Comparing the stochastic nonlinear wave and heat equations: a case study

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order α>0) of a space-time white noise. In particular, we show that the well-posedness theory breaks at α=1/2 for SNLW and at α=1 for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for 0<α<1/2. We first revisit this argument and establish multilinear smoothing of order 1/4 on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of α. On the other hand, when α≥1/2, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for α≥1/2. (ii) As for SNLH, we establish analogous results with a threshold given by α=1. These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values (α=3/4 in the wave case and α=2 in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).