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

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and heat (SNLH) with a quadratic nonlinearity, forced by fractional derivative (of order $\alpha > 0$) of space-time white noise. In particular, we show that well-posedness theory breaks at = \frac 12$ for SNLW 1$ SNLH. This provides first example showing behaves less favorably than (i) As SNLW, Deya (2020) essentially proved its local $0 < \alpha 12$. revisit this argument establish multilinear smoothing $\frac 14$ on second term in spirit recent work Gubinelli, Koch, Oh (2018). allows us to simplify some range $. On other hand, when \geq 12$, is ill-posed sense not continuous function time values spatial distributions. shows standard method such as Da Prato-Debussche trick or variant, based higher expansion, down \ge (ii) SNLH, analogous results threshold given 1$. These examples case rough noises, existing singular PDEs before reaching critical ($\alpha 34$ 2$ case) predicted scaling analysis (due Deng, Nahmod, Yue (2019) due Hairer (2014) case).