Global Existence Of Solutions For Damped Wave Equation With Memeory Term

This paper consider the Cauchy problem in R" for a nonlinear damped wave equation with a nonlinear memory term (?) ds. For the noncompact initial data with small energy, a global in time existence results are proved under some conditions on the space dimensional and the power in the nonlinear memory term p. This generalizes a previous result due to Ahmad Z. Fino which dealt with a solution in the framework of compactly supported initial data in 2010. Firstly, the weighted function with noncompact support is introduced and also including its properties, and the weighted energy functional is defined Then the existence and uniqueness of the local solution for the problem is obtained by using the contraction mapping principle. Next, the nonlinear term is estimated in the weighted energy space, and a prior estimate of the energy is derived; finally, the existence and the uniqueness of the global solution has been proved; the decay properties of the solution are also given.