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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.14966 |
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| _version_ | 1866912385357316096 |
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| author | Pineau, Ben Taylor, Mitchell A. |
| author_facet | Pineau, Ben Taylor, Mitchell A. |
| contents | In this article we are concerned with evolution equations of the form \begin{equation*} \partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u}) \end{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 \emph{highest} Sobolev exponent $s=s(q,d)$ for which the above evolution is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ (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 \begin{equation*} \partial_tu-Δu=\pm |u|^{p-1}u, \hspace{5mm} p>1, \end{equation*} is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ when $\max\{0,s_c\}<s<2+p+\frac{1}{q}$ and is \emph{strongly ill-posed} when $s\geq \max\{s_c,2+p+\frac{1}{q}\}$ and $p-1\not\in 2\mathbb{N}$ 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 Schrödinger equation and the corresponding well-posedness result when $p\geq \frac{3}{2}$. 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 $d\gg p$ that there are nonlinear Schrödinger equations which are ill-posed in \emph{every} Sobolev space $H_x^s(\mathbb{R}^d)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_14966 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the optimal Sobolev threshold for evolution equations with rough nonlinearities Pineau, Ben Taylor, Mitchell A. Analysis of PDEs In this article we are concerned with evolution equations of the form \begin{equation*} \partial_tu-A(D)u=F(u,\overline{u},\nabla u, \nabla \overline{u}) \end{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 \emph{highest} Sobolev exponent $s=s(q,d)$ for which the above evolution is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ (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 \begin{equation*} \partial_tu-Δu=\pm |u|^{p-1}u, \hspace{5mm} p>1, \end{equation*} is well-posed in $W_x^{s,q}(\mathbb{R}^d)$ when $\max\{0,s_c\}<s<2+p+\frac{1}{q}$ and is \emph{strongly ill-posed} when $s\geq \max\{s_c,2+p+\frac{1}{q}\}$ and $p-1\not\in 2\mathbb{N}$ 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 Schrödinger equation and the corresponding well-posedness result when $p\geq \frac{3}{2}$. 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 $d\gg p$ that there are nonlinear Schrödinger equations which are ill-posed in \emph{every} Sobolev space $H_x^s(\mathbb{R}^d)$. |
| title | On the optimal Sobolev threshold for evolution equations with rough nonlinearities |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.14966 |