) > 0 depending only on E such that is an E -valued continuous martingale (or local martingale) and a Gaussian process.
The proof for Proposition 2.8 is very similar to one in [4]. So, we omit it. Let us finally restate the Kolmogorov continuity theorem. This theorem gives a sufficient condition for a stochastic process to be Hölder continuous.
When ζ is a Gaussian process, the condition (2.1) can be weakened.
Then, there exists a modification of ζ whose P -almost all trajectories are Hölder continuous functions with an arbitrarily smaller exponent than ǫ 2 .

Strict and mild solutions.
Let us restate the problem (1.1). Throughout this paper, we consider (1.1) in a UMD Banach space E of type 2, where (i) A is a sectorial operator on E.
(ii) W is a cylindrical Wiener process on a separable Hilbert space H, and is defined on a complete filtered probability space (Ω, F, F t , P).
Remark 2.14. A strict solution is a mild solution. The inverse is however not true in general ( [4]).
3. The deterministic case. In this section, we consider the deterministic case of (1.1), i.e. the equation (For this case, the UMD and type 2 properties are unnecessary.) Suppose that for some 0 < σ < β ≤ 1 and − ∞ < α 1 < 1.
Let us fist consider the case where the initial value ξ is arbitrary in E.
with the estimate Furthermore, if α 1 ≤ 0, then X becomes a strict solution of (3.1) possessing the regularity: and satisfying the estimate Here, the constant C depends only on the exponents.
Proof. The proof is divided into four steps.
We have The integral in the right-hand side of the latter equality is well-defined and continuous on (0, T ]. This is because by (2.5), (2.6) and (F1), Since A 1−α 1 is closed, we observe that Thus, The function X defined by Step 2. Let us verify the estimate (3.2).
Hence, in any case of α 1 , where C is some positive constant depending only on the exponents. Using (3.4) and (3.5), we observe that Due to (2.5), (2.6) and (2.8), it is then seen that Thus, (3.2) has been verified.
Step 3. Let us show that if α 1 ≤ 0, then • X is a strict solution of (3.1).
In view of (3.5), Since we obtain that . . ) be the Yosida approximation of A. Then, A n satisfies (A) uniformly and generates an analytic semigroup S n (t) (see e.g., [26]). Furthermore, for any 0 ≤ ν < ∞ and 0 < t ≤ T, Consider a function X n defined by We have Let us estimate A n X n (t) . When −1 ≤ α 1 ≤ 0, (2.5), (3.8) and (3.9) give A n X n (t) In the meantime, when α 1 < −1, A n X n (t) Therefore, in any case of α 1 , there exists C 1 > 0 independent of n such that where Let us verify that Y is continuous on (0, T ]. Take 0 < t 0 ≤ T . By using (2.5) and (2.6), for every t ≥ t 0 , Thus, it is easily seen that Similarly, we obtain that The function Y is hence continuous at t = t 0 and then at every point in On the other hand, due to (3.7) and (3.8), We thus arrive at As a consequence, Meanwhile, since A α 1 n is bounded, by some direct calculations, From this equation, for any 0 < ǫ ≤ T, Using (3.10), the Lebesgue dominate convergence theorem applied to (3.11) provides that This shows that X is differentiable on [ǫ, T ]. Since ǫ is arbitrary in (0, T ], we conclude that By combining (3.6) and (3.13), the first statement has been verified: On the other hand, taking ǫ → 0 in (3.12), we have the integral t 0 F (s)ds is well-defined. The latter equality then shows that t 0 AX(s)ds is well-defined and that Therefore, X is a strict solution of (3.1). We thus arrive at the second statement.
Step 4. Let us prove that X satisfies the estimate (3.3) when α 1 ≤ 0. Thanks to (3.10), This together with (3.12) gives Hence, there exists C > 0 depending only on the exponents and the constants in (2.6), (2.7) and (2.8) such that By Steps 1-4, the proof is complete.
Let us now consider the case where the initial value ξ belongs to a subspace of E, namely D(A β−α 1 ). The below theorem shows maximal regularity for both initial value ξ and function F .
In addition, X satisfies the estimate: Furthermore, when α 1 ≤ 0, X becomes a strict solution of (3.1) possessing the regularity: and with the estimate: Here, C is some positive constant depending only on the exponents.
Proof. The proof is divided into several steps.
The condition (2.2) is hence fulfilled. In addition, (2.6) and (2.7) give On the other hand, for 0 < s < t ≤ T , Therefore, (2.6) gives The conditions (2.3) and (2.4) are then satisfied. We hence conclude that Furthermore, thanks to (3.17) and (3.18), Proof for J 2 . The norm of J 2 is evaluated by using (2.5) and (F1): Therefore, We now observe that for 0 < s < t ≤ T, The norm of J 21 (t, s) is estimated by using (2.5) and (2.6): The norm of J 22 (t, s) is evaluated as follows: Using the decomposition: In order to handle the first integral of the latter equality, we have Note that for every s 2 ≤ u ≤ s, In the meantime, Thanks to the estimates (3.22), (3.23), and (3.24), there exists C 2 > 0 depending only on exponents such that The norm of the last term, J 23 (t, s), is evaluated by using (2.5) and (2.8): and Proof for J 3 . Since t 1−β A −α 1 F (t) has a limit as t → 0, Furthermore, (2.5), (2.7) and (2.8) give We now write The norm of the first term in the right-hand side of the equality is estimated by using (2.5), (2.7) and (2.8): Meanwhile, the norm of the second term is evaluated by: This means that there exists C 4 > 0 such that In addition, since s 1−β A −α 1 F (s) has a limit as s → 0, (2.9) gives According to (3.27), (3.28), (3.29) and (3.30), it is seen that We have thus proved that and that there exists C > 0 depending only on the exponents such that The estimate (3.14) then follows from (3.16) and (3.31).
Step 3. Let us show the remain of the theorem. Consider the case α 1 ≤ 0. Thanks to Theorem 3.1, X is a strict solution of (3.1) in C 1 ((0, T ]; E). On the account of (3.12), it is seen that In addition, (3.15) follows from (3.14) and the estimate: By Steps 1-3, the proof is now complete.

The stochastic case.
Let us consider the stochastic evolution equation (1.1), where F and G satisfy the following conditions: For some 0 < σ < β − 1 2 ≤ 1 2 and −∞ < α 1 < 1, With the σ and β as above and some −∞ < α 2 < 1 2 − σ, Throughout this section, the notation C stands for a universal constant which is determined in each occurrence by the exponents.
Denote by W G the stochastic convolution defined by The next two theorems show the regularity of W G . 1}. Then, for any 0 < γ < σ and 0 < ǫ ≤ T . In addition, where C is some constant depedning only on the exponents. Furthermore, if Proof. We divide the proof into three steps.
Step 2. Let us verify that there exists an increasing function m(·) defined on (0, T ] such that lim t→0 m(t) = 0 and
By Theorem 2.7, A κ 2 W G is a Gaussian process on (0, T ]. Thanks to the estimate in Step 2, Theorem 2.10 applied to A κ 2 W G provides that a.s. In order to prove that E A κ 2 W G ∈ F β,σ ((0, T ]; R), we again use the estimate in Step 2. We have Then, This imlies that On the other hand, repeating the argument as in (4.2) and (4.3), we have Thereby, Hence, By (4.4), (4.5) and (4.6), we conclude that Thanks to Steps 1 and 3, the proof of the theorem is now complete.
Using the Fubini theorem, we have Hence, The theorem has been thus proved.
Since κ ≤ 1 − α 1 , we obtain that This means that E A κ X satisfies (2.2). On the other hand, by Theorem 3.2, Hence, it is easily seen that In addition, Theorem 4.1 provides that Using the inequality: it is easily seen that E A κ X satisfies (2.3) and (2.4).
In addition, since we arrive at The proof is now complete.
The following corollary is a direct consequence of Theorem 4.3. 5. An application to heat equations. Consider the following nonlinear stochastic heat equation: is the Laplace operator.
Let us first make precise what we mean by ∂W ∂t (t, x). Since the process W (t, x) depends on both position x and time t, it is often chosen of the form where {e j } ∞ j=1 is an orthonormal and complete basis of some Hilbert space, say H 0 , and {B j } ∞ j=1 is a family of independent real-valued standard Wiener processes on a filtered, complete probability space (Ω, F, {F t } t≥0 , P).
It is known that (see, e.g., [4]) the series ∞ j=1 e j B j (t) converges to a cylindrical Wiener process on a separable Hilbert space H ⊇ H 0 . (The embedding of H 0 into H is a Hilbert-Schmidt operator.) We still denote the cylindrical Wiener process by {W (t), t ∈ [0, T ]}. The noise term σ(t, x) ∂W ∂t (t, x) in (5.1) is therefore considered as G(t) ∂W ∂t (t), where W is the cylindrical Wiener process on H and G(t), 0 ≤ t ≤ T, are linear operators from H to some Banach space.
Let A be a realization of the differential operator −∆ + a(x) in H −1 (R d ). Thanks to [26,Theorem 2.2], A is a sectorial operator on H −1 (R n ) with domain D(A) = H 1 (R n ). As a consequence, (−A) generates an analytical semigroup on H −1 (R n ).
Using A, F and G, the equation (5.1) is formulated as a problem of the form (1.1) in H −1 (R d ). Consider separately the deterministic case and the stochastic case.