For a class of scaling-critical reaction-diffusion equations with log-attenuated nonlinearities and white-noise initial data, the solutions converge to a Gaussian McKean-Vlasov process whose variance is set by an ODE.
Norm inflation for the cubic nonlinear heat equation above the scaling critical regularity
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abstract
We consider the ill-posedness issue for the cubic nonlinear heat equation and prove norm inflation with infinite loss of regularity in the H\"older-Besov space $\mathcal C^s = B^{s}_{\infty, \infty}$ for $ s \le -\frac 23$. In particular, our result includes the subcritical range $-1< s \le -\frac 23$, which is above the scaling critical regularity $s = -1$ with respect to the H\"older-Besov scale. In view of the well-posedness result in $\mathcal C^s$, $s > -\frac 23$, our ill-posedness result is sharp.
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McKean-Vlasov limits of scaling-critical reaction-diffusion equations with random initial data
For a class of scaling-critical reaction-diffusion equations with log-attenuated nonlinearities and white-noise initial data, the solutions converge to a Gaussian McKean-Vlasov process whose variance is set by an ODE.