Boundary behavior and interior Hölder regularity of the solution to nonlinear stochastic partial differential equation driven by space-time white noise

Han, Beom-Seok; Kim, Kyeong Hun

doi:10.1016/j.jde.2020.07.002

Cited by 11 publications

(16 citation statements)

References 14 publications

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“…The idea of proof follows from [10,Section 8.4] or [5]. First, we consider the Lipschitz case λ = 0.…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

“…It should be remarked that solution u has α times as much regularity in space as in time; see Corollary 3.1. In addition, it is worthwhile noting that there are analogous arguments for α = 2; see [5,Corollary 2.15] and [10,Section 8.4]. Moreover, the relation among parameters and conditions are provided.…”

Section: Introductionmentioning

confidence: 98%

“…For λ ∈ [0, 1/2), Eq. (1.3) is treated on a bounded domain and the whole space; see [5,10,20]. In [20], the author handles Eq.…”

Section: Introductionmentioning

confidence: 99%

“…(1.3) with L = , ξ = 1, and nonnegative continuous deterministic initial data u(0) = u 0 . In [5], also on a bounded domain, the uniqueness, existence, maximal Hölder regularity, and boundary behavior of a solution are obtained when Eq. (1.3) has L = a D 2 + bD + c, random and space and time dependent coefficients a, b, c, ξ, and random nonnegative initial data u(0) = u 0 .…”

Section: Introductionmentioning

confidence: 99%

“…It should be mentioned that if α ↑ 2, the condition on λ coincides the one for the usual Laplacian case (α = 2). For example, see [5], [10,Section 8.4], and [20].…”

Section: Introductionmentioning

confidence: 99%

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A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

Han

2021

Stoch PDE: Anal Comp

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We present the existence, uniqueness, and regularity of a strong solution to a superlinear stochastic partial differential equation (SPDE) with the random fractional Laplacians:whereẆ is a space-time white noise, α ∈ (1, 2), and λ ∈ [0, α/2 − 1/2). The leading coefficient a satisfies the ellipticity condition and depends on (ω, t). The lower-order coefficients b, c, and ξ depend on (ω, t, x). The coefficients a ,b, c, and ξ are bounded. The initial data u 0 depends on (ω, x). The unique existence of local solutions to the SPDE follows from the unique solvability of a general Lipschitz case. We prove a Hölder embedding theorem for solution space H γ p (τ ) and maximum principle for SPDEs with the random fractional Laplacians to extend local solutions to a global one. The range of λ ∈ [0, α/2 − 1/2) depending on the highest order of the fractional Laplacian is given as a sufficient condition for the existence. When α ↑ 2, the condition is in accordance with the one for unique solvability of Laplacian case. Moreover, the Hölder embedding theorem provides maximal Hölder regularity of the solution u(ω, t, x), which has α times as much regularity in space as in time; for T ∈ (0, ∞) and small ε > 0, almost surely

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“…The idea of proof follows from [10,Section 8.4] or [5]. First, we consider the Lipschitz case λ = 0.…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

“…For λ ∈ [0, 1/2), Eq. (1.3) is treated on a bounded domain and the whole space; see [5,10,20]. In [20], the author handles Eq.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It should be mentioned that if α ↑ 2, the condition on λ coincides the one for the usual Laplacian case (α = 2). For example, see [5], [10,Section 8.4], and [20].…”

Section: Introductionmentioning

confidence: 99%

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A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

Han

2021

Stoch PDE: Anal Comp

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A regularity theory for stochastic generalized Burgers’ equation driven by a multiplicative space-time white noise

Han

2022

Stoch PDE: Anal Comp

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L-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

Han,

2024

Journal of Differential Equations

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Boundary behavior and interior Hölder regularity of the solution to nonlinear stochastic partial differential equation driven by space-time white noise

Cited by 11 publications

References 14 publications

A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

A regularity theory for stochastic partial differential equations driven by multiplicative space-time white noise with the random fractional Laplacians

A regularity theory for stochastic generalized Burgers’ equation driven by a multiplicative space-time white noise

L-regularity theory for the stochastic reaction-diffusion equation with super-linear multiplicative noise and strong dissipativity

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