Department of Mathematics, Zayed University, Dubai, UAE
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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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  study the stability of the mild solution. Walsh reats the stochastic heat equations in one dimension. Chow  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
with the initial condition u(x,0)=f(x) with f ∈ L2(D) (initial position) and and 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  and given by
Assume that with ux∈L2(D) and then for all t ≥ 0, x ∈ D=(0,lx), the Fourier-series solutions (2) have Fourier coefficients cn satisfying (a.s.)
Proof. See Schurz .
We need to truncate the infinite series (2) for practical computations. So, we have to consider finite-dimensional truncations of the form
with Fourier coefficients cn satisfying the naturally truncated system of stochastic differential equations (SDEs).
Assume that Define the Lyapunov functional VN as follows
This functional is a modification of a functional appeared in Schurz . 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
Consider the Lyapunov functional defined in equation (6), and let
Proof. See .
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,
(c) satisfies the condition of radial unboundedness
Proof. See .
Recall equation (5) governed by
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
whenever δ > 0.
If ∃ a 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 .
If 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,σ2λn – a1>0. Define the linear operator L as in Schurz 
The first and second partial derivative of VN(t) with respect to cn are
But by our assumption that
LVN(cn(t)) ≤ 0.
So by Lemma 4, applied to truncation of (8), the trivial solution of system (8) is stochastically stable.
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
∀p ≥ 2 and ∀0 ≤ t ≤ T, with σ2λ1–a1>0, we have ∀0 ≤ t ≤ T.
1) If a2 ≥ 0, then
2) If a2 > 0, then
Proof. 1) Note that we have Since λn is increasing in n,
Pull over expectation, then
By using Corollary 6, we have
2) From the definition of V(u(t)), it is clear that
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
∀u(0) ∈ D. The quantity of the left hand side of (12) is called the sample top Lyapunov exponent of u.
for some constants C, A. Then C ≥ 0 and
v(t) ≤ C exp(At), 0 ≤ t ≤ T (14)
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 (uN) ≤ 0.
Proof. Return to the analysis of finite N-dimensional equation (5). Recall that
But by our assumption that
where k ≥0.
using extended Gronwall lemma, Lemma 8, gives us
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
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(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).
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.