A note on nonexistence of global solutions to a nonlinear integral equation
The authors consider the equation \(u_t=-(-\Delta)^{\beta/2} u+h(t)u^{1+ \alpha}\) in \(\mathbb{R}^n \times (0,T)\), where \(0<\beta\leq 2\). They extend results of \textit{H. Fujita}, who treated the case \(\beta=2\) and \(h\equiv 1\) [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.34002)]. The main result of the present paper gives sufficient conditions so that a solution \(u\) of the considered equation blows up in finite time, that means there exists \(T_0\) so that \(u(x,t)=\infty\) for every \(t\geq T_0\) and \(x\in \mathbb{R}^n\).
scientific article; zbMATH DE number 1421340