Stability of ODE blow-up for the energy critical semilinear heat equation
The authors consider the energy critical semilinear heat equation \[ \begin{cases} \partial_t u =\Delta u +|u|^{p-1}u, \;p=\frac{d+2}{d-2} & (t,x)\in{\mathbb R}\times{\mathbb R}^d, d\geq 3,\\ u(0,x)=u_0(x)\in {\mathbb R}.\end{cases} \] They give a self-contained proof of the stability of solutions \(u\) blowing up in finite time with type-I ODE blow-up \[ \|u\|_{L^\infty}\sim k(T-t)^{\frac{d-2}{4}}, \;T>0, \;k=\big( \frac{d-2}{4} \big)^{\frac{d-2}4}, \] which adapts to the energy critical case. It is proved that e set of solutions blowing up in finite time with type-I blow-up is open in \(W^{3,\infty}({\mathbb R}^d).\)
semilinear heat equations, blow-up solutions
scientific article; zbMATH DE number 6673226
scientific article