Explosive solutions of parabolic stochastic partial differential equations with L$\acute{e}$vy noise

In this paper, we study the explosive solutions to a class of parbolic stochastic semilinear differential equations driven by a L$\acute{\mbox{e}}$vy type noise. The sufficient conditions are presented to guarantee the existence of a unique positive solution of the stochastic partial differential equation under investigation. Moreover, we show that the positive solutions will blow up in finite time in mean $L^{p}$-norm sense, provided that the initial data, the nonlinear term and the multiplicative noise satisfies some conditions. Several examples are presented to illustrated the theory. Finally, we establish a global existence theorem based on a Lyapunov functional and prove that a stochastic Allen-Cahn equation driven by L$\acute{\mbox{e}}$vy noise has a global solution.


Introduction
Fujita [21] considered the initial-boundary problem for a semilinear parabolic Fujita [22] showed that if D is bounded, a( We refer to [20] about the many developments on solutions of nonlinear parabolic equations may blow up in finite time.
Recent years, stochastic partial differential equations has attracted the attention of many researchers. It is of interest to study the non-existence of global solutions to parabolic stochastic partial differential equations perturbed by random noise as follows: u(x, 0) = g(x), x ∈ D, u(x, t) = 0, x ∈ ∂D. (1. 3) When f (u) ≡ 0, σ(u) = u γ (γ ≥ 1), Mueller [24] considered the equation u(t, 0) = u(t, J) = 0, (1.4) whereẆ =Ẇ (t, x) is 2-parameter white noise and u(x, 0) is nonnegative and continuous. The conclusion is that for 1 ≤ γ < 3 2 , u exists for all time. Mueller [25] showed that when u(x, 0) is a continuous nonnegative function on [0, J], vanishing at the endpoints, but not identically zero, then there is a positive probability that the solution u of (1.4) blows up in finite time if γ > 3/2. When σ(u) ≡ 1 and the Laplacian operator ∆ is replaced by the infinitesimal generator of a C 0 semigroup, Prato and Zabczyk [14] considered the stochastic semilinear equation du = (Au + F (u))dt + dW u(0) = ξ, (1.5) where A is the generator of semmigroup S(t) = e At on a Banach space E, and F is a mapping from E into E. W is a Wiener process defined on a probability space (Ω, F t , P ). ξ is an F 0 -measurable E-valued random variable. They assumed that F satisfies the Lipschitz condition on bounded sets of E. This property of F together with some other conditions ensure that (1.5) has a unique non-exploding solution.
When σ(u) = σ (positive constant), W is a 2-dimensional Brownian sheet, f is a nonnegative, convex function such that ∞ 0 1/f < ∞, Bonder and Groisman [26] proved that the solution to (1.3) blows up in finite time with probability one for every nonnegative initial datum u(x, 0) ≥ 0. Dozzi and López-Mimbela [27]  Chow [12,2] considered the initial-boundary value problem for the parabolic is a symmetric, uniformly elliptic operator with smooth coefficients, f and σ are given functions. For x ∈ R d , t ≥ 0, W (x, t) is a continuous Wiener random field defined in a complete probability space (Ω, F , P ) with a filtration F t . W (x, t) has mean EW (x, t) = 0 and covariance function q(x, y) defined by To consider positive (nonnegative) solutions, the author assume that the following conditions hold: (P1) There exists a constant δ ≥ 0 such that (P3) The initial datum g(x) on D is positive and continuous.
Chow [12] proved that the solution of Eq.(1.6) is positive. Under some suitable conditions, Chow [12,2] showed that the positive solutions of a class of stochastic reaction-diffusion equations will blow up in the L p -norm sense, p ≥ 1. Chow and Liu [3] considered the problem of explosive solutions in mean L p -norm sense of semilinear stochastic functional parabolic differential equations of retarded type.
Lv and Duan [1] considered the Eq.(1.6) with A = △, the Laplacian operator, the nonlinear term f is assumed to be satisfied by an inequality, which is weaker than the condition (P2), the noise intensity σ allows to be higher nonlinear than the square nonlinear (see [1, formula (3.2)]). They proved that the noise could induce finite time blow up of solutions.
Recent years, stochastic partial differential equations driven by Lévy noise have attracted many attentions (see, for example, [15,5,9,17,14,18,8,6,10] where O ⊂ R n is a bounded domain with C ∞ boundary ∂O, A = n i,j=1 ∂ ∂x i (a i,j (x) ∂ ∂x j ) be a symmetric, uniformly elliptic operator with smooth coefficients, W (x, t) is a Wiener random defined on the completed probability space (Ω, F , {F } t≥0 , P), N (dt, du) is the compensated Poisson measure. Under some conditions, they showed that the solution of (1.7) blows up in finite time. It should be pointed out that the nonlinear term b : [0, ∞) × R × O → R is assumed to be locally Lip-continuous w.r.t the second variable such that b(t, r, x) ≥ 0 for any r ≤ 0, however, there are many functions don't satisfy this condition, for example, b(r) = r(1 − r 2 ). And O ⊂ R n is assumed to be bounded, the proof of Theorem 2.1 in [7] depends on the boundedness of volume of O. The results of [7] can't be generalized to the case for unbounded domain, such as O = R n .
In this paper, we study the problem of explosive solutions to a class of semilinear stochastic parabolic differential equations driven by Lévy noise. The paper is organized as follows. In Section 2, we recall some basic results for semilinear stochastic parabolic equations with Lévy noise. In Section 3, under some assumptions, we prove that the existence of positive solutions of a semilinear stochastic reaction-diffusion equation. In Section 4, under some suitable conditions on the drift or diffusion term, we prove that the solutions of stochastic parabolic differential equations will blow up in a finite time in mean L p -norm sense, p ≥ 1. Some examples are presented to illustrate the theory. In Section 5, we establish a global existence theorem based on a Lyapunov functional. We show that the existence of global solution to stochastic Allen-Cahn equation driven by Lévy noise.

Preliminaries
Let D be a domain in R d , which has a smooth boundary if it is bounded.
Denote L 2 (D) by H, the usual L 2 real Hilbert space with the inner product (·, ·) and norm · , respectively. Let W (x, t) be a continuous Wiener random field defined on a complete probability space (Ω, F , P) with a filtration F t . W (x, t) has mean zero and covariance function q(x, y) such that The associated covariance operator Q in H with kernel q(x, y) is defined by In this paper, we assume that the covariance function q(x, y) is bounded, continuous and there is q 0 > 0 such that sup x,y∈D |q(x, y)| ≤ q 0 and Tr Q = Consider the initial-boundary problem of a semilinear stochastic reaction-diffusion is a symmetric, uniformly elliptic operator with smooth coefficients, that is, there exists a constant c > 0 such that b(x, ξ) : and W t = W (·, t), then we can rewrite the equation where A is regarded as a linear operator from To consider the positive solutions, we assume that (2.1) has a unique (strong) solution. In addition, we assume that (A3) There exist a a constant µ ∈ [2, β + 1) and mappings ψ : on D is positive and continuous.
As in [12], let η(r) = r − denote the negative part of r for r ∈ R, or η(r) = 0, if r ≥ 0 and η(r) = −r if r < 0. Set k(r) = η 2 (r) so that k(r) = 0 for r ≥ 0 and k(r) = r 2 for r < 0. For ε > 0, let k ε (r) be a C 2 -regularization of k(r) defined by It is easy to see that k ε (r) has the following properties.
is more general than the Laplacian operator △, Theorem 3.1 is the generalization of Theorem 3.1 in [1].

Remark 3.3.
If it is assumed that β ∈ (0, 1), for the case 1 + β ≤ m < 2 and 1 + β ≤ q < 2, by the L p interpolation inequality and Young inequality, we can get the corresponding results.

Explosive solutions
In this section, we consider the unbounded solutions of the equation (2.1).
For the elliptic equation: it is well known that all the eigenvalues of −A are strictly positive, increasing and the eigenfunction φ corresponding to the smallest eigenvalue λ 1 does not change sign in the domain D (see p. 355, [16]). We can normalize it in such a way that Theorem 4.1. Suppose the initial-boundary value problem (2.1) has a unique local solution and the conditions (A1)-(A4) hold. In addition, we assume that λ 1 > a 2 , and if λ 1 < a 2 , we assume that D g(x)φ(x)dx > 0, where λ 1 is the smallest eigenvalue of −A and φ is the corresponding eigenfunction. Then, for any p ≥ 1, there exists a constant T p > 0 such that That is, the solution explodes in mean L p -norm sense.
Proof. By Theorem 3.1, Eq. (2.1) has a unique positive solution. We will prove the theorem by contradiction. We suppose ( Taking the expectation to both sides of (4.5) and by Fubini's theorem, we have or, in the differential form, where ξ(t) = Eû(t), ξ 0 = (g, φ). By (A1) and Jensen's inequality, we obtain If λ 1 ≥ a 2 , for ξ 0 > λ 1 −a 2 a 1 1 β−1 , we can show that ξ(·) is strictly increasing. It follows from (4.7) that If λ 1 < a 2 , for ξ 0 > 0, we can show that ξ(·) is strictly increasing. We have Since T is arbitrary, either (4.8) or (4.9) results in a contradiction. Therefore, for where q = p/(p − 1), p ≥ 1. So the positive solution explodes at some time T ′ ≤ T e in the mean L p -norm for each p ≥ 1. The proof is complete.
The constants σ 0 , c 0 , a 0 , α are strictly positive and x · y = 3 i=1 x i y i . Here A = △, f = u then condition (A2) is satisfied. If Therefore, by Theorem 4.1, the solutions to the Eq. (4.11) will blow up in finite time in mean L p -norm for any p ≥ 1. Note that Theorem 3.1 in [7] is not suitable for Eq. (4.11).
To discuss the noise-induced explosion, we consider the following stochastic reaction-diffusion equation: which is a special case of Eq. (2.1), where σ is independent of ∇u. We assume that the noise terms satisfy the following conditions: (A1 ′ ) The correlation function q(x, y) is continuous and positive for x, y ∈ D such for any non-negative v ∈ H and some constant κ > 0.
(A3 ′ ) There exist continuous functions σ 0 , G such that they are both positive, convex and satisfy (A4 ′ ) There exist continuous functions ϕ 0 , K such that they are both positive, convex and satisfy (A5 ′ ) There exists a constant M > 0 such that κG(u) + K(u) > 2λ 1 u for u > M, (A6 ′ ) The initial datum satisfies the following there exists a constant T p such that That is, the solution explodes in mean L p -norm sense.
Remark 4.1. In [2], [3], the correlation function q(x, y) is assumed to satisfy the for any positive v ∈ H and for some q 1 > 0.
In fact, this assumption is not suitable. If the domain D is bounded and q ∈ (0, 1], by the Cauchy-Schwarz inequality, we have where µ(D) is the volume of D. By (4.21), (4.22), we have µ(D) ≥ q 1 . If the bounded domain D is small enough, then we get a contradiction.

Global solutions for a stochastic Allen-Cahn equation driven by a
Lévy type noise In this section, we consider the following stochastic Allen-Cahn equation driven by a Lévy type noise, u(x, t) = 0, t > 0, x ∈ ∂D, Let V be a real separable Hilbert space. We first consider the more general where the coefficients A, F t , Σ t and Γ t are assumed to be non-random or deterministic. W (x, t) is a Wiener random field, (Z, B(Z)) is a measurable space. N(dt, dz) is the compensated Poisson measure. Here we say that an F t -adapted V -valued process u t is a strong solution, or a variational solution, of the equation (5.2) if u ∈ L 2 ([0, T ]; V ), and for any ϕ ∈ V , the following equation holds for each t ∈ [0, T ] a.s.
Denote (1) Φ : U T → R is locally bounded and continuous such that its first two partial (2) The derivatives ∂ t Φ and Φ ′ ∈ V are locally bounded and continuous in U T .
Let U ⊂ V be a neighborhood of the origin. By a similar statement to that in [4, pp. 228], we present the definition of Lyapunov functional. Define the operator L t as follows: where Q is the covariance operator.
A strong Itô functional Φ : U × R + → R is said to be a Lyapunov functional for the equation (5.2), if (1) Φ(0, t) = 0 for all t ≥ 0, and, for any ε > 0, there is a δ > 0 such that inf t≥0, h ≥ε Φ(h, t) ≥ δ, and (2) for any t ≥ 0 and v ∈ U, Let u h t be a strong solution of the equation for any T > 0. If the above holds for T = ∞, the solution is said to be ultimately bounded.
Lemma 5.1. Let Φ : U × R + → R + be a Lyapunov functional and let u h t denote the strong solution of (5.2). For r > 0, let B r = {h ∈ V : h < r} such that B r ⊂ U.
with B c r = V \B r . We put τ = T if the set is empty. Then the process φ t = Φ(u h t∧τ , t ∧ τ ) is a local F t -supermartingale and the following Chebyshev inequality holds Proof. Let Φ(v, t) = e −αt Ψ(v, t). We have Therefore Φ is a Lyapunov functional. By Lemma 5.1, we have as r → ∞, for any T > 0.
Proof. In view of the proof of Theorem 3-6.5 in [4, pp. 86] and the proof of Theorem 3.2 in [6], we can show that the equation (5.1) has a local strong solution.