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One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE

$$\partial_t u=\Delta u-u^3+\xi,$$

where $\xi$ is space-time white noise. It is difficult to study because $u$ is distribution valued in $d\geq 2$, so the cubic nonlinearity is a-priori ill-posed. Typically, the way to solve this is to mollify the noise and solve the renormalized

$$\partial_t u_\varepsilon=\Delta u_\varepsilon-u_\varepsilon^3+C_\varepsilon u+\xi_\varepsilon,$$

and hope that the limit exists as $\varepsilon\to 0$, as $C_\varepsilon\to\infty$.

It seems like the SPDE

$$\partial_t u=\Delta u-u^2+\xi,$$

is a slightly easier prototype of a singular SPDE. Why has there been more work done for the cubic nonlinearity than the square? I know $\Phi_d^4$ comes from stochastic quantization but is the cubic or square nonlinearity more pathological purely from a stochastic analytic perspective?

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    $\begingroup$ I'm not too familiar with this, so take the following with a grain of salt. Ignoring the noise, $\partial_t u=\Delta u-u^3$ is better behaved than $\partial_t u=\Delta u-u^2$, as the latter can suffer from finite-time blowups of $u \to -\infty$ when the initial $u$ is negative. This pathology probably counts against the square nonlinearity. Furthermore, even if one could avoid this pathology in the case without noise (taking an initial condition with $u$ positive), adding noise likely can stochastically drive $u$ negative and again suffers the blowup. $\endgroup$
    – user196574
    Commented Sep 22 at 8:14

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