We establish the existence and uniqueness of solutions to a $(1+1)$-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equations of this type describe non-Markovian dynamics with long-range temporal and spatial dependencies. The main difficulty is that the solution to the linearized equation has low spatial H\"{o}lder regularity, rendering the singular nonlinearity $G(u)\partial_x u$ classically ill-defined. The proof relies on two main ingredients: first, combining the Da Prato-Debussche decomposition with rough path theory to rigorously interpret the nonlinear product; and second, deriving Schauder-type estimates for fractional heat kernels given in terms of slowly decaying Mittag-Leffler functions. The nonlocality of the fractional time derivative requires a strong assumption on the initial condition.
We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive fractional time-space white noise (ii) With multiplicative time-space Brownian noise The fractional time derivative is interpreted as the Caput...
We study the initial–boundary value problem for a fractional nonlinear Klein–Gordon equation posed on the hyperoctant Dn,n⩾2 with nonhomogeneous boundary data on each coordinate hyperplane. The equation involves a spatial fractional Laplacian of Caputo type with order β∈(3/2,2), which induces strong nonlocal dissipatio...
To quantitatively characterize the long-time dynamics of the periodic damped stochastic Korteweg--de Vries (sKdV) equation driven by additive noise, we investigate the uniform-in-time error estimates for a Lie--Trotter operator splitting approximation. This splitting combines the exact deterministic KdV flow with the e...
Time-fractional diffusion equations describe subdiffusive transport in heterogeneous media, but their numerical treatment is complicated by the nonlocal Caputo derivative and by the weak singularity that the solution develops at the initial time. We study a global-in-time discretization that combines spectral collocati...
For weak solutions to quasilinear degenerate parabolic equations of $p$-Laplace type, a central obstacle in applying the method of intrinsic scaling to prove their H\"{o}lder regularity is the derivation of forward-in-time propagation estimate for the spatial measure of level sets. In this paper, we revisit and overcom...
Xia Hao, Yan Li, Zhi-Wen Zhao· 0 citations
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