A note on controllability of impulsive neutral stochastic functional differential equations driven by a Rosenblatt process in a Hilbert space

Hajji, Salah; Lakhel, El Hassan

doi:10.56947/gjom.v12i1.777

Cited by 1 publication

(1 citation statement)

References 18 publications

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“…For investigations into fourth-order degenerate parabolic equations in one spatial dimension, refer to the work in [8]. Given the significance of this topic, recent studies have delved into the existence and uniqueness of differential equations, as evidenced in [3,4,5,16,21,25,27,31,32,33,37].…”

Section: Introductionmentioning

confidence: 99%

Global existence of a pair of coupled Cahn-Hilliard equations with nondegenerate mobility and logarithmic potential

Harfash,

Al-Musawi

2024

GJOM

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We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qε). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qε). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qε). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces.

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