This work establishes the existence of weak solutions for p-Laplacian-like equations in weighted Sobolev spaces under Dirichlet boundary conditions, assuming the data is in the weighted Lebesgue space.

We prove in this paper some existence and uniqueness results of weak solutions for some nonlinear degenerate parabolic problems arising from capillarity phenomena with the Dirichlet type boundary condition.

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.

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 invariant finite-dimensional linear spaces for a class of nonlinear evolution equations with second-order time dependence of the type ${\partial }_{tt}u=-{\partial }_x\left(A(x,u)u\right)+{\partial }_{xx}\left(B(x,u)u\right)+f\left(u\right),$ where the coefficients may depend nonlinearly on the solution. Using direct algebraic verification and invariant subspace reductions, we derive several exact solutions and reduce the governing equations to finite-dimensional systems of ordinary differential equations. Various examples are presented to illustrate the applicability of the method to nonlinear transport, diffusion, and reaction models.