Non existence of global solutions of parabolic equation in conical domains (Q1904458)

In this paper is studied the initial-boundary value problem \[ u_t = \Delta u + |x |^\sigma u^p \text{ in } D \times (0,T), \quad u(x,t) = 0 \text{ on } \partial D \times (0,T),\;u(x,0) - u_0 (x) \text{ in } D, \] where \(\sigma \geq 0\), \(p > 1\), \(u_0 \geq 0\), \((1 + |x |^2)^{\sigma/2 (p - 1)} u_0\) is continuous and bounded in \(\overline D\) and \(u_0 = 0\) on \(\partial D\). Two theorems about nonexistence of nontrivial global solutions are proved.