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Capacity solutions for anisotropic variable exponent parabolic-elliptic systems with degenerate term

  • Hichem Khelifi , Youssef El Hadfi ORCID logo EMAIL logo and Said El Manouni
Published/Copyright: August 28, 2025
Analysis
From the journal Analysis

Abstract

This paper focuses on establishing the existence of a capacity solution for the following anisotropic elliptic-parabolic system with variable exponent:

{ u t i = 1 d [ | u x i | p i ( x ) 2 u x i ( 1 + | u | ) γ ( p i ( x ) 1 ) ] x i = Θ ( u ) | v | 2 in Q T = Ω × ( 0 , T ) , div ( Θ ( u ) v ) = 0 in Q T , u = 0 , v = v 0 on Γ = Ω × ( 0 , T ) , u ( , 0 ) = u 0 in Ω ,

where Ω is an open bounded subset of R d with d > 2 , Q T is the cylinder Ω × ( 0 , T ) with T > 0 , 0 < γ < 1 , p i C ( Ω ̄ ) for all i = 1 , , d , Θ C ( R ) , u 0 L 2 ( Ω ) , and v 0 L 2 ( 0 , T ; H 1 ( Ω ) ) L ( Q T ) .

MSC 2020: 35K55; 35B45; 35D30; 35K65; 46E35

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Received: 2025-03-30
Accepted: 2025-08-07
Published Online: 2025-08-28

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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