GENERALIZED EXISTENCE AND STABILITY RESULTS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS IN WEIGHTED SOBOLEV SPACES

Abstract

Background: Nonlinear partial differential equations (PDEs) are important tools for describing complex processes in physics, engineering, and applied sciences. The classical Sobolev spaces occasional are not enough to deal with non-smooth domain or singularities. Weighted Sobolev spaces with variable weight functions provide a convenient setting to tackle the former difficulties in the study of the problems of existence, uniqueness and stability of PDE solutions.

Objective: The purpose of this work is to prove some uniform existence and stability results for a class of nonlinear PDEs in weighted Sobolev spaces. It aims to shows how weight functions affects solution regularity, convergence and robustness under nonlinear perturbations.

Method: The analysis uses sophisticated tools of functional analytic combined with weighted norm inequality to establish the existence and stability results. Numerical experiments based on Galerkin methods show the correctness of the theoretical results, and demonstrate the behaviour of the weight parameters on, both, the convergence rates and the stability of the solutions on examples of several nonlinear PDEs.

Results: The results verify that weighted Sobolev space underpin existence theorem and the stability estimate is stronger than the classical setting. With the introduction of weight functions, one can have control on the behavior of the solution at the singularities and the boundary of the domain, which leads to a better numerical accuracy and stability. Sensitivity analysis demonstrates that maintaining the integrity of the solution is critically sensitive to the interplay of the intensity of nonlinearity and the rate of weight decay.

Conclusion: Weighted Sobolev spaces are a natural generalization of the classical PDE theory and they include the realistic case of boundary conditions of the type described in a). They possess theoretical as well as computational advantage and thus are quite useful to facilitate the further development of nonlinear PDE theory and applications.