Blowup solutions for stochastic parabolic equations

TL;DR

This paper investigates the blowup behavior of solutions to stochastic parabolic equations, utilizing comparison principles and deterministic results to establish conditions under which solutions become unbounded.

Contribution

It introduces new blowup results for stochastic parabolic equations by adapting comparison techniques from deterministic cases.

Findings

Blowup occurs under certain initial conditions.

Comparison principles are effective in stochastic settings.

Results extend deterministic blowup theory to stochastic equations.

Abstract

In this short paper, we are concerned with the blowup phenomenon of stochastic parabolic equations. By using comparison principle and the results of deterministic parabolic equations, we obtain blowup results of solutions for stochastic parabolic equations.

Equations

$$\displaystyle\left\{\begin{array}[]{lll}u_{t}=\Delta u+u^{p},\quad t>0,\ \ p>0,&x\in\mathbb{R}^{d},\\ u(0,x)=u_{0}(x),&x\in\mathbb{R}^{d}.\end{array}\right.$$

$$\mathcal{U}=\left\{v(x)|v(x)\in BC({\mathbb{R}}^d,{\mathbb{R}}_{+}),v(x)\le \delta e^{-k|x{|}^2},k>0,\delta =\delta (k)>0\right\},$$

$${\mathcal{U}}_1=\left\{v(x)|v(x)\in BC({\mathbb{R}}^d,{\mathbb{R}}_{+}),v(x)\ge c{e}^{-k|x{|}^2},k>0,c\gg 1\right\}.$$

$$\displaystyle\left\{\begin{array}[]{lll}du=(\Delta u+f(u))dt+\sigma(u)dW_{t},\ \ \qquad t>0,&x\in D,\\[4.2679pt] u(x,0)=u_{0}(x)\geq 0,&x\in D,\\[4.2679pt] u(t,x)=0,\qquad\qquad\qquad\qquad\qquad\qquad t>0,&x\in\partial D.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{llll}du=\Delta udt+\sigma(u,x,t)dB_{t},\ \ t>0,&x\in\mathbb{R},\\ u(x,0)=u_{0}(x)\gneqq 0,&x\in\mathbb{R}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{lll}u_{t}=\Delta u+a(x)u^{p},\ \ t\in(0,T),&x\in\mathbb{R}^{d},\\[4.2679pt] u(0,x)=u_{0}(x)\geq 0,&x\in\mathbb{R}^{d},\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{llll}du=\Delta udt+b(t)|u|^{p}dW(t,x),\ \ t>0,&x\in\mathbb{R},\\ u(0,x)=u_{0}(x)\gneqq 0,&x\in\mathbb{R},\end{array}\right.$$

$$u(t,x)={\int }_{\mathbb{R}}K(t,x-y)u_0(y)dy+{\int }_0^t{\int }_{\mathbb{R}}K(t-s,x-y)b(s)|u{|}^p(s,y)W(ds,dy),$$

$$b_{\infty }:={\int }_0^{\infty }(b^{-1}(t){)}^{2}dt<+\infty .$$

$$\mathbb{E}|u(t,x){|}^2$$

$${\left({\int }_0^t{\int }_{\mathbb{R}}K(t-s,x-y)a(y){\left[\mathbb{E}|u{|}^2(s,y)\right]}^{\frac{p}{2}}dyds\right)}^2$$

$$\mathbb{E}|u(t,x){|}^2$$

$$\displaystyle\left\{\begin{array}[]{lll}v_{t}=\Delta v+\frac{1}{\sqrt{2(a_{\infty}b_{\infty}\vee 1)}}a(x)v^{p},\ \ \ t\in(0,T),&x\in\mathbb{R},\\[4.2679pt] v(0,x)=\frac{1}{\sqrt{2(a_{\infty}b_{\infty}\vee 1)}}u_{0}(x)\geq 0,&x\in\mathbb{R}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{lll}u_{t}=(-\Delta)^{\frac{\alpha}{2}}u+u^{p},\ \ \qquad t>0,&x\in\mathbb{R}^{d},\\[4.2679pt] u(0,x)=u_{0}(x)\geq 0,&x\in\mathbb{R}^{d}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{llll}du=-(-\Delta)^{\frac{\alpha}{2}}udt+b(t)a(x)|u|^{p}dW(t,x),\ \ t>0,&x\in\mathbb{R},\\ u(0,x)=u_{0}(x)\gneqq 0,&x\in\mathbb{R},\end{array}\right.$$

$$u(t,x)={\int }_{\mathbb{R}}p(t,x-y)u_0(y)dy+{\int }_0^t{\int }_{\mathbb{R}}b(s)a(y)p(t-s,x-y)|u{|}^p(s,y)W(ds,dy),$$

$$b_{\infty }:={\int }_0^{\infty }(b^{-1}(t){)}^{2}dt<+\infty ,a_{\infty }:={\int }_0^{\infty }(a^{-1}(x){)}^{2}dx<+\infty .$$

$$\displaystyle\left\{\begin{array}[]{llll}du_{t}=\Delta udt+b(t)a(x)|u|^{p}dB_{t},\ \ t>0,&x\in\mathbb{R}^{d},\\ u(x,0)=u_{0}(x)\gneqq 0,&x\in\mathbb{R}^{d},\end{array}\right.$$

$$u(t,x)={\int }_{{\mathbb{R}}^d}K(t,x-y)u_0(y)dy+{\int }_0^t{\int }_{{\mathbb{R}}^d}K(t-s,x-y)a(y)b(s)|u{|}^p(s,y)dyd{B}_s,$$

$$b_{\infty }:={\int }_0^{\infty }(b^{-1}(t){)}^{2}dt<+\infty .$$

$$\mathbb{E}|u(t,x){|}^2$$

$${\int }_0^{t}f(s)ds\le {\left({\int }_0^t(b^{-1}(s){)}^{2}ds\right)}^{\frac{1}{2}}{\left({\int }_0^t(b(s)f(s){)}^{2}ds\right)}^{\frac{1}{2}}.$$

$$v(t,x)={\left(\mathbb{E}|u(t,x){|}^2\right)}^{\frac{1}{2}},0<t\le T,$$

$$\displaystyle\left\{\begin{array}[]{lll}v_{t}=\Delta u+\frac{1}{\sqrt{2(b_{\infty}\vee 1)}}a(x)v^{p},\ \ \qquad t>0,&x\in\mathbb{R}^{d},\ t\in(0,T),\\[4.2679pt] v(0,x)=\frac{1}{\sqrt{2(b_{\infty}\vee 1)}}u_{0}(x)\geq 0,&x\in\mathbb{R}^{d}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{llll}du=\Delta udt+(1+t)^{q}a(x)|u|^{p}dB_{t},\ \ t>0,&x\in\mathbb{R}^{d},\\ u(x,0)=u_{0}(x)\gneqq 0,&x\in\mathbb{R}^{d}.\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{llll}du=\Delta udt+|u|^{p}dt+\sigma(t,x)dB_{t},\ \ t>0,&x\in\mathbb{R}^{d},\\ u(x,0)=u_{0}(x)\gneqq 0,&x\in\mathbb{R}^{d}.\end{array}\right.$$

$${\left({\int }_{{\mathbb{R}}^d}K(t,x-y)u_0(y)dy\right)}^2\ge \frac{4}{1-\beta }{\int }_0^t{\left({\int }_{{\mathbb{R}}^d}K(t-s,x-y)\sigma (s,y)dy\right)}^{2}ds,\forall t>0,$$

$$\mathbb{E}|u(t,x){|}^2$$

$$(a+b{)}^2\ge \beta a^2-\frac{\beta }{1-\beta }{b}^2,a,b\in \mathbb{R}.$$

$$\mathbb{E}|u(t,x){|}^2\ge \frac{\beta }{4}{\left({\int }_{{\mathbb{R}}^d}K(t,x-y)u_0(y)dy+{\int }_0^t{\int }_{{\mathbb{R}}^d}K(t-s,x-y){\left(\mathbb{E}|u{|}^2(s,y)\right)}^{\frac{p}{2}}dyds\right)}^2.$$

Full text

Keywords: Blowup; Stochastic parabolic equation; Comparison principle.

AMS subject classifications (2010): 35K20, 60H15, 60H40.

1 Introduction

For deterministic partial differential equations, finite time blowup phenomenon has been studied by many authors, see the book [10]. There are two cases to study this problem. One is bounded domain and the other is whole space. On the bounded domain, the $L^{p}$-norm of solutions ($p>1$) will blow up in finite time. The other is the whole space. The following ’Fujita Phenomenon’ has been attracted in the literature. Consider the following Cauchy problem

[TABLE]

It has been proved that:

(i) if $0<p<1$, then every nonnegative solution is global, but not necessarily unique;

(ii) if $1<p\leq 1+\frac{2}{d}$, then any nontrivial, nonnegative solution blows up in finite time;

(iii) if $p>1+\frac{2}{d}$, then $u_{0}\in\mathcal{U}$ implies that $u(t,x,u_{0})$ exists globally;

(iv) if $p>1+\frac{2}{d}$, then $u_{0}\in\mathcal{U_{1}}$ implies that $u(t,x,u_{0})$ blows up in finite time,

where $\mathcal{U}$ and $\mathcal{U_{1}}$ are defined as follows

[TABLE]

Here $BC=\{$ bounded and uniformly continuous functions $\}$, see Fujita [8, 9].

It is easy to see that for the whole space, there are four types of behaviors for problem (1.3), namely, (1) global existence unconditionally but uniqueness fails in certain solutions, (2) global existence with restricted initial data, (3) blowing up unconditionally, and (4) blowing up with restricted initial data. The occurrence of these behaviors depends on the combination effect of the nonlinearity represented by the parameter $p$, the size of the initial datum $u_{0}(x)$, represented by the choice of $\mathcal{U}$ or $\mathcal{U_{1}}$, and the dimension of the space.

Now, we recall some known results of stochastic partial differential equations (SPDEs). In this paper, we only focus on the stochastic parabolic equations. It is known that the existence and uniqueness of global solutions to SPDEs can be established under appropriate conditions ([4]). For the finite time blowup phenomenon of stochastic parabolic equations, we first consider the case on a bounded domain. Consider the following equation

[TABLE]

Da Prato-Zabczyk [18] considered the existence of global solutions of (1.7) with additive noise ($\sigma$ is a constant). Dozzi and López-Mimbela [5] studied equation (1.7) with $\sigma(u)=u$ and proved that if $f(u)\geq u^{1+\alpha}$ ($\alpha>0$) and the initial data is large enough, the solution will blow up in finite time, and that if $f(u)\leq u^{1+\beta}$ ($\beta$ is a certain positive constant) and the initial data is small enough, the solution will exist globally. A natural question arises: If $\sigma$ does not satisfy the global Lipschitz condition, what can we say about the solution? Will it blow up in finite time or exist globally? Chow [2] answered part of this question. Lv-Duan [12] described the competition between the nonlinear term and noise term for equation (1.7). Bao-Yuan [1] and Li et al.[11] obtained the existence of local solutions of (1.7) with jump process and Lévy process, respectively. For blowup phenomenon of stochastic functional parabolic equations, we refer to [3, 6]. In a somewhat different case, Mueller [15] and, later, Mueller-Sowers [16] investigated the problem of a noise-induced explosion for a special case of equation (1.7), where $f(u)\equiv 0,\,\sigma(u)=u^{\gamma}$ with $\gamma>0$ and $W(t,x)$ is a space-time white noise. It was shown that the solution will explode in finite time with positive probability for some $\gamma>3/2$.

For the whole space, Foondun et al. [7] considered the nonexistence of global solutions for the Cauchy problem of stochastic fractional parabolic equations. Comparing with the deterministic parabolic equations, they only obtained the result similar to type (4), also see [21]. In paper [13], we established the similar results to types (1) and (3). The method used there is the properties of heat kernel. But in [13], we only obtained the results for one dimension in the whole space. More precisely, we obtained the following result.

**   Proposition 1.1**

Suppose $\sigma^{2}(u,x,t)\geq C_{0}u^{4}$, $C_{0}>0$, then the solutions of the following equation will blow up in finite time for any nontrivial nonnegative initial data $u_{0}$,

[TABLE]

In this paper, we consider the blowup phenomenon of SPDEs included fractional diffusion equations, and generalize of the result Proposition 1.1. There are a lot of work to do for SPDEs comparing with the deterministic case, see [22].

This paper is arranged as follows. In Sections 2, we state out the main results and the proofs. Throughout this paper, we write $C$ as a general positive constant and $C_{i}$, $i=1,2,\cdots$ as a concrete positive constant.

2 Main results and proofs

In this section, we recall some known results and state out the main results.

**   Proposition 2.1**

[17]** Consider the following equation

[TABLE]

where $0\lneqq a\in C^{\beta}(\mathbb{R}^{d})$, $\beta>0$ and $p>1$.

1. Assume that for large $|x|$ and constants $c_{1},c_{2}>0$, $c_{1}|x|^{m}\leq a(x)\leq c_{2}|x|^{m}$, $m>-1$. If $1<p\leq 1+(2+m)/d$, then (2.3) does not possess global solutions, for any choice of initial data $u_{0}\gneqq 0$;

2. Assume that for large $|x|$ and constants $c_{2}>0$, $a(x)\leq c_{2}|x|^{-2}$. If $d=1$ and $1<p\leq 2$, then (2.3) does not possess global solutions, for any choice of initial data $u_{0}\gneqq 0$.

We now consider the following Cauchy problem

[TABLE]

where $W(t,x)$ is a white noise both in time and space, and $0<b\in C(\mathbb{R}_{+})$. In the rest of paper, we always assume that the initial data is a nonnegative continuous function. A mild solution to (2.6) in sense of Walsh [20] is any $u$ which is adapted to the filtration generated by the white noise and satisfies the following evolution equation

[TABLE]

where $K(t,x)$ denotes the heat kernel of Laplacian operator. We get the following results.

**   Theorem 2.1**

Let $0<b\in C(\mathbb{R}_{+})$ such that

[TABLE]

If $1<p\leq 2$, then (2.6) does not possess global solutions, for any choice of initial data $u_{0}\gneqq 0$.

Proof. By taking the second moment and using the Walsh isometry, we get for any $0<t\leq T$ ($T>0$ is any fixed number)

[TABLE]

Let $0\lneqq a\in C^{\beta}(\mathbb{R})$ such that for large $|x|$, there is a positive constant $c_{1}>0$ satisfying $a(x)\leq c_{1}|x|^{-2}$. Then $a_{\infty}:=\int_{\mathbb{R}}a^{2}(y)dy<+\infty$. Observing that

[TABLE]

We deduce that

[TABLE]

For $0<t\leq T$, we set $v(t,x)=\left(\mathbb{E}|u(t,x)|^{2}\right)^{\frac{1}{2}}$, then $v$ is an upper solution of the following equation

[TABLE]

By Proposition 2.1, if $1<p\leq 2$, then (2.9) does not possess global solutions, for any choice of initial data $u_{0}\gneqq 0$. So does the equation (2.6). The proof is complete. $\Box$

**   Remark 2.1**

In deterministic parabolic equations, we will assume that $d\geq 1$, which is different from the stochastic case. The main reason is that the Itô isometry may be not right for $d>1$.

Now, we extend Theorem 2.1 to stochastic fractional parabolic equations. Initially, let us consider the following deterministic equation:

[TABLE]

Sugitani [19] proved that if $p\in(1,1+\alpha/d]$ and the initial data is non-trivial and non-negative, the solution of (2.12) will blow up in finite time. Next, let us consider the following stochastic fractional Laplacian equation

[TABLE]

where $W(t,x)$ is described in (2.3). We will study the nonexistence of global solutions to (2.15). Here, a solution to (2.15) is defined by

[TABLE]

where $p(t,x)$ denotes the heat kernel of fractional Laplacian operator. By a similar discussion as in Theorem 2.1, we have

**   Corollary 2.1**

Assume $0<\alpha<2$ and $1<p\leq 1+\alpha$. Assume further that

[TABLE]

Then (2.15) does not possess global solutions, for any choice of initial data $u_{0}\gneqq 0$.

Next, we consider the following stochastic parabolic equation

[TABLE]

where $B_{t}$ is one-dimensional Brownian motion, $a\in C(\mathbb{R}^{d})$ and $0<b\in C(\mathbb{R}_{+})$. A mild solution to (2.18) in sense of Walsh [20] is any $u$ which is adapted to the filtration generated by the white noise and satisfies the following evolution equation

[TABLE]

where $K(t,x)$ denotes the heat kernel of Laplacian operator.

**   Theorem 2.2**

Let $p\geq 2$ and for large $|x|$, $c_{1}|x|^{m}\leq|a(x)|\leq c_{2}|x|^{m}$ with $c_{1},c_{2}>0,m\in\mathbb{R}$. Assume that $m>d-2$ and $2\leq p\leq 1+(2+m)/d$, that

[TABLE]

Then (2.18) does not possess global solutions for any choice of initial data $u_{0}\gneqq 0$.

Proof. By taking the second moment and using the Walsh isometry, we get for any $0<t\leq T$ ($T>0$ is any fixed number)

[TABLE]

where we used the fact that

[TABLE]

Set

[TABLE]

then $v(t,x)$ is an upper solution of the following equation

[TABLE]

Under the condition of Proposition 2.1 that the solution of (2.21) does not possess global solutions for any choice of initial data $u_{0}\gneqq 0$, so does the equation (2.18). $\Box$

**   Remark 2.2**

(i) It follows from Theorem 2.2 that we consider the equation (2.21) in any dimension, which is different from that of [13].

(ii) About the stochastic Fujita index, we have the following example. Consider the following equation

[TABLE]

Theorem 2.2 shows that if $2\leq p\leq 1+(m+2)/d$, then (2.24) does not possess global solutions for any choice of initial data $u_{0}\gneqq 0$.

Lastly, we consider the impact of additive noise on parabolics. That is to say, we consider the following Cauchy problem

[TABLE]

We have the following result.

**   Theorem 2.3**

(i) If $d=1$ and $2\leq p\leq 3$ and the initial data satisfies

[TABLE]

for some real number $\beta\in(0,1)$. Then the solution of (2.27) will blow up in finite time.

(ii) For general $d$, we assume that $u_{0}\in\mathcal{U_{1}}$ and (2.28) holds. If $p>1+d/2$ and $p\geq 2$, then the solution of (2.27) will blow up in finite time.

Proof. By taking the second moment, we get for any $0<t\leq T$ ($T>0$ is any fixed number)

[TABLE]

where we used

[TABLE]

In view of (2.28), from (2.29), we arrive at

[TABLE]

From (2.30), by using the comparison principle, we get the desired result. $\Box$

**   Remark 2.3**

It follows from Theorem 2.3 that the additive noise prevents the finite time blowup.

(ii) One can use a similar method to deal with the case of bounded domain if the operator admits a heat kernel. For references in this direction for the deterministic equations, we refer to [14].

Acknowledgment The first author was supported in part by NSFC of China grants 11771123, 11501577.