Medical, Pharma, Engineering, Science, Technology and Business

**Hazaimeh MH ^{*}**

Department of Mathematics, Zayed University, Dubai, UAE

- *Corresponding Author:
- Hazaimeh MH

Department of Mathematics

University College, Zayed University

P.O.Box 19282, Dubai, UAE

**Tel:**+971 4 402 1111

**E-mail:**[email protected]

**Received Date:** September 22, 2016; **Accepted Date:** October 05, 2016; **Published Date:** October 15, 2016

**Citation: **Hazaimeh MH (2016) Stability of Fourier Solutions of Nonlinear Stochastic Heat Equations in 1D. J Appl Computat Math 5:323. doi: 10.4172/2168- 9679.1000323

**Copyright:** © 2016 Hazaimeh MH. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

The main focus of this article is studying the stability of solutions of **nonlinear** stochastic heat equation and give conclusions in two cases: stability in probability and almost sure exponential stability. The main tool is the study of related Lyapunov-type functionals. The analysis is carried out by a natural *N*-dimensional truncation in isometric Hilbert spaces and uniform estimation of moments with respect to *N*.

Nonlinear stochastic **heat equation**, additive space-time noise, Lyapunov functional, Fourier solution, finitedimensional approximations, moments, stability.

Nonlinear stochastic heat equation; Additive spacetime noise; Lyapunov functional; **Fourier solution**; Finite-dimensional approximations; Moments; Stability

In this article we study the **stability** of solutions of semi-linear stochastic heat equations

with cubic nonlinearities *A*(*u*) in one dimensions in terms of all systems parameters, i.e., with non-global Lipschitz continuous nonlinearities. Our study focusses on stability of analytic solution *u*=*u*(*x*,*t*) under the **geometric** condition

where 0 ≤ *x* ≤ 1 such that D=[0,*l*].

Many authors have treated stochastic heat equations (e.g. [1,2]), semi-linear stochastic heat equations (e.g. [1-3]) or nonlinear stochastic evolution equations (e.g. [4,5]). Also, some authors study the stability of stochastic heat equations like Fournier and Printems [6] study the stability of the mild solution. Walsh reats the stochastic heat equations in one dimension. Chow [1] studies that the null solution of the stochastic heat equation is stable in probability by using the definition. Recall that:

Where *μ* is the Lebesgue measure in one dimensions. The paper is organized as follows. Section 2 states that the strong Fourier solution of equation (1) is proved. We write the solution using the finite-**dimensional** truncated system verifies properties of finitedimensional Lyapunov functional. Section 3 discusses the stability of the strong solution of equation (1) is stable in probability and almost sure exponential stability. Eventually, Section 4 summarizes the most important conclusions on the well-posedness and behaviour of the original infinite-dimensional system (1).

Consider the stochastic nonlinear heat equation with **additive** noise

(1)

with the initial condition *u*(*x*,0)=*f*(*x*) with *f* ∈ L^{2}(D) (initial position) and and driven by *i.i.d*. standard Wiener processes *W _{n}* with E[

(2)

**Theorem 1**

*Assume that* *with* *u _{x}*∈L

(3)

*Proof*. See Schurz [3].

We need to truncate the infinite series (2) for practical computations. So, we have to consider finite-dimensional truncations of the form

(4)

with Fourier coefficients *c _{n}* satisfying the naturally truncated system of stochastic differential equations (SDEs).

(5)

where

Assume that Define the **Lyapunov functional** *V _{N}* as follows

(6)

This functional is a modification of a functional appeared in Schurz [7]. It is clear that this function is of Lyapunov-type because it is nonnegative and smooth as long as *a*_{2} ≥ 0, **radially** unbounded if additionally σ^{2}π^{2}>*a*_{1}*l*^{2}. Equipped with Euclidean norm

**Lemma 2**

*Consider the Lyapunov functional defined in equation (6), and let*

Then ∀*u*∈L^{2}(D):

(7)

*Proof*. See [7].

**Lemma 3**

*Assume that* *a*_{2} ≥ 0. *Then*, ∀*N* ∈ N, *the functional V _{N} is*

(a) *nonnegative and positive semi-definite if* σ^{2}π^{2}>*a*_{1}*l*^{2} or *a*_{2} ≥ 0.

(b)* positive-definite if*σ^{2}π^{2}>*a*_{1}*l*^{2},

and

(c) satisfies the condition of radial unboundedness

Proof. See [7].

Recall equation (5) governed by

(8)

(9)

To simplify, let

**Definition:** *The trivial solution of system *(*8*) (*in terms of norm *)* is said to be stochastically stable or stable in probability, if for *0<* ε *< 1* and r *> 0*, *∃* a δ=δ*(*ε,r*)* such that, *∀*t *≥* δ, we have*

(10)

whenever δ > 0.

**Lemma 4**

*If* ∃ *a positive-definite function* *V*∈C^{2,1}(R^{d}×[0. ∞),R_{+}) *such that* LV(*x*,*t*) ≤ 0 *and* ∀( *x*,*t*)∈ R* ^{d}*×[0. ∞),

*dX(t)=f(x(t),t)dt + g(x(t),t)dw(t) * (11)

*is stochastically stable.*

*Proof*. See Arnold [8].

**Theorem 5**

*Let*

If then the **trivial solution** of equation (8) is stochastically stable i.e., stable in probability.

*Proof*. From Lemma 3, we know that V_{N}(*u*(*t*)) is positive-definite if ∀*n*N,σ^{2}λ_{n} – *a*_{1}>0. Define the linear operator L as in Schurz [3]

The first and second partial derivative of *V _{N}*(

But by our assumption that

Then thus

LV_{N}(*c _{n}*(t)) ≤ 0.

So by Lemma 4, applied to truncation of (8), the trivial solution of system (8) is *stochastically stable*.

**Corollary 6**

*Let p*≥*2 and let V be as above. Imposing the same assumptions as in Theorem 5 with N*→+∞, *then we have* ∀0 ≤ *t* ≤ *T*,

*Proof*. We know, from the definition of V(u), and Lemma 2 that it is easy to show that

**Corollary 7**

∀*p* ≥ 2 *and* ∀0 ≤ *t* ≤ *T*, with σ^{2}λ_{1}–*a*_{1}>0, *we have* ∀0 ≤ *t* ≤ *T*.

1) If *a*_{2} ≥ 0, then

2) If *a*_{2} > 0, then

*Proof*. 1) Note that we have Since λ_{n} is increasing in *n*,

So,

Pull over expectation, then

By using Corollary 6, we have

2) From the definition of V(*u*(*t*)), it is clear that

so

Now, take the expectation to both sides, and we get

**Remark:** The corollary 7 means that ∀*t* ≥ 0:

**Definition:** *The trivial solution of system (8) is said to be a.s. exponentially stable if*

(12)

∀u(0) ∈ D. The quantity of the left hand side of (12) is called *the sample top Lyapunov exponent of u*.

**Lemma 8**

*Let v(t) be a nonnegative integrable function such that* [9]

(13)

for some constants C, A. Then C ≥ 0 and

v(t) ≤ *C* exp(*At*), 0 ≤ t ≤ T (14)

**Theorem 9**

*Let V(u(t)) as in Theorem 5*. If *then the norm of the trivial solution of N-dimensional system (8) is a.s. exponentially stable with sample top Lyapunov exponent*

*θ* (*u _{N}*) ≤ 0.

*Proof*. Return to the analysis of finite *N*-dimensional equation (5). Recall that

But by our assumption that

so

where *k* ≥0.

Using Dynkin’s formula, we find that [10-16]

so

using extended Gronwall lemma, Lemma 8, gives us

hence

thus

(15)

If then the left side of identity (12) is negative and the trivial solution of the velocity *v* of N-dimensional system (8) is *a.s*. exponential stable.

Finally, we observe that all the previous estimates are uniformly bounded as N→∞. Hence, we arrive at

(16)

**Corollary 10**

*Let V(u(t)) as in Theorem 5*. *If* *then the norm of the v-component of the trivial solution of infinite-dimensional system (1) is a.s. exponentially stable with sample top Lyapunov exponent*

*θ*(*v _{N}*) ≤ –

*Proof*. Return to the proof of previous Theorem 9 and take the limit N to +∞ after the estimation process (16) in the sample Lyapunov exponent *θ*(*v _{N}*).

By analyzing appropriate *N*-dimensional truncations of the original semi-linear heat equations (1), we can verify the asymptotic stability of random Fourier series solutions with strongly unique, Markovian, continuous time Fourier coefficients under the presence of cubic nonlinearities. For this purpose, we introduced and studied an appropriate Lyapunov. The analysis is basicly relying on the fact that all estimations of **moments** of Lyapunov functional are made independent of dimensions N of their finite-dimensional truncations. Thus, the techniques of our proof are finite-dimensional in character, however the conclusions can be drawn to the original infinite-dimensional semilinear equation.

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