We study a class of nonlinear evolution equations of second order emerging from elastodynamics. In particular, we will examine the fine interplay of monotonicity and growth conditions that are being used to show existence of weak solutions.
The situation under strong growth and weak monotonicity assumptions of the underlying nonlinear term is well studied. We will present a novel approach with the opposite situation of weak growth and strong monotonicity assumptions.
Therefore we will drop polynomial growth restrictions and instead work in the setting of anisotropic Orlicz spaces. The nonlinearity is then assumed to satisfy a growth condition in terms of a generalized N-function.
Existence of solutions under the assumptions mentioned above is shown via convergence of a subsequence of approximate solutions arising from a full discretization. The numerical method combines the backward Euler method for the time discretization with a generalized internal approximation scheme for the spacial discretization.