| A class of nonlinear elliptic-parabolic thermally coupled system has been considered in this thesis. The main properties of the solution to scalar or vector systems such as existence, uniqueness, regularity and blow-up are discussed. This thesis is mainly composed of four parts of contents:In chapter 1, the basic application background, basic knowledge, advanced studies and main idea of this considered problem are introduced.Chapter 2 is devoted to investigate the main properties of the solution to the coupled system in scalar forms. In studying the nonlinear coupled scalar system, firstly, we define the variational formulation of the considered problem, then applying decoupled method, Maximum principle and basic theories of nonlinear elliptic and parabolic equations to get the existence result, without additional regularity assumption in 3D in [19,23,38], etc. By applying Meyers' estimate in [39], the uniqueness is obtained without added assumptions on the regularities of the solution as (4.2) and (4.20) in [7]. And a non-trivial extension of the blow-up analysis in [7] to the case of diffusion-convection-reaction system is presented following the idea from [33].In chapter 3, a global solution to the scalar nonlinear coupled system is studied under the assumptions thatσ(s),κ(s)∈W1,∞ (R), (b|-)∈[L∞(Ω)]2,c∈L∞(Ω) and c-1/2▽·(b|-)≥-(κ1-α)λ1 (λ1 denotes the first eigenvalue of -ΔinΩ,α>0). We apply Faedo-Galerkin method to construct an approximate solution to the problem that exists in a local time, then prove some priori estimates to show that the approximate solution can be extended to the interval [0,T]. By using the compact theorem, we get the global solution as n→∞.In chapter 4, the vector case is studied, and existence of the solution is established. We get the result when the temperature equation contains convective terms, and the thermally conductivity is a function ofθ,▽θ, extending the one in [40]. Thus our results may be more applicable in practical situations.
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