Abstract
There are two contributions in this paper. The first is that the abstract result for the existence of the unique solution of certain nonlinear parabolic equation is obtained by using the properties of H-monotone operators, consequently, the proof is simplified compared to the corresponding discussions in the literature. The second is that the connections between resolvent of H-monotone operators and solutions of nonlinear parabolic equations are shown, and this strengthens the importance of H-monotone operators, which have already attracted the attention of mathematicians because of the connections with practical problems.
Highlights
Introduction and preliminaries1.1 Introduction Nonlinear boundary value problems involving the generalized p-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extraction, flow through porous media and non-Newtonian fluids, just to name a few
We would mention the books of Lieberman [, ] where in [ ] the theory of linear and quasilinear parabolic second-order partial differential equations is elaborated, with emphasis on the Cauchy-Dirichlet problem and the oblique derivative problem in bounded space-time domains; while in [ ] a detailed qualitative analysis of second-order elliptic boundary value problems that involve oblique derivatives is presented
F is everywhere defined, monotone and hemi-continuous, which implies that F is maximal monotone in view of Lemma
Summary
Introduction and preliminaries1.1 Introduction Nonlinear boundary value problems involving the generalized p-Laplacian operator arise from many physical phenomena, such as reaction-diffusion problems, petroleum extraction, flow through porous media and non-Newtonian fluids, just to name a few. Let Jr denote the duality mapping from X into X∗ , which is defined by Let B : X → X∗ be a maximal monotone operator such that [ , ] ∈ G(B), the equation J(ut – u) + tBut has a unique solution ut ∈ D(B) for every u ∈ X and t > .
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