Chaotic characterization of one dimensional stochastic fractional heat equation

We study the Cauchy problem of the nonlinear stochastic fractional heat equation ∂ t u = − ν 2 ( − ∂ x x ) α 2 u + σ ( u ) W ˙ ( t , x ) on real line R driven by space-time white noise with bounded initial data. We analyze the large- | x | fixed- t behavior of the solution u t ( x ) for Lipschitz continuous function σ : R → R under three cases. (1) σ is bounded below away from 0. (2) σ is uniformly bounded away from 0 and ∞ . (3) σ ( x ) = c x (the parabolic Anderson model). From the sensitivity to the initial data of stochastic fractional heat equation, we describe that the solution to the Cauchy problem of stochastic fractional heat equation exhibits chaotic behavior at fixed time before the onset of intermittency.