Initial value problems for the nonlinear heat equation in Triebel-Lizorkin spaces (Q1887994)

The paper deals with the Cauchy problem \[ \begin{cases} u_t-\Delta u=F(u,D_xu,D^2_xu),\quad x\in \mathbb R^n,\;t>0,\\ u(0,x)=\varphi(x) \end{cases} \] with initial data \(\varphi\) belonging to suitable Triebel-Lizorkin space. The right-hand side \(F(\cdot)\) is supposed to be smooth in a neighborhood of zero and \(F(\cdot)=O(| \cdot| ^{1+\alpha}),\) \(\alpha\in \mathbb N,\) \(\alpha>n/2.\) Global-in-time existence is proved for the above problem with small initial data.