The Application Of Iterative Method And Differential Method In Studying The Dynamic Properties Of Petrovsky Equation With Memory

In this paper,we investigate asymptotic behavior for the solution to the Petrovsky equation with memory（?）with initial conditions and boundary conditions,where2)is a memory kernel function.The related energy has been shown to decay exponentially or polynomially as t→+∞by the theorem established under the assumption g^′（t）≤-kg^1+1/p（t） with p∈（2,∞）,k>0（in Journal of Functional Analysis,254（5）（2008）1342–1372,[1]）.Using interative technique,we prove the general decay result under the assumption g^′（t）+H（g（t））≤0 with thatis a C¹convex function（this condition presented in Journal of Differential Equations 259（12）（2015）7610–7635,[4]and extend the assumption in[1]）such that the exponential and polynomial decay results in[1]are only special cases.Moreover,applying differential technique and integration technique,we prove the general decay result under the assumption g^′（t）≤-ξ（t）g（t） with that ξ（t）∈L_loc¹[0,+∞) （the condition can not be covered by the condition in[4]）,and show that the exponential and polynomial decay results are the special cases with choosing special（）.