We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H 1 to the solution of the original three-dimensional boundary value problem. (2000): 74K20, 74K25. Mathematics Subject Classifications

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We consider one multidimensional version of the Cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part. Depending on the exponent of nonlinearity and spatial dimension of equation, we investigate the problem on the nonexistence of global solutions of the Cauchy characteristic problem. The question on the local solvability of that problem is also considered. Statement of the problemConsider the nonlinear equation of the typewhere λ and α are the given positive constants, F is the given and u is an unknown real functions,(1), we consider the Cauchy characteristic problem on finding in the truncated light cone of the future D T : |x| < t < T , x = (x 1 , . . . , x n ), T = const > 0, a solution u(x, t) of that equation by the boundary conditionswhere S T : t = |x|, t T , is the characteristic manifold which is, in fact, a conic portion of the boundary of the domain D T , ∂ ∂ν is the derivative in the direction of the outer normal to ∂D T . For the case T = +∞ we assume that D ∞ : t > |x| and S ∞ = ∂D ∞ : t = |x|.

SUMMARYWe study the well posedness of boundary value problems for elastic cusped prismatic shells in the N th approximation of I. Vekua's hierarchical models under (all reasonable) boundary conditions at the cusped edge and given displacements at the non-cusped edge and stresses at the upper and lower faces of the shell.

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