Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

作者：Emmrich Etienne^*; Thalhammer Mechthild

来源：Journal of Differential Equations, 2011, 251(1): 82-118.

DOI：10.1016/j.jde.2011.02.014

• Evolution equation of second order
• Integro-differential equation
• Monotone operator
• Weak solution
• Time discretization
• Variable time grid
• Internal approximation
• Convergence

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization. The discretization combines a time stepping on a non-uniform time grid, which generalizes the well-known Stormer-Verlet scheme, with an internal approximation scheme.
The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Holder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces.
Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.

• 出版日期2011-7-1