Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded
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Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain M. Salins1 Received: 4 March 2020 / Revised: 27 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution. Keywords Stochastic partial differential equations · Stochastic heat equation · Kolmogorov continuity theorem
1 Introduction We prove the existence and uniqueness of a mild solution to the nonlinear stochastic heat equation ⎧ 1 ⎨ ∂u (t, x) = u(t, x) + f (u(t, x)) + σ (u(t, x))W˙ (t, x), t > 0, x ∈ Rd , ∂t 2 ⎩u(0, x) = u (x) 0 (1.1)
B 1
M. Salins [email protected] Boston University, Boston, MA, USA
123
Stoch PDE: Anal Comp
in the case where f : R → R is not globally Lipschitz continuous. Instead we assume that there exists κ ∈ R such that for any u 1 < u 2 f (u 2 ) − f (u 1 ) ≤ κ(u 2 − u 1 ),
(1.2)
and f satisfies a growth condition. As a motivating example, let f : R → R be an odd-degree polynomial with negative leading coefficient f (u) = −αu 2m+1 +
2m
ak u k
k=0
where m ∈ N, α > 0 and ak ∈ R. Such polynomials are not globally Lipschitz continuous, but they do satisfy (1.2) where κ = supu f (u), which is finite because f (u) is an even polynomial with negative leading term. Our set-up also allows us to consider some pathological drift terms that are not locally Lipschitz continuous such as decreasing functions that are not absolutely continuous with respect to Lebesgue measure. In (1.1), W˙ is a Gaussian noise that is white in time and spatially homogeneous satisfying the strong Dalang condition (see Assumption 2.1 below). We assume that σ is globally Lipschitz continuous. A mild solution to (1.1) is defined to be an adapted random field solution to the integral equation (see [28]) t G(t, x − y)u 0 (y)dy + G(t − s, x − y) f (u(s, y))dyds Rd 0 Rd t + G(t − s, x − y)σ (u(s, y))W (dyds), (1.3)
u(t, x) =
0
Rd
|x|2
where G(t, x) := (2π t)− 2 e− 2t is the fundamental solution of the heat equation in Rd . The existence and uniqueness of solutions to (1.1) when f and σ are both globally Lipschitz continuous was proved by Dalang [9] using a Picard it
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