Global Well-posedness Of Two Classes Of Coupled Reaction-diffusion Systems With Two Nonlinear Source Terms And Two Types Of Parabolic Equations Under Dynamical Boundary Conditions

This work carries out qualitative studies on two types of coupled reaction-diffusion systemswith double nonlinear source terms and two classes of parabolic equations under dynamicalboundary conditions.Firstly, we study two types of coupled reaction-diffusion systems with double nonlinearsource terms. These systems often arise in population dynamics, fluid dynamics, electron flow,chemical reactions, heat propagation, etc. The difficulties in this part lie in the processingof two special source terms (the product of power type and exponential ones) in the systems.We overcome the impact of complex source terms and analyze the relationship between theproperties of solutions and the exponents of source terms, initial conditions. Then we enrichand perfect the theory of qualitative studies on two-component reaction-diffusion systems.Next, we focus on two classes of parabolic equations under dynamical boundary condi-tions. The dynamical boundary conditions are very natural in many mathematical models asheat transfer in a solid in contact with moving fluid, fluid dynamics problems, etc. The dynam-ical boundary conditions make the space nature of solutions change, such as the invariant sets.In addition, the original methods used to study qualitative studies are no longer entirely appli-cable. To overcome the above difficulties, we first redefine the functional space of solutions,analyze the properties of Nehari manifold and the invariance of solutions in some sets. Later,applying potential well method and concavity method, we obtain the sharp conditions for globalexistence and nonexistence of solutions at low and critical initial energy. For sufficiently largeinitial data, we study the global wellposedness of solutions for linear dynamical boundary caseby comparison principle and variational methods. And when the dynamical boundary turns tononlinear case, combined with concavity method, we introduce the estimates of Sobolev imbed-ding constants, and finally obtain the finite time blow up of solutions with high initial energy.We reveal the influence of the structure of dynamical boundary on the global well-posedness ofparabolic equation by assuming the restrictions for the initial data and exponents of nonlinearterms. Finally, we enrich and develop the potential well theory.