The Attenuation Problem Of The Moore-Gibson-Thompson Equation

In this paper,we study the problem of third order MGT ?Moore-Gibson-Thompson?equation with a memory term in Hilbert, space.The equation like this ?uttt+?utt-c2?u-b?ut+?₀^tg?t-s??u?s?ds=0,highest order for the third order term is uttt, nonlinear internal dissipation, indirect memory dissipation, we use multiplier techniques to obtain the general decay rate of the energy functional. First the above equation on both sides at the same time multiplied by ut, we can get E1?t? and E1'?t?after a series of calculation. We know that the development of the general equation of the highest order is two order. We applied a multiplier technology can get the expression of energy and the energy derivative. In order to obtain the total energy expressions we use a multiplier of technology on both sides of the equation. We multiply utt on both sides of equation at the same time. We can get E2?t? and E2'?t?. Thus we define the total energy functional E?t? = kE1?t? + E2?t?. Because energy expressions of E?t? is multifarious, we might as well to find the main body energy of R?t?. In order to help the energy of E?t? decay, we constructed two auxiliary energy functional??t?=???ututt, ??t?=-???utt?otg?t - s??u?t? - u?s??dsdx.Under the condition of g'?-??t?g?t?,we first prove ?t??R?t?,E'?t??S?t? and obtain the derivative of two auxiliary function expression, we can get though a series of calculation and estimate to obtain the similar expression of E'?t?. we construct L?t?=N1R?t? + N2??t?+??t? and prove the L?t??R?t? Through a series of calculation and estimate, we can obtain L'?t? ? -mR?t? + a?go?Vu??t?. On both ends of the above inequality and multiplied by ??t?. On the other hand, we use polishing technology to deal with the differential inequalities. Finally, we get E'?t?? -m??t?E?t?. We can prove that the energy functional is uniformly reduced to zero when time tends to infinity.Under the condition of q'?-?-gp?t?,we first prove E?t??R?t?,E'?t??S?t? and obtain the derivative of two auxiliary function expression, which contain the form of?gpo?u??t?.we construct L?t?=R?t? +e1??t?+e2??t? and prove the L?t??R?t?.Finally, though a series of calculation and estimate, we can get L'< -k1||utt|22-k2||?u||22-k3||?ut||22-k6?gpo?u??t?. In case of p = 1, it is general differential equation. Though the differential equation, we can get the energy form of exponential.In case of p > 1, we can get results of the polynomial decay.