Methods and Applications of Analysis

Volume 30 (2023)

Number 1

Life span of solutions to a PDE model for lithium-ion batteries in high space dimensions

Pages: 27 – 52



Xiangsheng Xu (Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Ms., U.S.A.)


In this paper we study a system of partial differential equations which models lithium-ion batteries. The system describes the conservation of Lithium and conservation of charges in the solid and electrolyte phases, together with the conservation of energy. The mathematical challenge is due to the fact that the reaction terms in the system involve the hyperbolic sine function along with possible degeneracy in one of the high-order terms. We obtain a local existence assertion for the initial boundary problem for the system. In particular, a lower bound for the blow-up time can be derived from our result. We hope that our analysis can lead to a deeper understanding of battery life.


lithium-ion batteries, de Giorgi iteration scheme, existence and boundedness of a solution

2010 Mathematics Subject Classification

Primary 35A01. Secondary 35B50, 35J57, 35K20, 35M10.

Received 25 August 2022

Accepted 18 May 2023

Published 21 July 2023