Doubly nonlinear equation involving $p(x)$-homogeneous operators: local existence, uniqueness and global behaviour
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 -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.
Keywords
Cite
@article{arxiv.1912.02510,
title = {Doubly nonlinear equation involving $p(x)$-homogeneous operators: local existence, uniqueness and global behaviour},
author = {Rakesh Arora and Jacques Giacomoni and Guillaume Warnault},
journal= {arXiv preprint arXiv:1912.02510},
year = {2019}
}