Existence and blow-up for a degenerate parabolic equation with nonlocal source
The authors investigate positive solutions of the nonlinear degenerate equation \(u_t=u^p(\Delta u+au\int_\Omega u^q dx)\) in \(x\in\Omega\), \(t>0\), with Dirichlet boundary conditions \(u(x,t)=0\) on \(x\in\partial\Omega\), \(t>0\) and \(u(x,0)=u_0(x)\). \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with a smooth boundary, \(p>1\), \(q\geq 1\) and \(a>0\). In order to find conditions on the existence of global solutions and blow-up solutions, in addition to compatibility assumptions, the first eigenvalue \(\lambda_1\) of the homogeneous Dirichlet problem associated to \(\Omega\) plays a crucial role. If \(\phi\) is the corresponding eigenfunction normalized so that \(\max_\Omega \phi(x)=1\), then the following results are proved. 1) If \(a\leq \lambda_1/\int_\Omega\phi^q dx\) and \(u_0(x)\leq \phi(x)\) then a solution exists globally. 2) If \(a> \lambda_1 \int_\Omega\phi^q dx\) and \(u_0(x)\geq \phi(x)\) then the solution blows-up in a finite time. 3) All solutions blow-up in a finite time provided that \(u_0(x)\) is sufficiently large. 4) If \(u_0(x)\) satisfies suitable conditions and if the solution \(u(x,t)\) blows up at finite time \(T^*\) then there exist two positive constants \(C_1\) and \(C_2\) such that \(C_1(T^*-t)^{-1/(p+q)}\leq \max_\Omega(x,t)\leq C_2(T^*-t)^{-1/(p+q)}\).
scientific article; zbMATH DE number 1879964