Finite time blow-up for the fractional critical heat equation in \(\mathbb{R}^n\)
The authors study finite time blow-up for the fractional critical heat equation in \(\mathbb{R}^n\), \[ u_t+(-\Delta)^su=u^{\frac{n+2s}{n-2s}} \quad \text{in }\mathbb{R}^n\times (0,T), \quad \left. u\right\vert_{t=0}=u_0(x) \] by the method of inner-outer gluing. If \(0<s<1\), \(4s<n<6s\) there is a solution that blows up at \(q_1, \dots, q_k\) at finite time \(T\), where \(q_1, \dots q_k\) are distinct \(k\) points. It takes the form \[ u(x,t)=\sum_{j=1}^kU_{\lambda_j(t), \xi_j(t)}(x)+Z_0^\ast (x)+\Theta(x,t), \] where \(Z_0^\ast \in C_0^\infty(\mathbb{R}^n)\) satisfies \(Z_0^\ast(q_j)<0\), \[ U_{\lambda, \xi}=\lambda^{-\frac{n-2s}{2}}(t)U\left( \frac{x-\xi(t)}{\lambda(t)}\right) \] is a rescaled bubble for \[ U(y)=\text{constant}\times \left( \frac{1}{1+\vert y\vert^2}\right)^{\frac{n-2s}{2}} \] satisfying \[ (-\Delta)^sU=U^{\frac{n+2s}{n-2s}} \quad \text{in \(\mathbb{R}^n\)}, \] and \(\lambda_j(t)=\kappa_j(T-t)^{\frac{2}{6s-n}}(1+o(1))\rightarrow 0\) with \(\kappa_j>0\), \(\xi_j(t)\rightarrow q_j\), and \(\Vert \Theta\Vert_\infty\leq T^c\) for some \(c>0\). Several suggestions are then derived from this result for related problems.
scientific article; zbMATH DE number 7187823
scientific article