Blowup for the nonlinear heat equation with small initial data in scale-invariant Besov norms (Q1729713)

The authors study the Cauchy problem for the semilinear heat equation \[ \begin{cases} \partial_t u = \Delta u + u^p, & \quad (x,t) \in \mathbb{R}^n \times (0,T),\\ u(x,0) = u_0(x), & \quad x \in \mathbb{R}^n, \end{cases} \] where $n \in \mathbb{N}$ and $p \in \mathbb{N}$ with $p \ge 2$ is such that $\frac{n(p-1)}{2} >1$, i.e. $p > p_F$, where $p_F = \frac{n+2}{n}$ is the Fujita exponent. \par The authors prove that for any $\delta >0$ and $q \in (p,\infty]$ there exists $u_0$ in the Schwartz class $\mathcal{S} (\mathbb{R}^n)$ satisfying $\| u_0 \|_{\dot{B}^{-2/p}_{np(p-1)/2,q}(\mathbb{R}^n)} \le \delta$ such that the maximal existence time $T^\ast$ of the solution $u \in C([0,T^\ast); L^{n(p-1)/2}(\mathbb{R}^n))$ is finite, i.e. $u$ blows up in finite time. \par This result shows that the value $r= \frac{np(p-1)}{2}$ in the homogeneous Besov space $\dot{B}^{-2/(p-1) + n/r}_{r,q} (\mathbb{R}^n)$ is critical in the following sense. For any $r \in (\frac{n(p-1)}{2}, \frac{np(p-1)}{2})$ and $q \in [1,\infty]$ there exists $\delta >0$ such that for any $u_0 \in L^{n(p-1)/2} (\mathbb{R}^n)$ with $\| u_0 \|_{\dot{B}^{-2/(p-1) + n/r}_{r,q} (\mathbb{R}^n)} \le \delta$ the solution $u$ exists globally in time (see [\textit{C. Miao} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 5, 1329--1343 (2007; Zbl 1124.35029)]), while the authors' result described above shows that this is not true for $r= \frac{np(p-1)}{2}$. \par Important ingredients of the proof are Fourier transform and the Littlewood-Paley decomposition.