Stability of Fourier Solutions of Nonlinear Stochastic Heat Equations in 1D

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. 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


Introduction
In this article we study the stability of solutions of semi-linear stochastic heat equations Many authors have treated stochastic heat equations (e.g. [1,2]), semi-linear stochastic heat equations (e.g. [1][2][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).

Truncated Fourier Series Solution and Finitedimensional Lyapunov Functional
Consider the stochastic nonlinear heat equation with additive noise with the initial condition u(x,0)=f(x) with f ∈ L 2 (D) (initial position)
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 c n satisfying the naturally truncated system of stochastic differential equations (SDEs).
is stochastically stable.

Theorem 5
Let then the trivial solution of equation (8) is stochastically stable i.e., stable in probability.
The first and second partial derivative of V N (t) with respect to c n are ( ) But by our assumption that 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
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 Consider the Lyapunov functional defined in equation (6), and let Proof. See [7].
Since λ n is increasing in n, Pull over expectation, then By using Corollary 6, we have , 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

Theorem 9
Let V(u(t)) as in Theorem 5. If Proof. Return to the analysis of finite N-dimensional equation (5). Recall that But by our assumption that Using Dynkin's formula, we find that [10][11][12][13][14][15][16] using extended Gronwall lemma, Lemma 8, gives us Finally, we observe that all the previous estimates are uniformly bounded as N→∞. Hence, we arrive at

Corollary 10
Let V(u(t)) as in Theorem 5. If ( ) 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 ).

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 (14)