Some Problems On Multi-coupled Nonlinear Parabolic Equations

In this paper, the author studies a nonlinear parabolic system multi-coupled via nonlinear boundary flux and nonlinear inner source terms. The main problems focus on occurrence and development of singularities to the solutions. Especially, a common error is pointed out and corrected for proving the simultaneous blow-up rates. By introducing the characteristic matrix equation associated to the multi-coupled system, the conditions are determined to identify non-simultaneous and simultaneous blow-up. In particular, four different simultaneous blow-up rates with blow-up sets are obtained also.In addition, the author considers a nonlinear parabolic equations with convections. The existence and non-existence of global solutions is established. It is shown that for the critical case, the existence or non-existence of global solutions depends on the size of the domain.Problem I Consider the following nonlinear parabolic equations multi-coupled via nonlinear boundary flux and nonlinear inner source terms:where the parameters l_ij, p_ij > 0 (i,j = 1,2). The complicated and interesting model contains eight nonlinear parameters to describe the multi-nonlinearities there. Always assume that the coupling is complete, namely, at least one of the following four conditions holds: l₁₂l(21) > 0, l₁₂P₂₁ > 0,p₁₂l₂₁ > 0 and p₁₂p₂₁ > 0. There are many subjects to be considered for the coupled system, such as, the blow-up critical exponent, simultaneous and non-simultaneous blow-up conditions, and all possible blow-up rates. Clearly, a precise classification for the nonlinear parameters is needed here. For this reason, the characteristic matrix equation for Problem I is introduced to reach thefollowing topics:(a) the necessary and sufficient conditions for simultaneous blow-up only;(b) the conditions for non-simultaneous blow-up only;(c) the regions for coexistence of simultaneous and non-simultaneous blow-up;(d) the blow-up rate with boundary flux dominating;(e) the blow-up rate with inner source dominating;(f) the blow-up rate with cross-dominating;(g) blow-up set.Problem II Consider coupled nonlinear diffusion equations with convectionsut = aiAu"11 - biumi²\Vu\2 + uPlvqi, (x, t) e Q x (0, T),vt = a2Avm2 - b2vm2'2\Vv\2 + uP2vq\ (x,t) eflx (0,T),u(x, t) = ï¿¡0, v(x, t) = eQ, (x, t) e dQ x (0, T),u(x,0) = Uo(x), v(x,Q) — vo(x), x € fl,where Q C RjV is a bounded domain with smooth boundary;parameters P2,Qi > 0, Pi,<72 ^ 0, ï¿¡o > 0, rrij > 1, Oj > 0, bi > 0 (i = 1, 2). Due to the varieties of the exponent parameters and the coefficients, Problem II covers some nonlinear diffusion systems of different forms, such asut = uhl(Auni + uklvh), vt = vh2(Avn2 + uk2vh) with boundary condition u = v = €q, andu = AeniU + eklU+llV y — Aen2V + ek2U+l2Vwith u = v = 0 on the boundary.The existence and non-existence of global solutions is discussed. If ax = a2 = 1 and b\ = i>2 = 0, the results in this paper cover those in [66] with the following extensions:(a) the parameter region for the coexistence of global and non-global solutions;(b) in the critical case, the influence of the domain's size to the existence (global boundedness in fact) or non-existence of global solutions.