Large Time Behavior Of Smooth Soultions To A Simplified Version Of An Energy Transport Model

The well posedness of smooth solution to a simplified version of an energytransport model is discussed in this paper. We have proved the existence, uniqueness,and large time behaviour of solution to the equations with hybrid cross-diffusion.The solution of the equations with a given initial condition is local, and smoothsolution is global while the doping profile function has positive upper and lowerbounds in semiconductors, and the initial-data is a small perturbation of anisothermal stationary solution. Furthermore, the smooth solution converges to astationary solution at an exponential rate as time tends to infinity.In chapter1, we introduce the significance of researching for hydrodynamicsmodel, and theoretical derivation of the reduced energy transport model ment ionedin this article. Some existing results for this class of models are recalled. Finally, ourmain results in this paper are stated.Chapter2is devoted some basic knowledge, related notations, significantinequalities, and mathematical symbols which will be useful later in the followingchapters.In chapter3, we mainly study the local existence solution of the simplifiedenergy transport model with mixed cross-diffusion. Firstly, we establish a prioriestimates of electron density in equations with the help of basic energy methods.Then, according to contraction mapping theorem and parabolic regularizationprocess, we prove equations of those models have a unique local smooth solution.In chapter4, we discuss the global existence and large time behaviour of smoothsolution. With the help of energy methods, Gagliardo-Nirenberg inequalities, andGronwall inequalities, we get the results as follows, when the initial-data is a smallperturbation of an isothermal stationary solution, the smooth solution of the class ofmodels converges to a steady-state solution at an exponential rate of decay as timegoes to infinity.Finally, a brief summary of this class of models is given, and some issuesinvolved in high dimensional space are presented.