Well-posedness And Decay Characteristics Of Solutions For The Moore-Gibson-Thompson Equation With Degenerate And Time-dependent Memory

Moore-Gibson-Thompson(MGT)equation is one of the nonlinear acoustic models describ-ing the propagation of acoustics wave in gases and liquids,it has a wide range of applications in medical and industry.This paper discusses the MGT equation without memory,with degenerate memory term,and time-dependent memory term respectively.The well-posedness and decay rates of the solution of the equation are obtained.The main contents are arranged as follows:In the first chapter,we mainly introduce the research background and some development of the MGT equation and introduces related research with degenerate memory terms and time-dependent memory kernel,analyze and summarizes the main work of this article.In Chapter 2,we mainly study the MGT equation without memory term.Starting from the existing exponential decay results,constructing a conservation problem,selecting appropriate multipliers and using some inequalities,we obtain the internal observation inequality.Then,with the help of an auxiliary problem and overcoming the difficulty of unequal energy between the conservation problem and the original problem,the equivalence between exponential decay and internal observation inequality is proved.In Chapter 3,we investigate well-posedness and decay rates of solution for Moore-Gibson-Thompson equation with infinite degenerate memory,where the memory kernel g(s)satisfies (?) By using the Faedo-Galerkin approximations together with some energy estimates,we prove that the above system is well-posedness.Besides,under appropriate assumptions,we establish the exponential or general decay results of energy via constructing suitable Lyapunov functionalsIn Chapter 4,we study the well-posedness and exponential decay result of Moore-Gibson-Thompson equation with time-dependent memory kernel.The main feature is that the memory kernel ?t(s)depends on time and satisfies?t(s)+?t'(s)?-?k(t)?t(s).By using the linear semigroup theory,we prove that the above system is well-posedness.In addition,we establish the exponential decay result.In the last chapter,we summarize the research contents of this paper and give some prob-lems which could be studied in the future.