Global Existence, Uniqueness And Exponential Decay For A Nonlinear Thermoelasticity System

In this paper, we study the existence, uniqueness and exponential decay of the global solution for the systemwith initial conditionsHere |·|2 is the norm in L~2(0,L) , u is the transverse displacement of the string, d is the temperature and M,N and R are real nonlinear functions.The paper will discuss the existence, uniqueness and exponential decay of the strong solution for the initial problem (1.1) —(1.2). The method we used in this paper is the Galerkin method.The following will introduce the main contents of this paper.In the first Chapter we give an introduction about the domestic and international development and investigative meaning for thispaper in brief.In the second Chapter, we list some necessary thesis that have been proved before but will be used in this paper.In the third Chapter, we give a proof for the global existence, uniqueness and exponential stability under certain initial and boundary conditions for the system (1.1)—(1.2).In the chapter when the boundary conditions are as following we discuss the global existence, uniqueness and exponential decay for the system (1.1)—(1.2).We use the same technique as the paper [8], and only if we confine the initial value and the nonlinearities (M,N and R), we get the global existence, uniqueness andexponential decay. But for the difference between the problem we discuss and one discussed by Clark, the certification for the estimation of the solution becomes more complicate.In the fourth Chapter, when the boundaries are as following we also get the global existence, uniqueness and exponential decay for the initial problem(l.l)—(1.2). To our attention, the transform of the boundaries make the space in which we discuss the problem vary, so the chapter is not superfluity, but the proof gets simpler.In every chapter, we eventually get the exponential decay for the system, and at last we give the certification of the uniqueness.