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The instability in the cosmic-ray precursor of a supernova shock moving ininterstellar medium is studied. The level of magnetohydrodynamic turbulence inthis region determines the maximum energy of particles accelerated by thediffusive shock acceleration mechanism. The high efficiency of cosmic rayacceleration is accepted, and the consideration is not limited by the case ofweak turbulence. It is assumed that Kolmogorov type nonlinear wave interactionstogether with the ion-neutral collisions restrict the amplitude of randommagnetic field. As a result, the maximum energy of accelerated particlesstrongly depends on the age of supernova remnant. The maximum energy can be ashigh as $\sim10^{17}Z$ eV in young supernova remnants and falls down to about$\sim10^{10}Z$ eV at the end of Sedov stage. Thus the standard estimate ofmaximum particle energy based on the Bohm limit calculated for the interstellarmagnetic field strength is not justified in this case. This finding may explainwhy the supernova remnants with the age more than a few thousand years are notprominent sources of very high energy gamma-rays.Comment: 10 pages, 2 figure

Limits on diffusive shock acceleration in supernova remnants in the presence of cosmic-ray streaming instability and wave dissipation