Study of a class of nonlinear heterogeneous diffusion with mixed phases under L∞ − data

Abstract

In this paper we investigate a class of nonlinear degenerate parabolic equations involving heterogeneous (p,q)-Laplacian operators and subject to Dirichlet boundary conditions. These equations model complex diffusion phenomena with mixed-phase behavior in heterogeneous media. Our aim is to establish existence and uniqueness results for weak solutions under minimal regularity assumptions on the source term f, without requiring any control at infinity. The main difficulties stem from the degeneracy of the operator, the non-standard (p,q)-growth conditions, and the discontinuity of material phases. To overcome these challenges, we develop a variational framework based on Orlicz–Sobolev space theory and employ a generalized version of the Minty–Browder theorem to ensure the surjectivity of the nonlinear operator. Our approach yields new energy estimates, compactness results in non-reflexive settings, and stability under L∞-perturbations of the data. This work provides a rigorous mathematical foundation for analyzing nonlinear diffusion problems in complex and irregular environments.