Weak solutions for p-Laplacian equation

Abstract

In this work we consider the p-Laplacian type parabolic equation with Dirichlet boundary condition and establish the existence of weak solutions using Leray-Schauder's fixed point theorem and semi-discretization process.

Existence of weak solutions to a certain homogeneous parabolic Neumann problem involving variable exponents and cross-diffusion

“…Moreover, in [26] the existence, uniqueness and stability of a weak solution to the equation where ≥ 0 and the vector-field a(x, t, ⋅) satisfies certain p(x, t)-growth and monotonicity conditions, cf. [25], was shown, see also [14] for p =constant. Additionally, in [8] it is shown that the solutions of a similar problem may vanish in finite time even if the equation combines the directions of slow and fast diffusion, and the extinction moment is estimated in terms of the data.…”

Existence of Entropy Solution for a Nonlinear Parabolic Problem in Weighted Sobolev Space via Optimization Method

“…For example, in [25], Xu and Zho studied the existence and uniqueness of weak solution for the initial-boundary value problem of a fourth-order nonlinear parabolic equation. In [4], Bhuvaneswari, Lingeshwaran and Balachandran established the existence of weak solution for the degenerate p-Laplacian parabolic by using semi-discretization process. In the case where p(.)…”

Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

“…Xu and Zhou [26] studied the stability and regularity of weak solutions for a generalized thin film equation for the corresponding homogeneous equation of type (1) with p as a constant. In this connection, Bhuveneshwari et al [10] established the existence of weak solutions for p-Laplacian equation. Bertsch et al [9] proved the existence of weak solutions for a class of fourth-order degenerate equation.…”