Asymptotic Analysis To Singular Solutions Of Multi-nonlinear Parabolic Systems With Inner Absorptions

This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic systems with inner absorptions and coupled boundary fluxes, such as multiple blow-up rates under different dominations of nonlinearities, non-simultaneous versus simultaneous blow-up, and so on. Firstly, a multi-nonlinear model with inner absorptions and coupled boundary fluxes of mixed type nonlinearities is discussed. The critical exponent is obtained, and clearly described via a so called characteristic algebraic system. In particular, another characteristic algebraic system with two new parameters is introduced to simply show the multiple blow-up rates. It is interesting to observe that two of the multiple blow-up rates obtained here do depend on the absorption exponents, unlike all the known results that the blow-up rates in the current literatures, to our knowledge, are all absorption-independent. Furthermore, in order to explore the reason for the new phenomenon (i.e., it is due to the mixed type nonlinearities or the coupling mechanism), the same problem is considered for the case of single type nonlinearities, and the blowup rates of the same propeties are obtained also. This is to confirm that the coupling mechanism plays the key role for the absorption-relevant blow-up rates. In addition, the thesis discusses non-simultaneous blow-up of solutions for coupled parabolic systems with negative-negative sources and positive-negative sources, respectively. Thereby, the influence of the sign of sources to non-simultaneous blow-up is determined.The main results obtained in this thesis can be summarized as follows:(â… ) Multiple blow-up rateIn Chapter 2, for the system with mixed type nonlinearities u_t =Δu - a₁u^m, v_t =Δv-a₂e^nv in (0,1)×(0,T), (?) =e^pv, (?) = u^q on d (?)Ω×(0, T), the critical blow-up exponent is established by using the comparision principle. Furthermore, multiple simultaneous blow-up rates of solutions with N = 1 (N is the space dimension) are established by Green's identity and the Scaling method . It should be mentioned that in previous literatures, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates, while here some absorption-erlevant simultaneous blow-up rates are obtained. In Chapter 3, the system u_t = u_xx - a₁u^m, v_t = v_xx - a₂vⁿ in (0,1)×(0, T) with coupled boundary fluxes u_x(1,t) = v^p, v_x(1,t) = u^q, u_x(0,t) = v_x(0,t) = 0 is considered. The multiple simultaneous blow-up rates obtained with a complete classification for all the nonlinear parameters of the model, where two absorption-relevant ones are observed also. This is to say that the absorption-relevant blow-up rates should be caused by the coupling mechanism. If p = q, m = n with u₀(x) = v₀(x), the system reduces to a scalar problem, which belongs to the class of absorption-independent blow-up rate. This shows a substantial difference between the coupled systems and the scalar equations with inner absorptions.(â…¡) Non-simultaneous versus simultaneous blow-upChapter 4 deals with the initial-boundary problem for u_t = u_xx -λ₁u^α₁, v_t = v_xx -λ₂v^β₁ in (0,1)×(0, T) coupled via u_x(1, t) = u^α₂v^p, v_x(1, t) = u^qv^β₂, u_x(0, t) = v_x(0, t) = 0, t∈(0, T). Firstly, by introducing an auxilary problem and a cut-off function, a basic lemma is proved. Then, combining with Green's identity and the Scaling method, the necessary-sufficient conditions for non-simultaneous blow-up of solutions under suitable initial data as well as the sufficient conditions under which any blow-up of solutions would be non-simultaneous are established.Chapter 5 considers non-simultaneous blow-up of solutions for the system with positive-negative sources u_t = u_xx + u^α₁ v_t = v_xx - v^β₁ in (0,1)×(0,T), coupled via boundary conditions u_x(1,t) = u^α₂v^p, v_x(1,t) = u^qv^β₂, u_x(0,t) = vx_0,t = 0, t∈(0,T). The non-symmetry of components u and v leads to a more complicated discussion. Comparing with the corresponding conclusions for the two models in Chapters 4 and 5, the contributions of the signs of sources to the non-simultaneous blow-up of solutions are shown here clearly.