An Evolutionary Boundary Value Problem

作者：Benseghir Aissa; Sofonea Mircea

来源：Mediterranean Journal of Mathematics, 2016, 13(6): 4463-4480.

DOI：10.1007/s00009-016-0756-y

• Nonlinear boundary value problem
• evolutionary inequality
• weak solution
• convergence results
• dual variational formulation

We consider a nonlinear initial boundary value problem in a two-dimensional rectangle. We derive variational formulation of the problem which is in the form of an evolutionary variational inequality in a product Hilbert space. Then, we establish the existence of a unique weak solution to the problem and prove the continuous dependence of the solution with respect to some parameters. Finally, we consider a second variational formulation of the problem, the so-called dual variational formulation, which is in a form of a history-dependent inequality associated with a time-dependent convex set. We study the link between the two variational formulations and establish existence, uniqueness, and equivalence results.

• 出版日期2016-12