Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations

Abstract

ABSTRACT. We consider the weakly coupled system of reaction-diffusion equations ut = Au + \x\ ai vP, vt=Av+ |a:r 2 it 9 ,where x E R^ (AT > 1), t > 0, p, q > 1 with pq > 1 and 0 < ai < N(jp -1), 0 < (72 < N(q -1). Put

Existence and nonexistence of global solutions of degenerate and singular parabolic systems

“…Fujita-type results for system (1.2) were obtained in [8] for k 1 = k 2 = 0 and in [9] for σ 1 = σ 2 = 0. Theorem 4.2 includes the blowup results of [8,9], giving a unique nonexistence condition containing all the parameters. This answers a question posed in Deng-Levine [1].…”

“…The nonexistence condition given in Theorem 4.2 includes all the parameters k i and σ i (i = 1, 2) and reduces to those found in [8,9] in the particular cases k i = 0 and σ i = 0, respectively. This solves a question posed in Deng and Levine [1].…”

Existence and Nonexistence of Global Solutions in Time for a Reaction-Diffusion System with Inhomogeneous Terms

“…Moreover, Fujita's result were also extended by Escobedo-Herrero [2] and Mochizuki [11] to the system (1) with K 1 ðx; tÞ ¼ K 2 ðx; tÞ ¼ 1, by Uda [18] to the system (1) with K i ðx; tÞ ¼ t q i ði ¼ 1; 2Þ, and by Mochizuki-Huang [12] to the system (1) with K i ðx; tÞ ¼ jxj s i with s i A ½0; nð p i À 1ÞÞ ði ¼ 1; 2Þ.…”

“…Here, we take the same strategy as in [12] and [13]. We will show that for an appropriate choice of r, ðF m ðtÞ; G m ðtÞÞ is not global in time, thereby contradicting the assumption that ðu; vÞ is a global solution.…”

“…Although the Fujita type critical exponent to the system (1) was established by Escobedo-Herrero [2] and Uda [18], their proofs were rather complicated. Mochizuki [11] and Mochizuki-Huang [12] simplified their proof and also determined the ''second critical exponent'' on the decay rate of the initial data. Our results are natural extensions of Mochizuki-Huang [12], and are proved by applying the arguments of Mochizuki-Huang [12], Pinsky [13] and Umeda [19,20].…”