A nonlinear kinetic theory of cosmic ray (CR) acceleration in supernova remnants (SNRs) is employed to re-examine the nonthermal properties of the remnant of SN 1987A for an extended evolutionary period of 5-100 yr. It is shown that an efficient production of nuclear CRs leads to a strong modification of the outer SNR shock and to a large downstream magnetic field B-d approximate to 20 mG. The shock modification and the strong field are required to yield the steep radio emission spectrum observed, as well as the considerable synchrotron cooling of high-energy electrons which diminishes their X-ray synchrotron flux. These features are also consistent with the existing X-ray observations. The expected gamma-ray energy flux at TeV energies at the current epoch is nearly epsilon(gamma) F-gamma approximate to 4 x 10(-13) erg cm(2) s(-1) under reasonable assumptions about the overall magnetic field topology and the turbulent perturbations of this field. The general nonthermal strength of the source is expected to increase roughly by a factor of two over the next 15-20 years; thereafter, it should decrease with time in a secular form.