Critical exponents for the heat-conduction equation with nonlocal, nonlinear perturbations (Q812298)

The author deals with the existence and nonexistence of a nonnegative solutions (globally defined in time \(t> 0\)) of the problem \[ u_t(x,t)- \Delta_x u(x,t)= u^p(x,t) \Biggl(\int_{\mathbb{R}^N} u^q(y, t)\,dy\Biggr),\quad x\in \mathbb{R}^N,\quad t> 0,\tag{1} \] \[ u(x,0)= u_0(x)\geq 0,\quad x\in\mathbb{R}^N,\quad p> 1,\quad q> 0,\tag{2} \] where \(u(x,t)\) is an unknown function, \(N\geq 1\), \(\Delta_x\) is the Laplace operator with respect to the \(x\in\mathbb{R}^N\). The author provides conditions on \(p\) and \(q\) that guarantee the existence or nonexistence of global in \(t\), nonnegative solutions for (1)-(2) which do not vanish identically.