Estimates of solutions for parabolic differential and difference functional equations and applications

Abstract

The theorems on the estimates of solutions for nonlinear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for suitable implicit finite difference functional schemes are proved. The proofs are based on the com- parison technique. The convergent and stable difference method is considered without the assumption of the global generalized Perron condition posed on the functional variable but with the local one only. It is a consequence of our estimates theorems. In particular, these results cover quasi-linear equations. However, such equations are also treated separately. The functional dependence is of the Volterra type. The aim of the paper is to prove theorems on the estimates of solutions for non- linear second-order partial differential functional equations of parabolic type with Dirichlet's condition and for generated by them implicit finite difference functional schemes. We also give the applications of the results. More precisely, we prove the theorem on the convergence of a difference method to a classical solution for the differential functional problem, which by the given estimates, may be treated in the subspace C (,R) ⊂ C (,R), where R ⊂ R is an interval. It is a new idea in area of nonlinear implicit difference methods which was studied for explicit methods by K. Kropielnicka and L. Sapa (14). This considerably extends the class of problems which are solvable by the described method. Therefore, the Lipschitz, Perron or generalized Perron conditions posed onf with respect toz need not be global, inC (,R), as in the papers due to M. Malec, Cz. Mączka, W. Voigt, M. Rosati and L. Sapa (15-19,24,25),