Asymptotic profile of solutions for strongly damped Klein-Gordon equations

We consider the Cauchy problem in R-n for strongly damped Klein-Gordon equations. We derive asymptotic profiles of solutions with weighted L-1,L-1(R-n) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L-2-norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low-dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.