2025/05/20 by Ben Pineau, Pineau, Ben, Mitchell A. Taylor +1 · 1 citation
Mathematics · Engineering · #Nonlinear Partial Differential Equations #Advanced Numerical Methods in Computational Mathematics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2505.14966
In this article we are concerned with evolution equations of the form ∂tu-A(D)u=F(u,u,∇ u, ∇ u) 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 Schrödinger and heat equations. More precisely, we will prove that the nonlinear heat equation ∂tu-Δu=± |u|p-1u, \hspace5mm pgt;1, is well-posed in Wxs,q(ℝd) when max\0,sc\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 Schrödinger equations which are ill-posed in every Sobolev space Hxs(ℝd).