Blow-up behaviors for semilinear parabolic systems coupled in equations and boundary condi\-tions
The paper deals with the nonlinear parabolic system \[ u_t = \Delta u + v^p, \qquad v_t = \Delta v + u^k \qquad (x \in\mathbb{R}^N_+, \quad t>0) \] equipped by the initial conditions \[ u(x,0) = u_0(x), \qquad v(x,0) = v_0(x) \qquad (x \in\mathbb{R}^N_+) \] and the boundary conditions \[ -\frac{\partial u}{\partial x_1} = v^q, \qquad -\frac{\partial v}{\partial x_1} = u^m \qquad (x_1 = 0, \; t>0), \] where \(\mathbb{R}^N_+ = \{ (x_1, x'): x_1>0, \; x' \in \mathbb R^{N-1} \}\). It is supposed that the initial data are bounded nonnegative \(C^1\)-functions such that \[ \lim_{| x| \to \infty} u_0(x) = \lim_{| x| \to \infty} v_0(x) = 0, \qquad \frac{\partial u_0}{\partial x_1} \leq 0, \quad \frac{\partial v_0}{\partial x_1} \leq 0 \qquad (x \in \mathbb{R}^N_+) \] and that the compatibility conditions hold. It is proved, under suitable assumptions on \(p,q\), that the blow up occurs only on the boundary of \(\mathbb R^N_+\), moreover the rate of the blow up of the solution is estimated. The authors have mentioned that the paper is closely related to the earlier papers [\textit{M. Chlebik} and \textit{M. Fila}, Rend. Mat. Appl., VII. Ser. 19, 449--470 (1999; Zbl 0980.35057); \textit{K. Deng, M. Fila} and \textit{H. A. Levine}, Acta Math. Univ. Comen. 63, 169--192 (1994; Zbl 0824.35048); \textit{S. Wang, C. H. Xie} and \textit{M. X. Wang}, J. Math. Anal. Appl. 218, 313--324 (1998; Zbl 0891.35052)].
scientific article; zbMATH DE number 2157556
scientific article