Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations

Abstract

In this paper we deal with the equation L(d2u/dt2)+B(du/dt)+Au∋f, where L and A are linear positive selfadjoint operators in a Hilbert space H and from a Hilbert space V⊂H to its dual space V′, respectively, and B is a maximal monotone operator from V to V′. By assuming some coerciveness on L+B and A, we state the existence and uniqueness of the solution for the corresponding initial value problem. An approximation via finite differences in time is provided and convergence results along with error estimates are presented.

Keywords

• nonlinear second-order evolution equations
• Cauchy problem
• existence and uniqueness
• time discretization
• convergence and error estimate.