Ahmed Jamea

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.

In this paper, we prove some existence results of weak solutions for some nonlinear degenerate elliptic problem in weighted variable exponent Sobolev space with Dirichlet type boundary condition.

We investigate nonlinear elliptic equations that describe double-phase phenomena with a reaction term that is independent of the gradient (i.e., it only satisfies the $L^{1}$-data condition), subject to homogeneous Dirichlet boundary conditions. Using the

Neural networks have emerged as powerful tools for constructing numerical solution methods for partial differential equations (PDEs). This review article provides an accessible introduction to recent developments in the field of scientific machine learning, focusing on methods such as Physics-Informed Neural Networks (PINNs), Deep Galerkin Methods (DGM), Deep Ritz Methods, and Neural Operator Methods. We compare these approaches, highlighting their strengths, limitations, and potential areas for improvement. Furthermore, we explore a variety of real-world applications where these neural networkbased PDE solvers have been successfully implemented. Finally, we discuss future directions and the ongoing challenges in this rapidly evolving research area.

We demonstrate the uniqueness of entropy solutions for a class of quasilinear elliptic equations governed by the double-phase operator, with Dirichlet-type boundary conditions and $L^{1}$-data on the right-hand side. We use the framework of Musielak–Orlic

In the present paper, we prove the uniqueness of entropy solution to a class of nonlinear degenerate weighted elliptic p(·)-Laplacian problems. These particular problems are delineated by Dirichlet-type boundary conditions and L1 data. We use a priori estimates in the framework of weighted variable exponent Sobolev spaces theory.

Our aim in this paper is to study the existence result of entropy solution for a specific type of nonlinear degenerate weighted elliptic p(.)-Laplacian problems with Dirichlet-type boundary condition and L1 data. In the framework of the theory of weighted Sobolev spaces with variable exponents, we use the regularization approach combined with a priori estimates.

This paper investigates the existence result of entropy solution for some nonlinear degenerate parabolic problem in weighted Sobolov space with Dirichlet type boundary conditions and L1 data.

Institute of Mathematics of the Czech Academy of Sciences