A nonlinear degenerate parabolic system with non-local source and crosswise-diffusion (Q1404184)

The authors study the asymptotic behaviour of the initial-boundary value problem \[ u_t= u^{r_1} (u_{xx} + a_1 u \|u \|_{q_1}^{p_1-1}), \quad v_t= v^{r_2} (v_{xx} + a_2 v \|v \|_{q_2}^{p_2-1}) \] on a bounded interval \(\Omega\) with Dirichlet zero boundary conditions and suitable initial data \(u_0(x)\) and \(v_0(x)\). The motivation comes from extending work on single equations of this form and from mathematical biology. The results concern finite-time blow-up of solutions and global existence. Using essentially comparison methods and results for scalar equations of the above form, the authors prove that if \(\lambda < \lambda_1\), all positive solutions of the system of equations blow up in finite time, while if \(\lambda \geq \lambda_2\), all positive solutions exist globally. Here \(\lambda\) is the first eigenvalue of \(-\Delta\) on \(\Omega\) with Dirichlet boundary conditions; the corresponding normalised eigenfunction is denoted by \(\phi(x)\). To define the quantities \(\lambda_1\) and \(\lambda_2\) we need some notation. Let \(\Pi(x)\) be defined as \(u_0(x)/\phi(x)\) for \(x \in \Omega\) and as \(\lim_{y \rightarrow x} u_0(y)/\phi(y)\) for \(x \in \partial \Omega\), and let \(k_1= \max_{x \in \overline{\Omega}} \Pi(x)\), \(k_2= \min_{x \in \overline{\Omega}} \Pi(x)\). Similarly, let \(\Psi(x)\) be \(v_0(x)/\phi(x)\) for \(x \in \Omega\) and \(\lim_{y \rightarrow x} v_0(y)/\phi(y)\) for \(x \in \partial \Omega\); \(k_3 = \max_{x \in \overline{\Omega}} \Psi(x)\), \(k_4= \min_{x \in \overline{\Omega}} \Psi(x)\). Then we set \[ \lambda_1 = \min \{ a_1 \|k_2 \phi\|^{p_1 -1}_{q_1}, a_2 \|k_4 \phi\|^{p_2 -1}_{q_2} \} \quad \lambda_2 = \max \{ a_1 \|k_1 \phi\|^{p_1 -1}_{q_1}, a_2 \|k_3 \phi\|^{p_2 -1}_{q_2} \}. \]