Abstract
This article studies the Cauchy problem for the damped wave equation with nonlinear memory. For a noncompactly supported initial data with small energy, global existence and asymptotic behaviour of solutions are obtained when 1 ≤ n ≤ 3. This result generalized the previous result by Fino [Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal. 74 (2011), pp. 5495–5505], which dealt with the solution with compactly supported initial data.
Acknowledgements
This research is supported by the National Natural Science Foundation of PR China (No. 71003082) and the Fundamental Research Funds for the Central Universities (SWJTU12CX61 and SWJTU12ZT13).