Asymptotic Analysis To Singular Solutions Of Multi-nonlinear Parabolic Equations

This thesis deals with asymptotic behavior of singular solutions for multi-nonlinear parabolic equations (systems). The topics include the critical exponent for a nonlinear diffusion system, the influences of the gradient perturbations on the blow-up properties of solutions for nonlinear parabolic equations, and the quenching behavior of solutions for parabolic problems with singular absorptions. Firstly, we consider a nonlinear diffusion system with inner absorptions and coupled nonlinear boundary fluxes. A precise analysis on interactions among the multi-nonlinearities in the system is given to determine the critical exponent. Secondly, we concern nonlinear parabolic models with convection so as to explore whether and in what extent the gradient terms influence blow-up behavior of solutions. Finally, in the studying of quenching phenomena, we determine simultaneous versus non-simultaneous quenching for a nonlinear parabolic system with coupled absorptions subject to positive Dirichlet boundary conditions, and characterize the asymptotic behavior of quenching time and set of solutions for heat equations with weighted nonlinear absorptions.Chapter 1 is to summarize the background of the related issues and to briefly introduce the main results of the present thesis.Chapter 2 deals with the initial-boundary problem for (u^m)_t =â–³u -α₁u^α₁, (vⁿ)_t =â–³v -α₂v^β₁ coupled via boundary flux (?). We introduce a so called characteristic algebraic system together with a complete classification for all the eight nonlinear parameters to obtain a simple and clear description to the critical exponent of the problem. Due to the generality of the model considered, this covers many known results on critical blow-up exponents. Comparing with those for scalar cases, the substantial effects of the coupling mechanism on critical exponents can be observed.Chapter 3 is devoted to the blow-up analysis for nonlinear parabolic equations with convection. The aim is to investigate the influences of gradient perturbations on the asymptotic behavior of solutions. For the semilinear parabolic equation u_t =â–³u + |â–½u|^r - ae^pu subject to nonlinear boundary flux (?) = e^qu, we obtain that the gradient term makes a substantial contribution to the formation of blow-up if and only if r≥2. In addition, the gradient term would significantly affect the blow-up rate as well whenever r > 2. Moreover, the blow-up rate may be discontinuous with respect to parameters included in the problem due to convection. However, the gradient perturbations have no essential effects on the spatial blow-up profile. For the nonlinear diffusion equation w_t = (e^(m-1)w)_xx -λe^(p-1)w with Neumann boundary conditions w_x(0,t) = 0, w_x(1,t) = e^(q-m)w(1,t) , using the scaling method, we establish the blow-up rate estimates for blow-up solutions. Under a transformation, this equation is equivalent to a porous medium type one with convection. We find that the gradient term just leads to a more complicated discussion without changing the blow-up rate of solutions.Chapter 4 studies two quenching problems, namely, coupled nonlinear parabolic system u_t =â–³u-v^-p, v_t =â–³v-u^-qinΩ×(0, T) with positive Dirichlet boundary conditions, and scalar heat equations with weighted nonlinear absorptions u_t = u_xx - Mf(x)u^-p subject to boundary conditions u(-1,t) = u(1,t) = 1 and initial dataφ(x). For the former problem, we characterize the non-simultaneous quenching criteria for radial quenching solutions withΩ= B_r: The quenching is simultaneous if p,q≥1, and non-simultaneous if p < 1≤q or q < 1≤p; If p,q < 1 with R > (?), then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. It should be mentioned that to get the coexistence result, we have to skillfully construct a set of initial data admitting required uniform lower estimates on quenching solutions. For the latter model, the asymptotic behavior of quenching time and set of solutions as M→+∞is established by local energy estimates. It is obtained that the quenching time T - (m/(p+1)). M^-1 with (?) as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/φ^p+1 for large M.