Concentration phenomena in the semilinear parabolic equation
This paper concerns asymptotic behavior of global solutions of the following semilinear parabolic equation \[ \begin{cases} u_t-\Delta u=u^p,\quad& (x,t)\in\Omega\times(0,+\infty),\\ u=0, &(x,t)\in\partial\Omega \times(0,+\infty),\\ u=\tau u_0, &(x,t)\in\Omega \times\{0\}, \end{cases} \] where \(\Omega\) is the unit ball in \(\mathbb R^N\) (\(N\geq 3\)), \(p=(N+2)/(N-2)\), \(u_0\not\equiv 0\) is nonnegative, radial, bounded function. The author proves that there exists \(\tau_1>\tau_0>0\) such that: (a) the problem possesses a unique global solution \(u(t)\) for every \(\tau\in(0,\tau_0)\), and \(u(t)\to 0\) uniformly as \(t\to\infty\); (b) the global solution \(u(t)\) for \(\tau=\tau_0\) is not uniformly bounded, and there exists a time sequence \(\{t_n\}\) so that \(\{u(t_n)\}\) and \(\{\nabla u(t_n)\}\) converge in certain sense to some concentration measures as \(t_n\to \infty\); (c) in the case \(\tau>\tau_1\) the solution \(u(t)\) blows up in finite time \(t\).
scientific article; zbMATH DE number 1782038
scientific article