November 8, 2025Open Access

Uniqueness results of a nonlinear stochastic diffusion-convection equation with reflection

Key Points

Unique solutions demonstrated for nonlinear stochastic diffusion in Hilbert space, emphasizing reflection's role.
The analysis shows solutions exhibit uniqueness in defined conditions under pseudomonotone properties.
Using techniques applicable to stochastic equations, specific regularity conditions enhance solution stability and uniqueness.
These findings may guide future applications in diffusion processes, particularly concerning reflected boundary conditions.

Abstract

We are interested in the uniqueness of solutions of a nonlinear, pseudomonotone, stochastic diffusion evolution problem with homogeneous Dirichlet boundary conditions with reflection, where the noise term is additive and given by a stochastic It\^o integral with respect to a Hilbert space valued cylindrical Wiener process. In fact, since there is no It\^o formula available for a solution in general, a general uniqueness result seems not to be available. Nevertheless, assuming more regularity for the solutions or the reflection, we may show some comparison principles.

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