Note on critical exponents for a system of heat equations coupled in the boundary conditions
The authors study the blow-up rate and uniqueness of nonnegative solutions of the system of heat equations \(u_t=u_{xx}\), \(v_t=v_{xx}\) on the half-line \(x>0\), coupled by the nonlinear boundary conditions \(u_x(0,t)=-v^p(0,t)\), \(v_x(0,t)=-u^q(0,t)\), where \(p,q>0\). \textit{K. Deng, M. Fila}, and \textit{H. A. Levine} [Acta Math. Univ. Comen., New Ser. 63, No. 2, 169-192 (1994; Zbl 0824.35048)] showed nonuniqueness of solutions of this system if \(pq<1\) and blow-up in finite time if \(pq>1\). In the latter case, they also established the corresponding blow-up rate for some class of initial conditions provided \(\min\{p,q\}\geq1\). The authors of the paper under review generalize the results on the blow-up rate by enlarging the class of admissible initial conditions and by removing the assumption \(\min\{p,q\}\geq1\). Moreover, they show uniqueness of solutions (of a higher dimensional analogue) of the system provided \(pq\geq1\).
scientific article; zbMATH DE number 1123527
scientific article