Well-posedness for p(x)-Laplacian parabolic equations under a multiple-regime setting on an annulus
Abstract
This paper proposes an energy-variational framework for the well-posedness of the initial boundary value problem of the parabolic equation with variable exponents on an annulus, under the multiple-regime setting consisting of the subcritical, critical and supercritical regimes of the variable exponents. The multiple-regime setting leads to the appearance of the critical and supercritical regions of positive measure, which cause the failure of the classical Sobolev embedding and the lack of compactness. To address the challenges posed by multiple regimes to well-posedness within the energy framework, we recover the compact Sobolev embedding for a class of radially symmetric functions by the Strauss inequality and obtain the local existence of the solution via the combination of the subdifferential technique and the Principle of Symmetric Criticality for any given radially symmetric initial data. Additionally, the uniqueness result is also given with the recovered Sobolev embedding. Moreover, examining the impact on well-posedness of solutions caused by the variable exponents crossing three sensitive regions, we establish a variational structure of the potential well theory for the evolution equation under the multiple-regime setting on the annulus. For subcritical and critical initial energy levels, we consider the non-positive and positive initial energy cases separately. We obtain the finite-time blow-up of the solution with non-positive initial energy and the threshold condition for global existence and finite-time blow-up of the solution with positive initial energy. For supercritical initial energy level, several sufficient criteria for the global existence and finite-time blow-up of the solution are given. Finally, the above results, including global existence and finite-time blow-up, are generalized to a more general locally symmetric domain containing an annulus.
Related
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BibTeX
@misc{wang2026wellposedness,
title = {{Well-posedness for $p(x)$-Laplacian parabolic equations under a multiple-regime setting on an annulus}},
author = {Yitian Wang and Runzhang Xu and Chao Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32083},
eprint = {2610.32083},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32083},
}
RIS
TY - UNPB
TI - Well-posedness for p(x)-Laplacian parabolic equations under a multiple-regime setting on an annulus
AU - Yitian Wang
AU - Runzhang Xu
AU - Chao Yang
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32083
UR - https://arxiv.org/abs/2610.32083
AB - This paper proposes an energy-variational framework for the well-posedness of the initial boundary value problem of the parabolic equation with variable exponents on an annulus, under the multiple-regime setting consisting of the subcritical, critical and supercritical regimes of the variable exponents. The multiple-regime setting leads to the appearance of the critical and supercritical regions of positive measure, which cause the failure of the classical Sobolev embedding and the lack of compactness. To address the challenges posed by multiple regimes to well-posedness within the energy framework, we recover the compact Sobolev embedding for a class of radially symmetric functions by the Strauss inequality and obtain the local existence of the solution via the combination of the subdifferential technique and the Principle of Symmetric Criticality for any given radially symmetric initial data. Additionally, the uniqueness result is also given with the recovered Sobolev embedding. Moreover, examining the impact on well-posedness of solutions caused by the variable exponents crossing three sensitive regions, we establish a variational structure of the potential well theory for the evolution equation under the multiple-regime setting on the annulus. For subcritical and critical initial energy levels, we consider the non-positive and positive initial energy cases separately. We obtain the finite-time blow-up of the solution with non-positive initial energy and the threshold condition for global existence and finite-time blow-up of the solution with positive initial energy. For supercritical initial energy level, several sufficient criteria for the global existence and finite-time blow-up of the solution are given. Finally, the above results, including global existence and finite-time blow-up, are generalized to a more general locally symmetric domain containing an annulus.
ER -