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Impacts of Noise on a Class of Partial Differential Equations

This paper is concerned with effects of noise on the solutions of partial differential equations. We first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic parabolic equations. Then, we prove that noise could induce singularities (finite time blow up of solutions). Finally, we show that a stochastic Allen-Cahn equation does not have finite time singularities and the unique solution exists globally.


Introduction
Stochastic partial differential equations (SPDEs) are playing an increasingly important role in modeling complex phenomena in physics, geophysics and biology. In recent years, existence, uniqueness, stability, blow-up phenomenon, invariant measures and other properties of the solutions to SPDEs have been extensively investigated [2,14,15]. It is known that the existence and uniqueness of global solutions to SPDEs can be established under appropriate conditions ( [2,5]).
It is also known that certain deterministic parabolic or hyperbolic partial differential equations (even with polynomial nonlinearity) tend to develop singularities in finite time [22,8]. These equations only have local solutions. For example, the following equation where α > 0 and D ⊂ R n is a bounded domain with smooth boundary ∂D. It was shown ( [8]) that for a nonnegative initial condition u 0 ∈ L 2 (D) satisfying the solution develops finite time blow-up. Here λ 1 is the first eigenvalue of the Laplacian operator −∆, with zero Dirichlet boundary condition on D, and φ is the corresponding eigenfunction normalized so that φ L 1 (D) = 1. Kaplan [13] showed that the solution of (1.1) will blow up if the initial state is large enough. Fujita [8,9] proved that the Cauchy problem (1.1), with D = R n , has no global positive nontrivial solutions if 0 < α < 2/n, and every solution with arbitrarily small initial data blows up. The same is true for α = 2/n as shown by Hayakawa [12]. When α > 2/n, solutions with small initial conditions tend to zero as time increases. In this paper, we will prove that noise can lead to finite time blow-up. As for stochastic parabolic equations, the existence of solutions has been well studied [14,15,24]. For instance, for the following equation Da Prato-Zabczyk [21] obtained the existence of global solutions with additive noise (σ is constant). Manthey-Zausinger [16] considered (1.3), where σ satisfied the global Lipschitz condition. Dozzi and López-mimbela [6] considered equation (1.3) with f (u) ≥ u 1+α and σ(u) = u. They proved that the solution will blow up in finite time if initial data is large enough and will exist globally if the initial data is small enough; also see [19]. A natural question arises: If σ does not satisfy the global Lipschitz condition, what can we say about the solution? Will it blow up in finite time or exist globally? In a somewhat different case, Mueller [17] and, later, Mueller-Sowers [18] investigated the problem of a noise-induced explosion for a special case of equation (1.3), where f (u) ≡ 0, σ(u) = u γ with γ > 0 and W (x, t) is a space-time white noise. It was shown that the solution will explode in finite time with positive probability for some γ > 3/2.
In the present paper, we shall provide separate sufficient conditions to ensure that the solutions of (1.3) remain positive, or blow up in finite time. Moreover, we will consider a special case, i.e., stochastic Allen-Cahn equation, whose solution will not blow up in finite time and thus exists globally.
This paper is arranged as follows. After some preliminaries in the next section, we prove that the solutions of (1.3) remain positive under some assumptions in Section 3. Section 4 is concerned with the blow-up phenomenon of solution to (1.3) and we will obtain a new result, which shows that noise can indeed lead to finite time blow-up. In Section 5, we consider the existence of global solution, with help of a Lyapunov functional, for a stochastic Allen-Cahn equation, where the noise intensity σ(u) = u 1+β (β > 0) is not globally Lipschitz continuous.

Preliminaries
To set the stage for our study, we recall Chow's recent works [3,4] on finite time blow-up for the following SPDE ) is a symmetric, uniformly elliptic operator with smooth coefficients, σ is a given function, and W (x, t) is a Wiener random field defined in a complete probability space (Ω, F, P) with a filtration F t . The Wiener random field has mean EW (x, t) = 0 and its covariance function q(x, y) is defined by where (t ∧ s) = min{t, s} for 0 ≤ t, s ≤ T . The existence of strong solution of (2.1) has been studied by many authors [2,20]. To consider positive solutions, they start with the unique solution u ∈ C(D × [0, T ]) ∩ L 2 ((0, T ); H 2 ) for equation (2.1). Under the following conditions (P1) There exists a constant δ ≥ 0 such that and such that f (r, x, t) ≥ 0 for r ≤ 0 and x ∈D, t ∈ [0, T ]; (P3) The initial datum u 0 (x) onD is positive and continuous, Chow obtained the following result [3]. From (P1), it follows that σ = ku (k is a constant) if we only consider the case that σ = σ(u). The similar result can be found in [6,23]. The previous results on existence of global solution to (2.1) require that σ(u) satisfies a global positive Lipschitz condition. A natural question is what the solution becomes if σ(u) does not satisfy the global Lipschitz condition. In Section 2, we shall study the positive solutions to (2.1) with σ(u) = u γ (for γ > 1). In Section 4, we will examine the existence of global solution to (2.1) with σ(u) = u γ (for γ > 1).
We consider the eigenvalue problem for the elliptic equation Then, all the eigenvalues are strictly positive, increasing and the eigenfunction φ corresponding to the smallest eigenvalue λ 1 does not change sign in domain D, as known in [10]. Therefore, we normalize it in such a way that In paper [4], Chow assumed that the following conditions hold, where λ 1 is the first eigenvalue of (2.2) with ∆ replaced by A.
(N1) There exist a continuous function F (r) and a constant r 1 > 0 such that F is positive, convex and strictly increasing for r ≥ r 1 and satisfies There exists a constant M 1 > r 1 such that F (r) > λ 1 r for r ≥ M 1 ; (N3) The positive initial datum satisfies the condition (N4) The following condition holds Alternatively, he imposes the following conditions S on the noise term: (S1) The correlation function q(x, y) is continuous and positive for x, y ∈D such that for any positive v ∈ H and for some q 1 > 0; (S2) There exist a positive constant r 2 , continuous functions σ 0 (r) and G(r) such that they are both positive, convex and strictly increasing for r ≥ r 2 and satisfy σ(r, x, t) ≥ σ 0 (r) and σ 2 0 (r) ≥ 2G(r 2 ) for x ∈D, t ∈ [0, ∞); (S3) There exists a constant M 2 > r 2 such that q 1 G(r) > λ 1 r for r ≥ M 2 ; (S4) The positive initial datum satisfies the condition (S5) The following integral is convergent so that Proposition 2.2 [4, Theorem 3.1] Suppose the initial-boundary value problem (2.1) has a unique local solution and the conditions (P1)-(P3) are satisfied. In addition we assume that either the conditions (N1)-(N4) or the alternative conditions (S1)-(S5) given above hold true. Then, for a real number p > 0, there exists a constant T p > 0 such that Looking at the conditions in Propositions 2.1 and 2.2, it is clear that the condition (P1) is very stringent. A noise intensity like σ(u) = u 1+β , β > 0, does not satisfy the condition (P1). But the condition (S5) implies that G(r) ≥ r 1+ε , where ε is a positive constant. Therefore, in order to prove that noise can lead to blow up, we should delete or change the condition (P1). Moreover, if we assume that σ = σ(u), we see that the term − n i,j=1 a ij (x)∇u i ∇u j will not play any role. Unfortunately, it can not help to resolve the difficulty unless the elliptic operator is replaced by the p-Laplace operator; see remark 3.1.

Positive solutions
In this section, we will consider the positive solution to (1.3), which will be used to examine the finite time blow-up phenomenon.
For simplicity, we first consider the following stochastic parabolic Itô equation where m > 1 and b ∈ R. We assume that the covariance function q(x, y) is bounded, continuous and there is a constant q 0 > 0 such that sup x,y∈D |q(x, y)| ≤ q 0 and R q(x, x)dx < ∞.
In addition, we assume that where a 2 ∈ R, b i , β ≥ 0, and Similar to [3,4], let η(r) = r − denote the negative part of r for r ∈ R. Set Then one can check that k ε (r) has the following properties.
Lemma 3.1 [3, Lemma 3.1] The first two derivatives k ′ ε , k ′′ ε of k ε are continuous and satisfy the conditions: k ′ ε (r) = 0 for r ≥ 0; k ′ ε ≤ 0 and k ′′ ε ≥ 0 for any r ∈ R. Moreover, as ε → 0, we have where θ(r) = 0 for r ≥ 0, θ = 1 for r < 0, and the convergence is uniform for r ∈ R. and With the aid of the above lemmas, we can obtain the following positivity result.
Then the solution of initial-boundary value problem (3.1) with nonnegative initial data remains positive so that u(x, t) ≥ 0, a.s. for almost every x ∈ D, ∀t ∈ [0, T ].
Proof. We remark that when m = 1, Theorem 3.1 has been proved by [3]. Let Taking expectation over the above equality, we get Note that lim ε→0 EΦ ε (u t ) = E η(u t ) 2 , by taking the limits termwise as ε → 0 and using Lemma 3.1, we have where · denotes the norm of L 2 (D). We remark that ∇u − exists by Lemma 3.2. Moreover, it follows from Lemma 3.2 that (3.3) is well defined. In paper [14], the authors proved that u ∈ L p (D) if u 0 ∈ L p (D), where p ≥ 1 and u is the solution of (1.3). We also remark that in [14] they assumed σ satisfied the linear growth. But one can use the method of [24] to obtain the desire results, that is, the results in [14] also hold for (3.1). By using (3.2) and η(u) ≥ 0, we obtain where we have used the condition on a 1 , that is, (−1) β a 1 = |a 1 |. Now, we will use ǫ-Yang inequality and the following interpolation inequality of L p to deal with the last two terms of (3.4), where 0 < θ < 1 and For simplicity, we write u instead of u − . Notice that 1 < m < r+1 2 and 0 ≤ q(x, x) ≤ q 0 , by using (3.5), we have which, by means of Gronwall's inequality, implies that It follows that η(u t ) = u − (x, t) = 0 a.s. for a.e. x ∈ D, ∀t ∈ [0, T ]. This completes the proof.
Remark 3.1 1. Comparing Theorem 3.1 with Proposition 2.1, it is easy to see that our assumption is weaker. For example, f (u) = u(1 − u 2 ), will not satisfy the condition (P2), but it is covered in our theorem. By using a similar method, one can deal with the nonlinearity term depending on the x and t. In this section, we only consider the case that m > 1 and it is possible to use the similar method to deal with the case that 0 < m < 1. 2.
Obviously, if f (u) ≡ 0, Theorem 3.1 will fail, that is, we can not obtain the positivity of solutions to (3.1) with f (u) ≡ 0, even for the one dimension. But we can obtain the positivity of solutions of (3.1) with f (u) ≡ 0 under the condition that ∆ is replaced by the P −Laplacian ∆ p (see the following discussions).
3. From the proof of Theorem 3.1, we know that the term −|∇u| 2 is no use. Actually, the term −|∇u| 2 is a good term, as we can use it to control the stochastic term. But when we use the embedding theorem and interpolation inequality, we find that the term u m L m would be changed to u ν L 2 , where ν > 2. Due to the convexity of the function x ν , we can not get the desired result. However, if we change −|∇u| 2 to −|∇u| p , p > n ≥ 2, we can show that the solution is positive; see Theorem 3.2 below.
It follows from Theorem 3.1 that the the value of m depends on the nonlinearity term f . The following result shows that the value of m may not depend on the nonlinearity term f . Now, we consider the following Itô parabolic equation with the p−Laplacian operator where ∆ p u = div(|∇u| p−2 ∇u). We assume that there exist positive constants α and β such that Theorem 3.2 Assume that p > max{2m, n} and (3.9) holds. Then the solution of initialboundary value problem (3.8) with nonnegative initial data remains positive: u(x, t) ≥ 0, a.s., for almost every x ∈ D, ∀t ∈ [0, T ].
Proof. Similar to the proof of Theorem 3.1, we get It follows from Lemma 3.2 that the above inequality is well defined. For simplicity, we write u instead of u − . By the Sobolev embedding inequality and for p > n, which implies that where ε > 0 and we have used the Young inequality. Noting that p > max{2m, n}, there exists a constant γ ∈ (0, 2) such that Letting kq 0 ε ≤ 1 and submitting (3.12) into (3.10), we get which, by means of Gronwall's inequality, implies that It follows that η(u t ) = u − (x, t) = 0 a.s. for a.e. x ∈ D, ∀t ∈ [0, T ]. This completes the proof. We remark that the value of m in Theorems 3.1 and 3.2 either depends on the nonlinear term f or the operator ∆ p . The reason is that we don't choose suitable test function. In the followings, we will give a new test function β ε (r) instead of k ε (r). Let Then by direct verification, we have the following result.
By Itô formula, we have Taking expectation over the above equality and using Lemma 3.3, we get Here and after, we denote · L 1 by · 1 . Let η(u) = u − denote the negative part of u for u ∈ R. Then we have lim ε→0 EΦ ε (u t ) = E η(u t ) 1 . It follows from Lemma 3.3 that which implies that lim ε→0 u 2m β ′′ ε (u) = 0 provided that 2m > 1. By taking the limits termwise as ε → 0 and using Lemma 3.3, we get which implies that u − = 0 a.s. for a.e. x ∈ D, ∀t ∈ [0, T ]. This completes the proof.
One can use the same test function to improve the result of Theorem 3.2.

Blow-up Phenomenon
In this section, we shall consider the solutions of (3.1) which blow up in finite time. We first show that a similar result to [8] that holds for (3.1), and then we examine how noise induces blow-up in finite time. We divide this section into three subsections. . We remark that when σ ≡ 0, then (3.1) becomes the determinative parabolic equation. Fujita [8] gave the existence and non-existence theorem for global solution of (3.1) with σ ≡ 0. The following result is similar to that in [8].

First result on blow-up
Theorem 4.1 Suppose the initial-boundary value problem (3.1) has a unique local solution. Assume that all the assumptions in Theorem 3.1 hold, where a 1 > 0. In addition, if λ 1 ≥ a 2 , we assume that
It is remarked that Proposition 2.2 covers a part of the above result. The following example shows that Theorem 4.1 generalizes Proposition 2.2.
Example Consider the following stochastic parabolic Itô equation where k ∈ R and D is defined as in (1.1). Fujita [8] obtained that the solution of (4.8) with k = 0 and u 0 ≥ 0 will blow up in finite time. By Theorem 3.1, we know that the solution of (4.8) remains positive if u 0 ≥ 0. It follows from Theorem 4.1 that the solution of (4.8) will blow up in finite time under the same assumptions as in [8]. We also remark that Proposition 2.2 is not suitable to (4.8).
Lastly, we give the following remarks.
Remark 4.1 1. From the proof of Theorem 4.1, it is easy to see that the stochastic term does not play any role because the first moment of white noise is zero. In other words, the first moment does not touch the white noise. Clearly, white noise can not prevent the blow-up of the solution. If we want to study the noise can prevent singularities (see [7]), perhaps we should consider the color noise or complex noise.
2. In the proof of Theorem 4.1, the positivity of solutions can assure thatû(t) > 0 and so it is important. Combining Proposition 2.2 and the fact that one can obtain a similar result to Theorem 4.1 under the assumptions of Proposition 2.2.
4. If a 1 = 1 and a 2 = 0, then the condition (4.1) becomes (1.2). That is, under the same conditions on initial data, the solutions of (1.1) and (3.1) will blow up in finite time. Thus we can say we obtain a similar result to [8].

Second result on blow-up
In this subsection, we consider the issue about how noise may induce finite time blow-up of the solution of stochastic partial differential equations.
Consider the following stochastic parabolic Itô equation where b ∈ R, α > 0 and 1 ≤ m < 1 + α 2 . When m = 1, Dozzi and López-mimbela [6] obtained the global solution of (4.9) if the initial data and the noise are small enough (see Theorem 5 in [6]), which is similar to the determinative case [8]. It is known that when b = 0 and the nonnegative initial data is small enough, (4.9) has a unique global solution [8]. In this subsection, we will show that noise can induce blow-up.
where λ 1 is defined as in (2.2) and q(x, y) is the correlation function. Then the solution of (4.9) will blow up in finite time, that is, there exists a constant T * > 0 such that Proof. By [2,14,20], we know that (4.9) has a unique local solution. It follows from Theorem 3.1 that the solution of (4.9) remains positive. Similar to the proof of Theorem 4.1, it suffices to show that Eû 2 (t) blows up in finite time, whereû(t) = (u, φ).
By applying Itô formula toû 2 (t) and making use of (2.2), we get Let η(t) = Eû 2 (t). By taking an expectation over (4.12), we obtain q(x, y)u m (x, s)φ(x)u m (y, s)φ(y)dxdyds, (4.13) or, in the differential form (4.14) By Jensen's inequality, (4.14) yields which implies that, for η . Hence this is a contradiction. This completes the proof. Before ending this section, we make the following remarks.
Remark 4.2 1. Theorem 4.2 contains a new result. First, we suppose there exists a positive constant q 1 such that inf x,y∈D q(x, y) ≥ q 1 . When b = 0, Fujita [8] showed that the solution of (4.9) will exist globally if the initial data is sufficiently small. Then we fixed the initial data sufficiently small such that (4.9) with b = 0 has a unique global solution. Finally, we take the suitable value of b such that (4.10) holds and it follows from Theorem 4.2 that the unique positive solution of (4.9) will blow up in finite time. Hence we can say that the noise induces the finite time blow-up.
2. From the proof of Theorem 4.2, we know that if one can prove that the solution of (2.1) is positive without using the property of f (u), then the solution of (2.1) with σ = u m (m > 1) will blow up in finite time under the condition that f (u) ≥ 0 for u ≥ 0. Similar to that in [4], one can prove that Theorems 4.1 and 4.2 also hold for D = R n (Theorem 3.2 in [4]).
3. From (4.10), we see that for m = 1 and b 2 q 1 /2 ≥ λ 1 , the solution of (4.9) will blow up in finite time for any nonnegative initial data. On the other hand, it follows from the proof of Theorem 4.2 that noise can make the existence time smaller.
(4.16) Theorem 4.3 Assume that all the assumptions in Theorem 3.3 hold. Assume further that u 0 is a nonnegative continuous function, f (u, x, t) ≥ 0 for u ≥ 0, x ∈ D, t > 0 and where λ 1 is defined as in (2.2) and q(x, y) is the correlation function. Then the solution of (4.16) will blow up in finite time, that is, there exists a constant T * > 0 such that Proof. By [2,14,20], we know that (4.16) has a unique local solution. It follows from Theorem 3.3 that the solution of (4.16) remains positive. Similar to the proof of Theorem 4.2, it suffices to show that Eû 2 (t) blows up in finite time, whereû(t) = (u, φ).

Global solution for a stochastic Allen-Cahn equation
In this section, we show that the solution of a stochastic Allen-Cahn equation does not have finite time singularities and it exists globally. This is an example of SPDEs whose coefficients are not globally Lipschitz continuous.
We consider the following stochastic Allen-Cahn equation.
where 1 < m < 2, b ∈ R. If b = 0, (5.1) becomes the well-known deterministic Allen-Cahn equation [1], which describes the process of phase separation in iron alloys, including order-disorder transitions. Hairer et al. [11] considered (5.1) with m = 0. The equation (5.1) with b = 0 has a global solution. We want to know when the solution of (5.1) exists globally and when the solution blows up. In this section, we shall use the Lyapunov functional method to prove that the solution of (5.1) exists globally, i.e., no finite time blow up. Throughout this section, we assume that H = L 2 (D), · = · H . Moreover, W (x, t) is a Wiener random field, and q(x, y) is its covariance function as defined in Section 2.
Let u(x, t; u 0 ) be a solution of (5.1) with the initial data u 0 . We first give the definition of global solution.
for any T > 0. If the above holds for T = ∞, the solution u(x, t) is said to be ultimately bounded, i.e., global solution.
We shall use Lyapunov functional method to obtain the existence of global solution to (5.1). In the following, we recall the definition of Lyapunov functional ( [2]). We do this for a more general stochastic partial differential equation where A, F and are assumed to be non-random or deterministic. Let V be a separate Hilbert space. Here we say that a F t -adapted V -valued process u is a strong solution of equation (5.2) if u ∈ L 2 (Ω × [0, T ]; V ), and for any φ ∈ V , the following equation holds for each t ∈ [0, T ] a.s. Recall that the generator for this stochastic partial differential equation is where Q is covariance operator. Let U ⊂ H be a neighborhood of the origin. A function Φ: U × R + → R is said to be a Lyapunov functional for the equation (5.2), if (1) Φ is locally bounded and continuous such that its first two partial derivatives ∂ t Φ(v, t), ∂ x Φ(v, t) and ∂ xx Φ(v, t) exist, and ∂ t Φ(v, t), ∂ x Φ(v, t) are locally bounded.
(2) Φ(0, t) = 0 for all t ≥ 0, and, for any r > 0, there is δ > 0 such that (3) For every t ≥ 0 and v ∈ U ∩ H 1 0 , In order to obtain the global solution of (5.1), we need the following two lemmas.
Lemma 5.1 [2, p. 199, Lemma 3.1] Let U × R + → R + be a Lyapunov functional and let u t denote the strong solution of 5.2 with initial data u 0 . For r > 0, let B r = {h ∈ H : h ≤ r} such that B r ⊂ U . Define with B c r = H B r . We put τ = T if the set is empty. Then the process φ t = Φ(u t∧τ , t ∧ τ ) is a local F t -supermartingale and the following Chebyshev inequality holds Now, we use the Lemmas 5.1 and 5.2 to study the global solution of (5.1).
Theorem 5.1 Assume that 1 < m < 2 and u 0 (x) ≥ 0 for x ∈D. Assume further that there exists a positive constant q 0 such that the covariance function q(x, y) satisfies the condition sup x,y∈D q(x, y) ≤ q 0 . Then (5.1) has a strong global solution.
Proof. It follows from [2,14,20] By using interpolation inequality (3.5), we have where θ = 2−m m satisfying (3.6). Substituting (5.5) into (5.4), we have , which implies that all the assumptions in Lemma 5.2 hold. Thus by Lemma 5.2 we know that the solution of (5.1) exists globally. This completes the proof.
Corollary 5.1 Assume that γ > 1, 1 < m < (γ + 1)/2, u 0 ≥ 0 and assume also that there exists a positive constant q 0 such that the covariance function q(x, y) satisfies the condition sup x,y∈D q(x, y) ≤ q 0 . Then (5.6) has a unique strong global solution.
Theorem 5.1 and Corollary 5.1 imply that if the nonlinearity f (u) = ku − u γ can control the stochastic term u m , i.e., m < (γ + 1)/2, the stochastic partial differential equation also has global solutions, which is different from the earlier results.