Monotonicity methods for nonlinear diffusion equations and their approximations with error estimates

This paper deals with the initial-boundary value problem (P) for the nonlinear diffusion equation∂u∂t+(−Δ+1)β(u)=ginΩ×(0,T) in a general domain Ω⊂RN with smooth bounded boundary, where N∈N, T>0 and β is a single-valued maximal monotone function on R, e.g., β(r)=|r|q−1r(q>0). The above equation represents a number of well-known models, e.g., porous media equation. Colli and Fukao [6] studied the above problem (P) and the approximate problem (P)ε, which consists of the Cahn-Hilliard system, with error estimates when Ω is a bounded domain, N=2 or 3 and −Δ+1 is replaced with −Δ. They considered one more approximation (P)ε,λ to solve (P)ε. They used compactness methods for abstract doubly nonlinear evolution equations to solve the second approximate problem (P)ε,λ. They established the error estimate saying that the solution of (P)ε converges to the solution of (P) as ε↘0. The present work asserts that one can solve the original problem (P) and the approximate problem (P)ε individually and directly even if Ω is a general domain. Moreover, this paper gives an error estimate between the solution of (P)ε and the solution of (P).