Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-018-0639-4

Well-Posedness of Mild Solutions to Stochastic Parabolic Partial Functional Differential Equations Chaoliang Luo1

· Shangjiang Guo2 · Aiyu Hou1

Received: 13 July 2017 / Revised: 18 May 2018 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Abstract In this paper, we study the well-posedness of mild solutions to stochastic parabolic partial functional differential equations with space–time white noise. Firstly, we establish an existence–uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence–uniqueness property under the global/local Lipschitz condition but without assuming the linear growth condition. In particular, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of nonnegative mild solutions under the linear growth condition without assuming the Lipschitz condition. Keywords Stochastic partial functional differential equation · Non-Lipschitz · Mild solution · Existence · Uniqueness Mathematics Subject Classification 60H15 · 60G52 · 34K50

Communicated by Ahmad Izani Md. Ismail. This research was partially supported by grants from the National Natural Science Foundation of China (No. 11671123), Natural Science Foundation of Hunan Province (No. 14JJ1025), and Science Research Project of Hunan Province Education Office (Nos. 17C0467 and 17C0468).

B

Chaoliang Luo [email protected]

1

College of Science, Hunan University of Technology, Zhuzhou 412007, Hunan, People’s Republic of China

2

College of Mathematics and Econometrics, Hunan University, Changsha 410082, People’s Republic of China

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1 Introduction It is known that stochastic partial differential equations (SPDEs) are appropriate mathematical models for many multiscale systems with uncertain and fluctuating influences. In the past few decades, the theoretical research of SPDEs has attracted a large number of research workers and has already achieved fruitful results. There are many interesting problems, such as well-posed problems, blow-up problems, stability, random attractors and ergodicity, which have been investigated for different kinds of SPDEs. We refer the readers to [2,4,15,20–22,24,25] for details. In paper [15], Shiga investigated the following SPDE  ∂u(t,x)

= u(t, x) + b (u(t, x)) + a (u(t, x)) W˙ (t, x), t ≥ 0, x ∈ R, ∂t (1.1) u(0, x) = f (x),

2 where  = ∂∂x 2 , W˙ (t, x) is a space–time white noise, and a(u) and b(u) : R → R are continuous functions. Such an equation arises in many fields, such as population biology, quantum field, statistical physics and neurophysiology, see [9,18]. Noting that support compactness of solutions propagates with passage of time (SCP), Shiga showed that the SCP property and the strong positivity are two contrasting properties of solutions for (1.1). Moreover, Shiga also showed that if the coefficients a(u) and b(u) satisfy the global Lipschitz condition or a(u) and b(u) are continuous and satisfy the linear growth condition, then (1.1) has a continuous solution and the pathwise uniqueness or uniqueness in law holds in some appropriate space, respectively. Xie [25] generalized the results of Shiga [15] and considered the Cauchy problem of the following stochastic heat equation



= 21 ∂ ∂u(t,x) + b (t, x, u(t, x)) + σ (t, x, u(t, x)) ∂ x2 u(0, x) = u 0 (x), ∂u(t,x) ∂t

2

2 W (t,x)

∂t∂ x

,

t ≥ 0, x ∈ R, (1.2)

where {W (t, x), t ≥ 0, x ∈ R} is a two-sided Brownian sheet and the coefficients b, σ : [0, ∞) × R × R → R are continuous, nonlinear functions. By utilizing the successive approximations, Xie [25] investigated the existence of mild solutions to (1.2) under the weaker conditions than the Lipschitz condition. The above results are based on the fact that the future state of the system is dependent of the past states and is determined solely by the present. However, the rate of change of the state variable of a more realistic model depends on not only the current but also the historical and even the future states of the system. Stochastic functional differential equations (SFDEs) give a mathematical formulation for such models and have been extensively studied in the literature [1,5,7,8,10,11,14,16,17,26–28]. Particularly, the existence and uniqueness of mild solutions to stochastic partial functional differential equations (SPFDEs) have received a great deal of attention. Under the global Lipschitz condition and the linear growth condition, Taniguchi [17] and Luo [12] studied the existence and uniqueness of mild solutions to SPFDEs by the wellknown Banach fixed-point theorem and strong approximating system, respectively.

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Govindan [6] investigated the existence, uniqueness and almost sure exponential stability of stochastic neutral partial functional differential equations under the global Lipschitz condition and the linear growth condition by the stochastic convolution. However, we know that the nonlinear terms of some systems perhaps do not obey the global Lipschitz condition or the linear growth condition, or even the local Lipschitz condition. Therefore, in this paper we shall promote the works of [15,25] and investigate the following SPFDE 

2 2 W (t,x) ∂u(t,x) = 21 ∂ ∂u(t,x) + b (t, x, u t (x)) + σ (t, x, u t (x)) ∂ ∂t∂ ∂t x , x2 u t0 = ξ = {ξ(θ, x) : −τ ≤ θ ≤ 0, x ∈ R},

t0 ≤ t ≤ T, x ∈ R,

(1.3) where u t (x) =: {u(t + θ, x), −τ ≤ θ ≤ 0, x ∈ R} is regarded as an Ft -measurable C([−τ, 0] × R; R)-valued stochastic process, u t0 ∈ C([−τ, 0] × R; R) is an Ft0 measurable random variable, and τ is a fixed positive constant. The coefficients b, σ : [t0 , T ] × R × C([−τ, 0] × R; R) → R are continuous Borel measurable functions, and {W (t, x), t ≥ t0 , x ∈ R} is a two-sided Brownian sheet. The main aim of this paper is to study the well-posedness and other properties of mild solutions to (1.3) under some appropriate conditions. The paper is organized as follows. Firstly, we establish an existence–uniqueness theorem under the global Lipschitz condition and the linear growth condition. Secondly, we show the existence– uniqueness property under the global Lipschitz condition without the linear growth condition. Then, some sufficient conditions ensuring the existence–uniqueness are given under the local Lipschitz condition without assuming the linear growth condition. Furthermore, we consider the existence and uniqueness under the weaker condition than the Lipschitz condition. Finally, we obtain the nonnegativity and comparison theorems and utilize them to investigate the existence of mild solutions without assuming the Lipschitz condition.

2 Preliminaries In this section, we introduce some notations and recall some basic definitions. Let (, F, {Ft }t≥t0 , P) be a complete probability space with the filtration {Ft }t≥t0 , which satisfies the usual condition. The random field {W (t, x), t ≥ t0 , x ∈ R} is a two-sided Brownian sheet if it is an Ft -adapted centered Gaussian random field with the covariance E [W (s, x)W (t, y)] =

1 min{s, t}(|x| + |y| − |x − y|), for s, t ≥ t0 , x, y ∈ R. 2

For the two-sided Brownian sheet, we refer the readers to the monograph [19]. In the sequel, we shall suppose that W generates an {Ft }t≥t0 -martingale measure and investigate the stochastic integral with respect to W introduced by Evans [3] and Walsh [23].

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Let Cr , r ∈ R, denote the family of all continuous Borel measurable functions ψ(x) defined on R with the norm ψr =: ess supx∈R |ψ(x)|e−r |x| < ∞. Let C([− τ, 0] × R; R) be the space of all continuous R-valued functions ϕ(θ, x) defined on [− τ, 0]×R 1 with the norm defined by ϕ =: sup−τ ≤θ≤0,x∈R e− 2 r |x| | ϕ(θ, x) |, where τ > 0 and r > 0. Let Mr, p,T , p ≥ 2, be the family of all random fields {X (t, x), t ≥ t0 , x ∈ R} defined on (, F, {Ft }t≥t0 , P) such that X r, p,T =:

sup

s∈[t0 ,T ],x∈R

  e−r |x| E |X (s, x)| p < ∞.

Following Borel–Cantelli lemma [3], Mr, p,T with the norm  · r, p,T is a Banach space for each r > 0 and p ≥ 2. For the reader’s convenience, if p = 2 we abbreviate Mr, p,T and X r, p,T as Mr,T and X r,T . In the sequel, unless otherwise specified, we always assume that ξ ∈ C([− τ, 0] × R; R) is an Ft0 -measurable random variable, ξ(0, x) is an Ft0 -measurable Cr -valued random variable, and r is a positive constant. Definition 2.1 [25] Let J = [t0 − τ, T ]. A predictable random field u(t, x) defined on J × R is called a mild solution of (1.3), if for every t0 ≤ t ≤ T , it satisfies the following stochastic integral equation (SIE)  u(t, x) =

R

+

g(t, x, y)ξ(0, y)dy +  t t0

R

 t t0

R

g(t − s, x, y)b(s, y, u s (y))dyds

g(t − s, x, y)σ (s, y, u s (y))W (dy, ds),

(2.1) a.s.

where g(t, x, y) = (2π t)− 2 exp{− (x−y) 2t } is the fundamental solution of the homogeneous Cauchy problem of (1.3). 1

2

Remark 2.1 For any t ∈ [t0 , T ], if an Ft -measurable functional u(t, ·) = u(t, ·, ω) is a Cr -valued stochastic process and satisfies (2.1), we say u(t, x) is a Cr -valued solution of (1.3). Definition 2.2 [25] The solution u(t, x) of (1.3) on J × R is said to be unique if every other solution u(t, ¯ x) on J × R is indistinguishable from it, that is, P(u(t, x) = u(t, ¯ x) for all t ∈ J and x ∈ R) = 1. Definition 2.3 [27] The sample path u(t, x) = u(t, x, ω) of (1.3) is said to be explosive at t¯(ω) if for some ω ∈ , t¯(ω) < T , it satisfies limt→t¯− |u(t, x)| = ∞.

3 Main Results In the sequel, the letter c represents a positive constant depending on some parameters which may change occasionally its value from line to line, while ci and ci will denote some fixed positive constants.

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For the reader’s convenience, we firstly provide some fundamental inequalities on the heat kernel g(t, x, y) of (2.1), and Gronwall’s inequality and lemma of Kolmogorov. Lemma 3.1 [25] (i) For each r ∈ R and T > 0, there exists a constant c depending only on r , t0 and T such that  g(t, x, y)er |y| dy ≤ cer |x| , ∀x ∈ R, ∀t ∈ [t0 , T ]. (3.1) R

(ii) If 0 < p < 3, then there exists a positive constant c such that  t t0

(iii) If

R

|g(t − s, x, y)| p dyds ≤ ct

3− p 2

, ∀x ∈ R, ∀t ∈ [t0 , T ].

(3.2)

< p < 3, then there exists a positive constant c such that for all t ∈ [t0 , T ],

3 2

 t R

t0

|g(t − s, x, y) − g(t − s, x  , y)| p dyds ≤ c|x − x  |3− p , ∀x, x  ∈ R. (3.3)

(iv) If 1 < p < 3, then there exists a positive constant c such that for all t and t  (t0 ≤ t ≤ t  ≤ T ),  t t0

R

|g(t − s, x, y) − g(t  − s, x, y)| p dyds ≤ c|t − t  |

3− p 2

, ∀x ∈ R, (3.4)

and 

t t

 R

|g(t  − s, x, y)| p dyds ≤ c|t − t  |

3− p 2

, ∀x ∈ R.

(3.5)

Lemma 3.2 [3] Let υ(t) and f (t) be nonnegative, continuous functions defined for t0 ≤ t ≤ T , and let υ0 denote a nonnegative constant. If  υ(t) ≤ υ0 +

t

f (s)υ(s)ds

f or all t0 ≤ t ≤ T,

t0

then υ(t) ≤ υ0 e

t t0

f (s)ds

f or all t0 ≤ t ≤ T.

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Lemma 3.3 [25] Assume that {u(t, x), t ≥ 0, x ∈ R} is a real-valued random field. If there exist constants p > 0, κ > 0 and δ > 2 such that for any compact subset K, E[|u(t, x) − u(s, y)| p ] ≤ κer |x| (|t − s|δ + |x − y|δ ) for all |t − s| ≤ 1 and x, y ∈ K, then u(t, ·) has a continuous Cr -valued version. In what follows, we begin to state our main results. We firstly show that the Lipschitz condition and the linear growth condition can guarantee the existence and uniqueness of mild solutions to (1.3). Theorem 3.1 Suppose that b and σ satisfy the global Lipschitz condition and the linear growth condition on [t0 , T ] × R × C([− τ, 0] × R; R), that is, there exist two positive constants K and K¯ such that for all ϕ, ψ ∈ C([− τ, 0] × R; R), t ∈ [t0 , T ] and x ∈ R, |b(t, x, ϕ) − b(t, x, ψ)|2 + |σ (t, x, ϕ) − σ (t, x, ψ)|2 ≤ K ϕ − ψ2 , (3.6)  |b(t, x, ϕ)|2 + |σ (t, x, ϕ)|2 ≤ K¯ 1 + ϕ2 . (3.7) Suppose further that the initial value ξ satisfies Eξ 2 < ∞. Then, there exists a unique continuous Cr -valued solution u(t, ·) to (1.3). Moreover, the solution u(t, x) belongs to Mr,T . To show this theorem, we shall prepare the following lemma. Lemma 3.4 Assume that the linear growth condition (3.7) holds. If u(t, x) is a solution to Eq. (1.3) with initial value ξ satisfying Eξ 2 < ∞, then 

√ ur,T ≤ 1 + c3 Eξ 2 ec1 (T −t0 )+2c2 T −t0 ,

(3.8)

where c1 , c2 , c3 are constants specified in the following proof. In particular, u(t, x) belongs to Mr,T . Proof From (2.1), for t0 ≤ t ≤ T we have

2  

E[|u(t, x)|2 ] ≤ 3E g(t, x, y)ξ(0, y)dy R

2   t 

+ 3E g(t − s, x, y)b(s, y, u s (y))dyds t0

R

t0

R

2   t 

g(t − s, x, y)σ (s, y, u s (y))W (dy, ds) + 3E =: 0 (t, x) + 1 (t, x) + 2 (t, x).

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(3.9)

Well-Posedness of Mild Solutions to Stochastic Parabolic…

By Lemma 3.1, we have  0 (t, x) ≤ 3E

2 g(t, x, y)|ξ(0, y)|dy

R

 ≤ 3E

R

g(t, x, y)e

≤ ce

sup e

− 12 r |y|

2

|ξ(0, y)| dy

(3.10)

y∈R



r |x|



 1 2 r |y|



E sup e

−r |y|

|ξ(0, y)|

≤ c0 er |x| Eξ 2 .

2

y∈R

Note that sup

t∈[t0 −τ,T ],x∈R

e−r |x| |u(t, x)|2 ≤ ξ 2 +

sup

t∈[t0 ,T ],x∈R

e−r |x| |u(t, x)|2 ,

then combining Lemma 3.1 with Hölder and Burkhölder’s inequality, we obtain 1 (t, x) ≤ c

 t 

 g(t − s, x, y)dyds

t0 R t

 E

t0

 t

R



g(t − s, x, y)b2 (s, y, u s (y))dyds

g(t − s, x, y)(1 + Eu s 2 )dyds (3.11) t0 R    t ≤c g(t − s, x, y)er |y| sup e−r |y| (1 + Eu s 2 ) dyds ≤c

t0

≤ c1 er |x|

R



y∈R t

(1 + Eξ 2 + ur,s )ds,

t0

and 2 (t, x) ≤ cE

 t  t0

 t

 g (t − s, x, y)σ (s, y, u s (y))dyds 2

R

2

1 g(t − s, x, y)(1 + Eu s 2 )dyds √ t −s t0 R  t 1 ≤ c2 er |x| (1 + Eξ 2 + ur,s )ds. √ t −s t0

≤c

(3.12)

From (3.9)–(3.12), it follows that for t0 ≤ t ≤ T , e−r |x| E[|u(t, x)|2 ] ≤ c0 Eξ 2 +

  t c2 (1 + Eξ 2 + ur,s )ds, c1 + √ t −s t0

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which implies ur,t

  t c2 ≤ c0 Eξ  + c1 + √ (1 + Eξ 2 + ur,s )ds. t −s t0 2

Now, the Gronwall’s inequality [3] yields that 

√ 1 + Eξ 2 + ur,t ≤ 1 + (1 + c0 )Eξ 2 ec1 (t−t0 )+2c2 t−t0 . Consequently, ur,T ≤ (1 + c3 Eξ 2 )ec1 (T −t0 )+2c2

√

T −t0

,

where c3 =: 1 + c0 . Meanwhile, we can obtain that u(t, x) is an element of Mr,T .  Remark 3.1 Obviously, it is not difficult to show the boundedness of ur, p,T ( p > 2) by the same methods as those of Lemma 3.4, which implies u(t, x) ∈ Mr, p,T . Proof of Theorem 3.1. Continuity. Assuming that u(t, x) is the solution of (1.3), u(t, x) satisfies the following SIE  u(t, x) =

R

+

g(t, x, y)ξ(0, y)dy +

t0

 t R

t0

 t R

g(t − s, x, y)b(s, y, u s (y))dyds

g(t − s, x, y)σ (s, y, u s (y))W (dy, ds),

=: 1 (t, x) + 2 (t, x) + 3 (t, x). It is easy to prove that 1 (t, ·) and 2 (t, ·) are continuous Cr -valued stochastic processes by the regularity of heat kernel g(t, x, y) and the assumptions of b and ξ(0, y). Therefore, it is sufficient to show that 3 (t, ·) is a continuous Cr -valued stochastic process. By Lemma 3.1 and Remark 3.1, we see that for any compact subset K of R and some constant p > 12, E[|3 (t, x) − 3 (t, x  )| p ]  2p  t  (g(t − s, x, y) − g(t − s, x  , y))2 σ 2 (s, y, u s (y))dyds ≤ cE R

t0 

≤ c|x − x |

p−4 2

 t t0

≤ c(er |x| + e

r |x  |

R

p 2 |g(t − s, x, y) − g(t − s, x  , y)|E K¯ (1 + u s 2 ) dyds

)|x − x  |

p−4 2



(3.13) t t0

≤ c4 er |x| |x − x  |

123

p−4 2

(1 + Eξ  p + ur, p,s )ds

Well-Posedness of Mild Solutions to Stochastic Parabolic…

and E[|3 (t  , x  ) − 3 (t, x  )| p ] p    2 t g 2 (t  − s, x  , y)σ 2 (s, y, u s (y))dyds ≤ cE R

t

+ cE

 t 



R

t0

≤ cer |x| |t  − t|





(g(t − s, x , y) − g(t − s, x , y)) σ (s, y, u s (y))dyds

p−4 4



t t

+ cer |x| |t  − t|

p−4 4



2 2

 2p (3.14)

(1 + Eξ  p + ur, p,s )ds t

(1 + Eξ  p + ur, p,s )ds

t0

≤ c5 er |x| |t  − t|

p−4 4

for all t, t  with t0 ≤ t < t  ≤ T , t  − t ≤ 1, x, x  ∈ K. Combining (3.13) with (3.14), we have E[|3 (t  , x  ) − 3 (t, x)| p ] ≤ 2 p−1 E[|3 (t  , x  ) − 3 (t, x  )| p ] + 2 p−1 E[|3 (t, x) − 3 (t, x  )| p ]

 p−4 p−4 ≤ cer |x| |x − x  | 2 + |t  − t| 4 for all t  − t ≤ 1, x, x  ∈ K. This, together with Lemma 3.3, implies that u(t, ·) has a continuous Cr -valued version. Uniqueness Assume that u(t, x) and u(t, ¯ x) are two solutions of (1.3). By Lemma 3.4, both of them belong to Mr,T , and then for any t ∈ [t0 , T ] and x ∈ R we have E[|u(t, x) − u(t, ¯ x)|2 ]   t 

2 

≤ 2E g(t − s, x, y)(b(s, y, u s (y)) − b(s, y, u¯ s (y)))dyds R

t0

(3.15)   t 

2 

+ 2E g(t − s, x, y)(σ (s, y, u s (y)) − σ (s, y, u¯ s (y)))W (dy, ds) t0

R

= I1 (t, x) + I2 (t, x). By (3.1), (3.3) and (3.6), applying Hölder’s inequality and Theorem 1.7.1 in [13] we have  t   g(t − s, x, y)dyds I1 (t, x) ≤ c t

E

R

0 t  t0

R

 g(t − s, x, y)|b(s, y, u s (y)) − b(s, y, u¯ s (y))|2 dyds

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≤c

 t

g(t − s, x, y)Eu s − u¯ s 2 dyds (3.16)    t r |y| −r |y| 2 ≤c g(t − s, x, y)e E|u(l, y) − u(l, ¯ y)| dyds sup e t0

R

t0

R

≤ c6 er |x|

t0 ≤l≤s,y∈R

 t sup

t0 ≤l≤s,y∈R

t0



e−r |y| E|u(l, y) − u(l, ¯ y)|2 ds,

and  t

1 g(t − s, x, y)Eu s − u¯ s 2 dyds t −s t0 R  t 1 ≤c g(t − s, x, y)er |y| √ t − s R t  0 

I2 (t, x) ≤ c

√

sup

t0 ≤l≤s,y∈R

≤ c7 er |x|



t t0

(3.17)

e−r |y| E|u(l, y) − u(l, ¯ y)|2 dyds 



1 √ t −s

sup

t0 ≤l≤s,y∈R

e−r |y| E|u(l, y) − u(l, ¯ y)|2 ds.

Combining (3.15), (3.16) and (3.17), we deduce that E[|u(t, x) − u(t, ¯ x)| ] ≤ e  2

r |x|

  t c7 c6 + √ t −s t0

sup

t0 ≤l≤s,y∈R



e−r |y| E|u(l, y) − u(l, ¯ y)|2 ds.

Therefore, we have u − u ¯ r,t ≤

  t c7 u − u ¯ r,s ds. c6 + √ t −s t0

From Lemma 3.2, it follows that u − u ¯ r,t = 0, t ∈ [t0 , T ], which implies u(t, x) = u(t, ¯ x) for all t ∈ [t0 , T ] and x ∈ R, a.s. Existence We utilize the following successive approximation scheme. Define 0 u 0 (t, x) = R g(t, x, y)ξ(0,  y)dy, u t0 (t, x) = R g(t, x, y)ξ(θ, y)dy. For n = n 1, 2, . . ., we set u t0 (t, x) = R g(t, x, y)ξ(θ, y)dy and define

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Well-Posedness of Mild Solutions to Stochastic Parabolic…

 u n (t, x) =

 t

g(t − s, x, y)b(s, y, u n−1 (y))dyds s (3.18)  t g(t − s, x, y)σ (s, y, u n−1 (y))W (dy, ds). + s R

g(t, x, y)ξ(0, y)dy +

t0

t0

R

R

Now, we show that {u n (t, x), n = 0, 1, 2, . . .} is uniformly bounded in Mr,T by induction. Since Eξ 2 < ∞, by the same technique as the proof of Lemma 3.4, for t ∈ [t0 , T ] we can obtain   t c2 E[|u (t, x)| ] ≤ c0 e Eξ  + e (1 + Eξ 2 )ds c1 + √ t −s t0  √ ≤ c0 er |x| Eξ 2 + c1 (t − t0 ) + 2c2 t − t0 (1 + Eξ 2 )er |x| . 1

r |x|

2

2

r |x|

It follows that  √ u 1 r,t ≤ c0 Eξ 2 + c1 (t − t0 ) + 2c2 t − t0 (1 + Eξ 2 ), for t ∈ [t0 , T ], which implies u 1 r,T < +∞. Therefore, we have u 1 (t, x) ∈ Mr,T . Let us assume u n (t, x) ∈ Mr,T for some n; then, there exists a constant c which satisfies u n r,t ≤ c for t ∈ [t0 , T ]. From (3.18), it follows that  t

 c2 (1 + Eu ns 2 )ds t −s t0  √ ≤ c0 er |x| Eξ 2 + c1 (t − t0 ) + 2c2 t − t0 (1 + c + Eξ 2 )er |x| ,

E[|u n+1 (t, x)|2 ] ≤ c0 er |x| Eξ 2 + er |x|

c1 + √

and hence that  √ u n+1 r,t ≤ c0 Eξ 2 + c1 (t − t0 ) + 2c2 t − t0 (1 + c + Eξ 2 ) for t ∈ [t0 , T ], which means that u n+1 (t, x) ∈ Mr,T . Therefore, {u n (t, x), n = 0, 1, 2, . . .} is uniformly bounded in Mr,T . For any two nonnegative integers m, n, using the similar methods as above, we obtain for all t ∈ [t0 , T ] and x ∈ R, sup

t∈[t0 ,T ],x∈R

e−r |x| E|u n (t, x) − u m (t, x)|2

 (3.19)   t c7 −r |y| n−1 m−1 2 c6 + √ ≤ e E|u (l, y) − u (l, y)| ds. sup t −s t0 l∈[t0 ,s],y∈R

Since {u n (t, x), n = 0, 1, 2, . . .} is uniformly bounded in Mr,T , we have  sup m,n

sup

t∈[t0 ,T ],x∈R



2 e−r |x| E u n (t, x) − u m (t, x) < ∞.

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Applying Fatou’s lemma to (3.19), we get  lim

 sup

m,n→∞

t∈[t0 ,T ],x∈R

e

−r |x|

E|u (t, x) − u (t, x)| n

m

2

    t c7 lim c6 + √ e−r |y| E|u n−1 (l, y) − u m−1 (l, y)|2 ds. ≤ sup t − s m,n→∞ l∈[t0 ,s],y∈R t0

  It is not difficult to see that limm,n→∞ sups∈[t0 ,t],y∈R e−r |y| E|u n (s, y) − u m (s, y)|2 is nonnegative and monotonic in t ∈ [t0 , T ]. Using Lemma 3.2, we deduce that lim u n − u m r,t = 0, t ∈ [t0 , T ],

m,n→∞

which means that {u n (t, x), n = 0, 1, 2, . . .} is a Cauchy sequence of the Banach space Mr,T . Let u(t, x) denote the limit of {u n (t, x), n = 0, 1, 2, . . .}. We pass to limits in the definition of u n (t, x), to prove that u(t, x), t ∈ [t0 , T ], x ∈ R satisfies (2.1) a.s. which implies that u(t, x) is the solution of (1.3). Therefore, the proof of Theorem 3.1 is completed.  Now, we give an existence–uniqueness theorem for (1.3) under the global Lipschitz condition without the linear growth condition, which will be a generalization of Theorem 3.1. Theorem 3.2 Suppose that for every a ∈ [t0 , T ) and t ∈ [t0 , a], b(t, ·, 0) ∈ Cr and σ (t, ·, 0) ∈ Cr .

(3.20)

And suppose further that b and σ satisfy the Lipschitz condition (3.6). Then, there exists a unique continuous Cr -valued solution u(t, ·) to (1.3) on [t0 − τ, a]. Remark 3.2 From (3.6), it follows that |b(t, x, u t (x))|2 ≤ 2|b(t, x, 0)|2 + 2|b(t, x, u t (x)) − b(t, x, 0)|2 ≤ 2|b(t, x, 0)|2 + 2K u t 2 , |σ (t, x, u t (x))|2 ≤ 2|σ (t, x, 0)|2 + 2|σ (t, x, u t (x)) − σ (t, x, 0)|2 ≤ 2|σ (t, x, 0)|2 + 2K u t 2 ,

(3.21) (3.22)

where t ∈ [t0 , a] and x ∈ R. Combining (3.21), (3.22) and the conditions of Theorem 3.2, it is not difficult to show the validity of Theorem 3.2 by the same method as that of Theorem 3.1. So we omit the details. It is well known that the global Lipschitz condition is somewhat restrictive, so the next question will be whether the conditions of Theorem 3.2 can be further relaxed. We may wonder if there is a local solution. If so, what is the maximum existence interval of the local solution? The following theorem is related to these questions.

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Well-Posedness of Mild Solutions to Stochastic Parabolic…

Theorem 3.3 Assume that the condition (3.20) holds and the initial value ξ satisfies Eξ 2 < ∞. If b and σ satisfy the local Lipschitz condition on [t0 , T ) × R × C([−τ, 0] × R; R), that is, for every integer n ≥ 1, there exists a positive constant K n such that |b(t, x, ϕ) − b(t, x, ψ)|2 + |σ (t, x, ϕ) − σ (t, x, ψ)|2 ≤ K n ϕ − ψ2 , for all t ∈ [t0 , T ), x ∈ R and ϕ, ψ ∈ C([−τ, 0] × R; R) satisfying ϕ ∨ ψ ≤ n. Then, there exists a stopping time η = η(ω) ∈ (t0 , a] such that (1.3) has a unique solution u(t, x) for t ∈ [t0 − τ, η) and the solution u(t, x) explodes at η if η(ω) < a. Otherwise, the solution u(t, x) exists globally in [t0 − τ, T ). Proof We shall utilize a truncation procedure to prove the theorem. For every integer n ≥ 1, we define truncation functions bn (t, x, u t (x)) and σn (t, x, u t (x)) as follows,

 n ∧ u t  u t (x) , σn (t, x, bn (t, x, u t (x)) = b t, x, u t 

 n ∧ u t  u t (x) , u t (x)) = b t, x, u t 

(3.23)

t where we set u u t  = 1 if u t ≡ 0. Then, for any a ∈ [t0 , T ), bn and σn satisfy the global Lipschitz condition (3.6) on [t0 , a] × R × C([−τ, 0] × R; R). Therefore, by Theorem 3.2, there exists a unique solution u n (t, x) in Mr,a to the following equation

 u n (t, x) =

 t

g(t − s, x, y)bn (s, y, (u n )s (y))dyds (3.24)  t g(t − s, x, y)σn (s, y, (u n )s (y))W (dy, ds) + R

g(t, x, y)ξ(0, y)dy +

t0

t0

R

R

with the initial condition  (u n )t0 =

ξ(θ, x), 0,

if ξ  ≤ n, if ξ  > n,

(3.25)

where − τ ≤ θ ≤ 0, x ∈ R. Define the following sequence of stopping time ηn = a ∧ inf{t ∈ (t0 , a] : |u n (t, x)| ≥ n for any x ∈ R}, where we set inf φ = ∞ if possible. From (3.23) and (3.25), it follows that bn+1 (s, y, (u n )s (y)) = bn (s, y, (u n )s (y)) = b(s, y, (u n )s (y)), σn+1 (s, y, (u n )s (y)) = σn (s, y, (u n )s (y)) = σ (s, y, (u n )s (y)),

(3.26)

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for all t ∈ [t0 , ηn ]. Thus, (3.24) and the following equation  u n+1 (t, x) =

g(t, x, y)ξ(0, y)dy  t + g(t − s, x, y)bn+1 (s, y, (u n+1 )s (y))dyds R

+

t0

R

t0

R

 t

g(t − s, x, y)σn+1 (s, y, (u n+1 )s (y))W (dy, ds)

have the same coefficients for t ∈ [t0 , ηn ]; moreover, their initial data conditions overlap in D = {u t (x) ∈ C([− τ, 0] × R; R) : u t  ≤ n}. Therefore, we obtain u n+1 (t, x) = u n (t, x), t ∈ [t0 − τ, ηn ], a.s. This further shows that ηn is increasing in n. Thus, we can define η(ω) = limn→∞ ηn . By utilizing the arguments of Xu [28], we can deduce that u(t, x) = limn→+∞ u n (t, x) satisfies the integral equation (2.1) for t ∈ [t0 , η). That is to say, (1.3) has a unique solution u(t, x) for t ∈ [t0 −τ, η) and the solution u(t, x) explodes at η(ω) if η(ω) < a. Otherwise, from the arbitrariness of a, the solution u(t, x) exists in [t0 − τ, T ). Thus, we complete the proof of Theorem 3.3.  In the following, we consider the existence and uniqueness of mild solutions to (1.3) under the weaker conditions than the Lipschitz condition. Theorem 3.4 Assume that the linear growth condition (3.7) holds. If there exist a strictly positive, nondecreasing and continuous function λ(t) defined on [t0 , T ] and a nondecreasing concave function φ(u) defined on R+ satisfying φ(0) = 0, φ(u) > 0  1 du = ∞, such that for all ϕ, ψ ∈ C([−τ, 0] × R; R), t ∈ for all u > 0, and 0+ φ(u) [t0 , T ] and x ∈ R, |b(t, x, ϕ) − b(t, x, ψ)|2 + |σ (t, x, ϕ) − σ (t, x, ψ)|2 ≤ λ(t)φ(ϕ − ψ2 ).(3.27) Then, there exists a unique solution u(t, x) to (1.3). Moreover, the solution belongs to Mr,T . To testify this theorem, for the reader’s convenience, we provide the following corollary of Bihari’s lemma. Lemma 3.5 [25] Assume that the functions λ(t) and φ(u) satisfy the conditions of Theorem 3.4. If there exists a nonnegative measurable function z(t) satisfying z(0) = 0 such that for some α ∈ (0, 21 ], 

t

z(t) ≤ 0

then z(t) ≡ 0 on [t0 , T ].

123

λ(s)φ(z(s)) ds, ∀t ∈ [t0 , T ], (t − s)α

Well-Posedness of Mild Solutions to Stochastic Parabolic…

Proof of Theorem 3.4. Uniqueness We suppose that u(t, x) and u(t, ¯ x) are two solutions of (1.3). From Lemma 3.4, it follows that both of them belong to Mr,T . Moreover, E[|u(t, x) − u(t, ¯ x)|2 ]   t 

2 

≤ 2E g(t − s, x, y)(b(s, y, u s (y)) − b(s, y, u¯ s (y)))dyds R

t

 0 t 

2 

+ 2E g(t − s, x, y)(σ (s, y, u s (y)) − σ (s, y, u¯ s (y)))W (dy, ds) R

t0

= U1 (t, x) + U2 (t, x). (3.28) By (3.1), (3.3) and (3.27), using Hölder’s inequality, we have  t   U1 (t, x) ≤ c g(t − s, x, y)dyds t

R

t0

R

 0t   t

≤c t 0

 

2 dyds g(t − s, x, y)E λ(s)φ u s − u¯ s 

g(t − s, x, y)er |y| λ(s)φ

R



sup

l∈[t0 ,s],y∈R

≤ ce

r |x|



t

e

−r |y|

≤ c8 e



λ(s)φ t

t0

and U2 (t, x) ≤ cE

 t  t0

 t

(3.29) dyds



t0

r |x|

E|u(l, y) − u(l, ¯ y)|

2

 sup

l∈[t0 ,s],y∈R

λ(s) φ √ t −s

e

−r |y|

E|u(l, y) − u(l, ¯ y)|

2

ds 



sup

l∈[t0 ,s],y∈R

e

−r |y|

E|u(l, y) − u(l, ¯ y)|

2

ds

 R

g 2 (t − s, x, y)(σ (s, y, u s (y)) − σ (s, y, u¯ s (y)))2 dyds



1 g(t − s, x, y)E λ(s)φ u s − u¯ s 2 dyds √ (3.30) t −s t0 R    t λ(s) ≤ c9 er |x| e−r |y| E|u(l, y) − u(l, ¯ y)|2 ds. φ sup √ t − s t0 l∈[t0 ,s],y∈R ≤c

From (3.28), (3.29) and (3.30), it follows that sup

t∈[t0 ,T ],x∈R



≤

t t0

e−r |x| E|u(t, x) − u(t, ¯ x)|2

(c8 + c9 )λ(s) φ √ t −s



 sup

l∈[t0 ,s],y∈R

e−r |y| E|u(l, y) − u(l, ¯ y)|2 ds.

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By Lemma 3.5, we have u − u ¯ r,t = 0, t ∈ [t0 , T ], which implies u(t, x) = u(t, ¯ x) for all t ∈ [t0 , T ] and x ∈ R, a.s. Existence Similar to the proof of Theorem 3.1, we introduce the same successive approximation sequence as (3.18). By the same arguments, we can easily deduce that u n (t, x) ∈ Mr,T and {u n (t, x), n = 0, 1, 2, . . .} is uniformly bounded in Mr,T . Therefore, for any two nonnegative integers m, n, we can obtain sup

t∈[t0 ,T ],x∈R



≤c

t t0

e−r |x| E[|u n (t, x) − u m (t, x)|2 ]

λ(s) φ √ t −s



 sup

l∈[t0 ,s],y∈R

e

−r |y|

E|u

n−1

(l, y) − u

m−1

(l, y)|

2

(3.31) ds.

Since {u n (t, x), n = 0, 1, 2, . . .} is uniformly bounded in Mr,T , we have  sup m,n

 sup

t∈[t0 ,T ],x∈R

e

−r |x|

E|u (t, x) − u (t, x)| n

m

2

≤ ∞.

Applying Fatou’s lemma to (3.31), we get  sup

lim

m,n→∞

 ≤c



t∈[t0 ,T ],x∈R t

t0

λ(s) φ √ t −s

e

−r |x|

E|u (t, x) − u (t, x)| n

m

2



 lim

m,n→∞

 sup

l∈[t0 ,s],y∈R

e

−r |y|

E|u

n−1

(l, y) − u

m−1

(l, y)|

2

ds.

It is easy to find that limm,n→∞ [sups∈[t0 ,t],x∈R e−r |x| E|u n (s, x) − u m (s, x)|2 ] is nonnegative and monotonic in t ∈ [t0 , T ]. Using Lemma 3.5, we deduce that lim u n − u m r,t = 0, t ∈ [t0 , T ],

m,n→∞

which implies that {u n (t, x), n = 0, 1, 2, . . .} is a Cauchy sequence of the Banach space Mr,T . Let u(t, x) denote the limit of {u n (t, x), n = 0, 1, 2, . . .}. We pass to limits in the definition of u n (t, x), to prove that u(t, x), t ∈ [t0 , T ], x ∈ R, satisfies (2.1) a.s. which implies that u(t, x) is the solution of (1.3). Therefore, the proof of Theorem 3.4 is completed.  Similar to Theorem 3.2, we can easily get the following result, which will be a generalization of Theorem 3.4. Theorem 3.5 In addition to condition (3.20), suppose that b and σ satisfy the nonLipschitz condition (3.27) with the functions λ(t), φ(u) given in Theorem 3.4. Then, there exists a unique solution u(t, x) to (1.3). Moreover, the solution belongs to Mr,T .

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Well-Posedness of Mild Solutions to Stochastic Parabolic…

Remark 3.3 Set ⎧ ⎪ ⎨ 0, 1 φ(u) = u(log(u −1 )) 2 , ⎪ 1 ⎩ ι(log(ι−1 )) 2 + φ  (ι)(u − ι), −

u = 0, 0 < u < ι, u ≥ ι,

 (ι) denotes the left derivative of φ(u) at the point where ι is sufficiently small and φ− ι. Then, φ(u) is a nondecreasing, concave function and satisfies the related conditions of Theorems 3.4 and 3.5.

To investigate the existence of mild solutions to (1.3) under the linear growth condition without the Lipschitz condition, we now prepare two results, i.e., nonnegativity and comparison theorems. Theorem 3.6 In addition to the assumptions of Theorem 3.1, suppose that b(t, x, u) and σ (t, x, u) ar e continuous in (x, u), a.s. b(t, x, 0) ≥ 0 and σ (t, x, 0) = 0, a.s.

(3.32) (3.33)

where (x, u) ∈ R × R. Let u(t, x) be the solution to (1.3) with nonnegative initial function ξ satisfying E[ξ 2 ] ≤ ∞. Then, P(u(t, x) ≥ 0 for all t ∈ [t0 , T ] and x ∈ R) = 1. Proof For any  > 0, we introduce a nonnegative, symmetric and dx (x) defined on R and satisfying  (x) = 0 for |x| >  and

 R

( (x))2 dx = 1.

Define a spatially correlated Brownian motion Wx (t)(x ∈ R) satisfying W˙ x (t) =

 R

 (x − y)W˙ (t, y)dy,

where W˙ (t, y) is a space–time white noise. In fact, for every x ∈ R, Wx (t) =    ˙ R W x (s)ds is a one-dimensional Brownian motion. Set  = (g(, x, y) − I )/ for  > 0. We consider the following equation 



u (t, x) = ξ(0, x) +  +

t t0

t t0

( u  (s, x) + b(s, x, u s (x)))ds

σ (s, x, 7u s (x))dWx (s), x ∈ R,

(3.34)

where the last term is a standard one-dimensional stochastic integral. Since b and σ satisfy the Lipschitz condition and the linear growth condition, it is not difficult to

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show that (3.34) has a unique solution for every nonnegative initial function ξ . We claim that P(u  (t, x) ≥ 0 for every t ∈ [t0 , T ] and x ∈ R) = 1.

(3.35)

Introduce a nonnegative function ϑ(u) = − min{u, 0}. By (3.7), there must be a positive constant K  such that b(s, x, u) ≥ −K  |u|. Applying Itô’s formula, for t ∈ [t0 , T ] and x ∈ R, we have E[ϑ(u  (t, x))] = −I (u  (t, x) ≤ 0) {ξ(0, x)   t    + E[ u (s, x) + b(s, x, u s (x))]ds 

t0

 t 1 E[ϑ(u  (s, x))]ds  t0   1 t g(, x, y)E[ϑ(u  (s, y))]dyds, +  t0 R

≤

K +

where I (·) denotes the indicator function of (·). Therefore, combining the Gronwall’s lemma with the nonnegativity of ϑ(u), we can obtain that E[ϑ(u  (t, x))] = 0 for every t ∈ [t0 , T ] and x ∈ R, which yields (3.35). Next, we define





g (t) = exp(t ) = e

− t

t

R  (t, x, y) = e− 

∞  n=0 ∞ 

 t n 

n!  t n

n=1



n!

t

g(n) =: e−  I + R  (t, x, y), g(n, x, y).

By an elementary calculation similar to [15], it is easy to verify that R  (t, x, y) satisfies the following estimates:   3 − 21 (i) R (R  (t, x, y))2 dy ≤ 2π t ; (ii) For some α > 0 and β > 0, 

t

R

|R  (t, x, y) − g(t, x, y)|dy ≤ e−  + α

(iii) For t ∈ [t0 , T ] and x ∈ R, lim→0

123

t 

t0 R [R

 1

 (s, x,

3

t

if 0 <

 ≤ β; t

(3.36)

y) − g(s, x, y)]2 dyds = 0.

Well-Posedness of Mild Solutions to Stochastic Parabolic…

By (3.34), we can obtain u  (t, x) which satisfies the following equation u  (t, x) =



g  (t, x, y)ξ(0, y)dy R  t + g  (t − s, x, y)b(s, y, u s (y))dyds R

t0 t

 +

e

− t−s 

t0

+

 t t0

R

σ (s,

y, u s (y))dWx (s)

(3.37)

R  (t − s, x, y)σ (s, y, u s (y))dWx (s)dy.

Assume that u(t, x) is the unique solution to (1.3) under the conditions of Theorem 3.1, by (2.1), (3.36) and (3.37); applying the methods in Lemma 3.4, we can get sup

sup sup e−r |x| E[|u  (t, x)|2 ] < ∞,

0<≤1 t0 ≤t≤T x∈R

sup sup e−r |x| E[|u(t, x)|2 ] < ∞.

t0 ≤t≤T x∈R

Thus,  E |u  (t, x) − u(t, x)|2   

2 



≤ c E (g (t, x, y) − g(t, x, y))ξ(0, y)dy

R

 

2 

t − t−s

 + E

e  b(s, x, u s (x))ds

t0   

2 

t

 

R (t − s, x, y)(b(s, y, u s (y)) − b(s, y, u s (y)))dyds

+E t0 R   

2 

t  (R (t − s, x, y) − g(t − s, x, y))b(s, y, u s (y))dyds

+ E

t0 R  t  2(t−s) + e−  E σ 2 (s, x, u s (x)) ds (3.38) t0

+

 t t0

R

 

E

R  (t − s, x, z) R

(σ (s, z, u s (z)) − σ (s, z, u s (z))) (y +

 t t0

R

t0

R

 

E

R  (t − s, x, z)

2  − z)dz dyds

R

2  (σ (s, z, u s (z)) − σ (s, y, u s (y))) (y − z)dz dyds 2  t   + R  (t − s, x, z) (y − z)dz − g(t − s, x, y) R

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  E |σ (s, y, u s (y))|2 dyds 8 n (, t, x). ≡ n=1

From (3.6)–(3.8), (3.32) and the above estimates of R  (t, x, y), u  (t, x) and u(t, x), it follows that lim sup sup n (, t, x) = 0 for 1 ≤ n ≤ 8 with n = 3, 6,

→0 t0 ≤t≤T x∈R

and 

t

3 (, t, x) + 6 (, t, x) ≤ c

t0

1 E[u s (y) − u s (y)2 ]ds √ t −s

for t ∈ [t0 , T ] and x ∈ R. Therefore, (3.38) can be translated into the form E[|u  (t, x) − u(t, x)|2 ] ≤ c



t

√

t0

1 E[u s (y) − u s (y)2 ]ds + o(, t), t −s

where t0 ≤ t ≤ T and o(, t) satisfies lim→0 supt0 ≤t≤T o(, t) = 0. When  → 0, we obtain sup

t∈[t0 ,T ],x∈R



≤c

t

t0

 ≤c

t

t0

E|u  (t, x) − u(t, x)|2

1 √ t −s 1 √ t −s



 sup

l∈[t0 ,s],y∈R

e−r |y| E|u  (l, y) − u(l, y)|2 ds 



sup

l∈[t0 ,s],y∈R



E|u (l, y) − u(l, y)|

2

ds.

So, by Lemma 3.2, we deduce that  lim

→0

 sup

t∈[t0 ,T ],x∈R

E|u  (t, x) − u(t, x)|2 = 0,

which, together with (3.35), implies that P(u(t, x) ≥ 0 for all t ∈ [t0 , T ] and x ∈ R) = 1. Thus, the proof of Theorem 3.6 is completed.



By Theorem 3.6, we immediately obtain the following comparison result. Corollary 3.1 Assume that bi (t, x, u) (i = 1, 2) and σ (t, x, u) satisfy the assumptions of Theorem (3.1). Let u i (t, x) be the solution to (1.3) associated with the

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Well-Posedness of Mild Solutions to Stochastic Parabolic…

coefficients bi (t, x, u) and σ (t, x, u), whose initial function ξ i satisfies E[ξ i 2 ] < ∞. Suppose further that bi (t, x, u) (i = 1, 2) and σ (t, x, u) ar e continuous in (x, u), a.s. b1 (t, x, u) ≥ b2 (t, x, u), f or t ∈ [t0 , T ] and x ∈ R, u ∈ R, a.s. ξ 1 (θ, x) ≥ ξ 2 (θ, x), −τ ≤ θ ≤ 0, x ∈ R. Then, P(u 1 (t, x) ≥ u 2 (t, x) f or all t ∈ [t0 , T ] and x ∈ R) = 1. Proof Set u(t, x) = u 1 (t, x) − u 2 (t, x) and ξ = ξ 1 − ξ 2 . Then, u(t, x) satisfies SIE (2.1), where b, σ : [t0 , T ] × R × C([−τ, 0] × R; R) → R are continuous Borel measurable functions defined by b(s, y, u s (y)) = b1 (s, y, u 1s (y)) − b2 (s, y, u 2s (y)), σ (s, y, u s (y)) = σ (s, y, u 1s (y)) − σ (s, y, u 2s (y)). It is easy to check that b(s, y, u s (y)) and σ (s, y, u s (y)) satisfy all conditions of Theorem 3.6; then, we can deduce that P(u(t, x) ≥ 0 for all t ∈ [t0 , T ] and x ∈ R) = 1, that is, P(u 1 (t, x) ≥ u 2 (t, x) for all t ∈ [t0 , T ] and x ∈ R) = 1, which completes the proof of Corollary 3.1.



To illustrate the efficiency of nonnegativity and comparison theorems of mild solutions to (1.3), here we give two examples to show that these results are applicable. Example 3.1 Consider the following SPFDE ⎧ 1 ∂u(t, x) 1 ∂ 2 u(t, x) ⎪ ⎪ ⎪ = + κ1 e− 2 r |x| u t (x) + κ0 ⎪ 2 ⎪ 2 ∂x ⎨ ∂t 1 ∂ 2 W (t, x) ⎪ , t0 ≤ t ≤ T, x ∈ R, + κ2 e− 2 r |x| u t (x) ⎪ ⎪ ∂t∂ x ⎪ ⎪ ⎩ u t0 = {ξ1 (θ, x) : −τ ≤ θ ≤ 0, x ∈ R},

(3.39)

where r > 0, κi (i = 0, 1, 2) are some nonnegative constants, and the initial function ξ1 (θ, x) satisfies E[ξ1 2 ] < ∞. In this equation, the functions b(t, x, u t (x)) = 1 1 κ1 e− 2 r |x| u t (x) + κ0 and σ (t, x, u t (x)) = κ2 e− 2 r |x| u t (x) satisfy the Lipschitz condition and the linear growth condition. Theorem 3.1 implies that there exists a unique solution to (3.39). It follows from Theorem 3.6 that the solution of (3.39) remains nonnegative if ξ1 (θ, x) ≥ 0.

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Example 3.2 Consider Eq. (3.39) and the following SPFDE ⎧ 1 ∂u(t, x) 1 ∂ 2 u(t, x) ⎪ ⎪ ⎪ = + κ1 e− 2 r |x| u t (x) + κ0 ⎪ 2 ⎪ 2 ∂x ⎨ ∂t 1 ∂ 2 W (t, x) ⎪ , t0 ≤ t ≤ T, x ∈ R, + κ2 e− 2 r |x| u t (x) ⎪ ⎪ ∂t∂ x ⎪ ⎪ ⎩ u t0 = {ξ2 (θ, x) : −τ ≤ θ ≤ 0, x ∈ R},

(3.40)

where r > 0, κi (i = 0, 1, 2) are nonnegative constants satisfying κ1 ≥ κ1 , κ0 ≥ κ0 , κ2 = κ2 , and the initial function ξ2 (θ, x) satisfies E[ξ2 2 ] < ∞. Suppose that ξ2 (θ, x) ≥ ξ1 (θ, x) for all −τ ≤ θ ≤ 0 and x ∈ R. Comparing the coefficients and initial condition of (3.39) with those of (3.40), we see that all conditions of Corollary 3.1 are satisfied. Therefore, the solution u  (t, x) of (3.40) and the solution u(t, x) of (3.39) satisfy P(u  (t, x) ≥ u(t, x) for all t ∈ [t0 , T ] and x ∈ R) = 1. Now, we prepare a lemma which will be used in the forthcoming proof. Lemma 3.6 [15] Assume that {u n (t, ·), 0 ≤ t ≤ 1, n ≥ 1} is a sequence of Cr -valued continuous processes. If there exist constants d > 0, c > 0 and δ > 2 such that E[|u n (t, x) − u n (s, y)|d ] ≤ c (|t − s|δ + |x − y|δ )

(3.41)

for all 0 ≤ t, s ≤ 1, n ≥ 1, and x, y ∈ R with |x − y| ≤ 1, then the sequence of probability distributions on C([0, 1] → Cr ) induced by {u n (t, ·)} is tight. We now give a theorem about the existence of mild solutions to (1.3) under the linear growth condition without the Lipschitz condition by using the above results. Theorem 3.7 Suppose that b, σ : [t0 , T ] × R × C([−τ, 0] × R; R) → R are continuous Borel measurable functions satisfying the linear growth condition. Assume further that b(t, x, 0) ≥ 0 and σ (t, x, 0) = 0. Then, for every nonnegative initial function ξ satisfying E[ξ 2 ] ≤ ∞, there exists a nonnegative solution u(t, x) to (1.3). Proof We first choose two sequences of functions bn (t, x, ϕ) and σ n (t, x, ϕ) (n ≥ 1), which are Lipschitz continuous for variable ϕ ∈ C([− τ, 0] × R; R) and satisfy |bn (t, x, ϕ)|2 + |σ n (t, x, ϕ)|2 ≤ K¯ (1 + ϕ2 ), bn (t, x, 0) ≥ 0 and σ n (t, x, 0) = 0,

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(3.42) (3.43)

Well-Posedness of Mild Solutions to Stochastic Parabolic…

and that bn (t, x, ϕ) and σ n (t, x, ϕ) converge to b(t, x, ϕ) and σ (t, x, ϕ) uniformly as n → ∞. Then, by Theorems 3.1 and 3.6, for each n ≥ 1 the following SPFDE ⎧ ∂u(t, x) 1 ∂ 2 u(t, x) ⎪ ⎪ ⎪ = + bn (t, x, u t (x)) ⎪ ⎪ 2 ∂x2 ⎨ ∂t ∂ 2 W (t, x) ⎪ , t0 ≤ t ≤ T, x ∈ R, + σ n (t, x, u t (x)) ⎪ ⎪ ∂t∂ x ⎪ ⎪ ⎩ u t0 = {ξ(θ, x) : −τ ≤ θ ≤ 0, x ∈ R}, has a unique nonnegative continuous Cr -valued solution u n (t, ·) satisfying  u n (t, x) =

+

R

g(t, x, y)ξ(0, y)dy  t + g(t − s, x, y)bn (s, y, u ns (y))dyds

 t t0

R

t0

(3.44)

R

g(t − s, x, y)σ n (s, y, u ns (y))W (dy, dx), a.s.

Using (3.42) and (3.43), it is easy to see that u n (t, x) satisfies (3.41). From Lemma 3.6, it follows that the family of probability distribution induced by u n (t, x) is tight, which implies that u n (t, x) converges to a process u(t, x). We pass to limits as n → ∞ in (3.44) to obtain the existence of solution to (1.3). By the assumptions of Theorems 3.6 and 3.7, we obtain P(u(t, x) ≥ 0 for all t ∈ [t0 , T ] and x ∈ R) = 1. This completes the proof of Theorem 3.7.  Remark 3.4 Under the assumptions of Theorem 3.7, we cannot obtain the uniqueness of mild solutions, but the result can guarantee the existence and nonnegativity of mild solutions of (1.3). The following example is given to illustrate the efficiency of Theorem 3.7. Here, the coefficients b(t, x, u t (x)) and σ (t, x, u t (x)) are not Lipschitz continuous for variable u t (x), but satisfy our framework. Example 3.3 Consider the following stochastic parabolic partial functional differential equation ⎧  1 ∂u(t, x) 1 ∂ 2 u(t, x) ⎪ ⎪ ⎪ = + e− 2 r |x| u t (x) ⎪ 2 ⎪ 2 ∂x ⎨ ∂t  1 ∂ 2 W (t, x) ⎪ , t0 ≤ t ≤ T, x ∈ R, + e− 2 r |x| u t (x) ⎪ ⎪ ∂t∂ x ⎪ ⎪ ⎩ u t0 = {ξ3 (θ, x) : −τ ≤ θ ≤ 0, x ∈ R},

(3.45)

where r > 0 and the initial function satisfying E[ξ3 2 ] ≤ ∞. It is obvious that the √ √ 1 1 functions b(t, x, u t (x)) = e− 2 r |x| u t (x) and σ (t, x, u t (x)) = e− 2 r |x| u t (x) are not Lipschitz continuous, but satisfy the linear growth condition if u t (x) ∈ C([−τ, 0] ×

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R; [0, 1]). Theorem 3.7 implies that there exists a nonnegative solution to (3.45) when ξ3 (θ, x) ≥ 0.

References 1. Bao, J., Hou, Z.: Existence of mild solutions to stochastic neutral partial functional differential equations with non-Lipschitz coefficients. Comput. Math. Appl. 59, 207–214 (2010) 2. Duan, J.: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, New York (2014) 3. Evans, L.: An Introduction to Stochastic Differential Equations. American Mathematical Society, New York (2013) 4. Guo, B., Zhou, G.: Ergodicity of the stochastic fractional reaction–diffusion equation. Nonlinear Anal. 109, 1–22 (2014) 5. Govindan, T.: Stability of mild solutions of stochastic evolution equations with variable delay. Stoch. Anal. Appl. 21, 1059–1077 (2003) 6. Govindan, T.: Almost sure exponential stability for stochastic neutral partial functional differential equations. Stochastics 77, 139–154 (2005) 7. He, M.: Global existence and stability of solutions for reaction diffusion functional differential equations. J. Math. Anal. Appl. 199, 842–858 (1996) 8. Hou, Z., Bao, J., Yuan, C.: Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps. J. Math. Anal. Appl. 366, 44–54 (2010) 9. Iwata, K.: An infinite dimensional stochastic differential equation with state space C(R). Probab. Theory Relat. Fields 74, 141–159 (1987) 10. Luo, C., Guo, S.: Existence and stability of mild solutions to parabolic stochastic partial differential equations driven by Lévy space–time noise. Electron. J. Qual. Theory Differ. Equ. 53, 1–25 (2016) 11. Luo, J.: Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays. J. Math. Anal. Appl. 342, 753–760 (2008) 12. Luo, J., Liu, K.: Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps. Stoch. Process. Appl. 118, 864–895 (2008) 13. Mao, X.: Stochastic Differential Equations and Applications. Horwood Publishing, Chichester (2007) 14. Niu, M., Xie, B.: Impacts of Gaussian noises on the blow-up times of nonlinear stochastic partial differential equations. Nonlinear Anal. RWA 13, 1346–1352 (2012) 15. Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math. 46, 415–437 (1994) 16. Taniguchi, T.: Successive approximations to solutions of stochastic differential equations. J. Differ. Equ. 96, 152–169 (1992) 17. Taniguchi, T., Liu, K., Truman, A.: Existence, uniqueness, and asymptotic behavior of mild solutions to stochastic functional differential equations in Hilbert spaces. J. Differ. Equ. 181, 72–91 (2002) 18. Pardoux, E.: Stochastic partial differential equations, a review. Bull. Sci. Math. 117, 29–47 (1993) 19. Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) 20. Wang, B.: Asymptotic behavior of non-autonomous fractional stochastic reaction–diffusion equations. Nonlinear Anal. 158, 60–82 (2017) 21. Wang, B.: Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations. Nonlinear Anal. 103, 9–25 (2014) 22. Waymire, E., Duan, J.: Probability and Partial Differential Equations in Modern Applied Mathematics. Springer, New York (2005) 23. Walsh, J.: An Introduction to Stochastic Partial Differential Equations. Springer, Berlin (1986) 24. Wang, W., Roberts, A.: Average and deviation for slow–fast stochastic partial differential equations. J. Differ. Equ. 253, 1265–1286 (2012) 25. Xie, B.: The stochastic parabolic partial differential equation with non-Lipschitz coefficients on the unbounded domain. J. Math. Anal. Appl. 339, 705–718 (2008) 26. Xu, D.: Further results on existence–uniqueness for stochastic functional differential equations. Sci. China Math. 56, 1169–1180 (2013) 27. Xu, D., Li, B., Long, S., Teng, L.: Corrigendum to Moment estimate and existence for solutions of stochastic functional differential equations. [Nonlinear Anal.: TMA 108 (2014) 128–143]. Nonlinear Anal. 114, 40–41 (2015)

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Well-Posedness of Mild Solutions to Stochastic Parabolic… 28. Xu, D., Zhou, W.: Existence–uniqueness and exponential estimate of pathwise solutions of retarded stochastic evolution systems with time smooth diffusion coefficients. Discrete Contin. Dyn. Syst. 37, 2161–2180 (2017)

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