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In this paper, the nonlinear behavior, instability and bifurcation phenomena in the response of a cylindrical shallow shell under lateral ‎white noise excitation are studied. The structure interacts with a general non aging viscous medium that can be modeled by relaxation or ‎creep kernels. Using the powerful FPK equation and some practical and logical simplifications, an exact solution for such complex system ‎including nonlinearity, viscoelasticity and randomness is obtained. Since all statistical properties of response such as mean, variance, ‎statistical moments, central moments, etc. can be obtained from the probability density function, the behavior of this function including the ‎number and sign of its roots and their effects on the stability, bifurcation phenomenon and the type of bifurcation is investigated and ‎studied. In this process, using some non dimensional quantities, the governing equation and the probability density function are rearranged ‎such that the results of simulations can be used for a broad band of cylindrical shallow shells. Finally, using some examples, the variations ‎of the non dimensional quantities on the whole behavior, stability and bifurcation type of response are studied. ‎