On the optimal Sobolev threshold for evolution equations with rough nonlinearities

In this article we are concerned with evolution equations of the form {equation*} ∂_tu-A(D)u=F(u,ū,∇ u, ∇ ū) {equation*} where $A(D)$ is a Fourier multiplier of either dispersive or parabolic type and the nonlinear term F is of limited regularity. Our objective is to develop a robust set of principles which can be used in many cases to predict the highest Sobolev exponent $s=s(q,d)$ for which the above evolution is well-posed in Wₓs,q(Rᵈ) (necessarily restricting to $q=2$ for dispersive problems). We will confirm the validity of these principles for two of the most important model problems; namely, the nonlinear Schr\"odinger and heat equations. More precisely, we will prove that the nonlinear heat equation {equation*} ∂_tu-Δ u=± |u|ᵖ⁻¹u, {5mm} p>1, {equation*} is well-posed in Wₓs,q(Rᵈ) when max\0,sc\<s<2+p+1/q and is strongly ill-posed when s≥ max,2+p+1/q\ and p-1∈ 2N in the sense of non-existence of solutions even for smooth, small and compactly supported data. When $q=2$, we establish the same ill-posedness result for the nonlinear Schr\"odinger equation and the corresponding well-posedness result when p≥ 3/2. Identifying the optimal Sobolev threshold for even a single non-algebraic $p>1$ was a rather longstanding open problem in the literature. As an immediate corollary of the fact that our ill-posedness threshold is dimension independent, we may conclude by taking d p that there are nonlinear Schr\"odinger equations which are ill-posed in every Sobolev space Hₓˢ(Rᵈ).