Doubly nonlinear equation involving $p(x)$-homogeneous operators: local existence, uniqueness and global behaviour

TL;DR

This paper studies the existence, uniqueness, and long-term behavior of solutions to a nonlinear parabolic PDE involving variable exponent operators, providing new comparison principles and a priori estimates.

Contribution

It introduces novel comparison principles and a priori estimates for solutions to p(x)-homogeneous operator equations, extending understanding of their qualitative properties.

Findings

Proved uniqueness and positivity of weak solutions.

Established comparison principles for the PDE.

Analyzed stabilization and regularity of solutions.

Abstract

In this work, we investigate the qualitative properties as uniqueness, regularity and stabilization of the weak solution to the nonlinear parabolic problem involving general $p(x)$-homogeneous operators: \begin{equation*} \left\{ \begin{alignedat}{2} {} \frac{q}{2q-1}\partial_t(u^{2q-1}) -\nabla.\, a(x, \nabla u) & {}= f(x,u) + h(t,x) u^{q-1} && \quad\mbox{ in } \, (0,T) \times \Omega; u & {}> 0 && \quad\mbox{ in }\, (0,T) \times \Omega ; u & {}= 0 && \quad\mbox{ on }\, (0,T) \times \partial\Omega; u(0,.)&{}= u_0 && \quad\ \mbox{in}\, \ \Omega. \end{alignedat} \right. \end{equation*} Thanks to the Picone's identity obtained in [10], we prove new results about comparison principles which yield a priori estimates, positivity and uniqueness of weak solutions.