Well-Posedness And Decay Rates Of Solutions For The Moore-Gibson-Thompson Equation With Memory

The Moore-Gibson-Thompson(MGT)equation with memory appears in high-frequency ultrasound.It considers heat flux and molecular relaxation time.According to the irreversible thermodynamics of re-expansion,heat flux relaxation leads to a third derivative in time,while molecular relaxation leads to the non-local effect of memory term control.The MGT equation is a hyperbolic type with viscous effect and has a wide range of applications in medicine and industry.This paper mainly discusses a kind of MGT equation with memory term.The well-posedness and general decay rates of the solution of the equation are obtained.The main contents are given as follows:In the first chapter,we review the background and some development of the related prob-lems and briefly describe the main work of this paper.In Chapter 2,we study the general decay result for third-order Moore-Gibson-Thompson equation with memory in the subcritical case and critical case,where relaxation function satisfies g'(t)?-?(t)H(g(t)).Moreover,in the critical case,we need to consider the boundness of A.We introduce suitable energy and perturbed Lyapunov functionals to establish the optimal explicit and general energy decay result,from which we can recover the earlier exponential rate in the special case.In Chapter 3,we investigate explicit and general decay results of energy for third-order Moore-Gibson-Thompson equation with infinite memory,where relaxation function satisfies?0? g(s)/G-1(-g;(s))ds+sup s?R+g(s)/G-1(-g'(s))<+?.In the subcritical case,we prove the general decay result.For the critical case,we obtain general decay results in the cases of A bounded and unbounded from which the exponential and polynomial decay results of[13]are only special cases.Our results allow a much larger class of the convolution kernels which improves the earlier related results.In Chapter 4,we study the well-posedness and general decay result of fourth-order Moore-Gibson-Thompson equation with memory.We show that the equation is well-posed by Faedo-Galerkin method.On the other hand,for a class of relaxation functions satisfying g'(t)?-?(t)H(g(t)),we establish the explicit and general energy decay result,from which we can improve the earlier related results.In the last chapter,we summarize the research contents of this paper and give some prob-lems which could be studied in the future.