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ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Stability of Fourier Solutions of Nonlinear Stochastic Heat Equations in 1D

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: haziem67@gmail.com

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.

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Abstract

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.

Keywords

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

Introduction

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

Equation

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

Equation

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:

Equation

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).

Truncated Fourier Series Solution and Finitedimensional Lyapunov Functional

Consider the stochastic nonlinear heat equation with additive noise

Equation   (1)

with the initial condition u(x,0)=f(x) with f ∈ L2(D) (initial position) andEquation and Equation driven by i.i.d. standard Wiener processes Wn with E[Wn(t)]2=0, E[Wn(t)]2=t. The solution of equation (1) in terms of Fourier series is proved by Schurz [3] and given by

Equation    (2)

Theorem 1

Assume that Equation with ux∈L2(D) andEquation then for all t ≥ 0, x ∈ D=(0,lx), the Fourier-series solutions (2) have Fourier coefficients cn satisfying (a.s.)

Equation   (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

Equation   (4)

with Fourier coefficients cn satisfying the naturally truncated system of stochastic differential equations (SDEs).

Equation     (5)

where Equation

Assume that Equation Define the Lyapunov functional VN as follows

Equation    (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 a2 ≥ 0, radially unbounded if additionally σ2π2>a1l2. Equipped with Euclidean norm

Equation

Lemma 2

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

Equation

Then ∀u∈L2(D):

Equation    (7)

Proof. See [7].

Lemma 3

Assume that a2 ≥ 0. Then, ∀N ∈ N, the functional VN is

(a) nonnegative and positive semi-definite if σ2π2>a1l2 or a2 ≥ 0.

(b) positive-definite ifσ2π2>a1l2,

and

(c) satisfies the condition of radial unboundedness

Equation

Proof. See [7].

Stability of Fourier Solutions

Recall equation (5) governed by

Equation   (8)

Equation   (9)

To simplify, let

Equation

Definition: The trivial solution of system (8) (in terms of norm Equation) 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

Equation    (10)

whenever δ > 0.

Lemma 4

Ifa positive-definite function V∈C2,1(Rd×[0. ∞),R+) such that LV(x,t) ≤ 0 and ∀( x,t)∈ Rd×[0. ∞), then the trivial solution of the equation.

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

is stochastically stable.

Proof. See Arnold [8].

Theorem 5

Let Equation

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

Proof. From Lemma 3, we know that VN(u(t)) is positive-definite if ∀nN,σ2λna1>0. Define the linear operator L as in Schurz [3]

Equation

The first and second partial derivative of VN(t) with respect to cn are

Equation

But by our assumption that

Equation

Then thus

LVN(cn(t)) ≤ 0.

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

Corollary 6

Let p2 and let V be as above. Imposing the same assumptions as in Theorem 5 with N→+∞, then we have ∀0 ≤ tT,

Equation

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

Equation

Corollary 7

p ≥ 2 and ∀0 ≤ tT, with σ2λ1a1>0, we have ∀0 ≤ tT.

1) If a2 ≥ 0, then

Equation

2) If a2 > 0, then

Equation

Proof. 1) Note that we have Equation Since λn is increasing in n,

Equation

So,

Equation

Pull over expectation, then

Equation

By using Corollary 6, we have

Equation

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

so

Equation

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

Equation

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

Equation

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

Equation   (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]

Equation   (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 Equation then the norm of the trivial solution of N-dimensional system (8) is a.s. exponentially stable with sample top Lyapunov exponent

θ (uN) ≤ 0.

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

Equation

Equation

But by our assumption that

Equation

so

Equation

where k ≥0.

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

Equation

so

Equation

using extended Gronwall lemma, Lemma 8, gives us

Equation

hence

Equation

thus

Equation    (15)

If Equation 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

Equation    (16)

Corollary 10

Let V(u(t)) as in Theorem 5. If Equation 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(vN) ≤ –k<0.

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(vN).

Conclusion

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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