On the steady states of a nonlocal semilinear heat equation (Q2724362)

The paper deals with existence and nonexistence of steady states for the following nonlocal semilinear heat equation NEWLINE\[NEWLINE u_t= u_{xx} + \varepsilon \Biggl(\int_0^1 |u(t,x)|dx\Biggr)^q \Biggl/ (1-u)^\betaNEWLINE\]NEWLINE for \(0<x<1\), \(t>0\), with the zero Dirichelet boundary condition and initial condition \(u(x,0)=u_0(x)\). By using the strong maximum principle and modifying a technique introduced by \textit{K. Deng} [SIAM J. Math. Anal. 26, 98-111 (1995; Zbl 0824.35010)], the authors obtain existence and nonexistence results of steady states, related to the value of the parameter \(\varepsilon>0\).