On Critical Exponents for a Semilinear Parabolic System Coupled in an Equation and a Boundary Condition

In this paper, we consider the system utsDuq¨ p, ¨tsD¨ xgRqN, t)0, ­ u ­ ¨ y s0, y suq x1s0, t)0, ­ x1 ­ x1 u x, 0.su0 x., ¨ x, 0.s¨0 x. xgRqN, where RqNs x1, x9.< x9gRNy1, x1)04, p, q)0, and u0 , ¨ 0 are nonnegative and bounded. We prove that if pqF1 every nonnegative solution is global. When pq)1 we let a s pq2.r2 pqy1., bs 2qq1.r2 pqy1.. We show that if max a, b.)Nr2 or max a, b.sNr2 and p, qG1, then all nontrivial nonnega- tive solutions are nonglobal; whereas if max a, b.-Nr2 there exist both global and nonglobal nonnegative solutions