arXiv subjects

Amjad Saef

Publications and source records attributed to Amjad Saef.

3 recordsLinked to original sources

arXiv2026

Well-posedness of stochastic parabolic equations with gradient nonlinearities and applications to phase-field models

We study well-posedness of stochastic parabolic equations with gradient nonlinearities. Our analysis is based on recent maximal-regularity frameworks for nonlinear stochastic parabolic equations in critical spaces. We extend the existing results by controlling drift and noise coefficient separately. This way we can allow for less regular driving noise in case of subcritical dispersion coefficients. Our approach, based on gluings of local solutions, moreover implies new continuation criteria. We then apply our existence result and the continuation criteria to show global well-posedness of phase-field models of moving boundary problems.

math.PR↗

arXiv2026

On a stochastic phase-field model of cell motility with singular diffusion

We study existence of solutions in the variational sense for a class of stochastic phase-field models describing moving boundary problems. The models consist of stochastic reaction-diffusion equations with singular diffusion forced by a phase-field. We investigate both the case of an independently evolving phase-field and of coupled phase-field evolution driven by a viscous Hamilton-Jacobi equation. Such systems are used in the modelling of single-cell chemotaxis, where the contour of the cell shape corresponds to a level set of the phase-field. The technical challenge lies in the singularities at zero level sets of the phase-field. For large classes of initial data, we establish global existence of probabilistically weak solutions in $L^2$-spaces with weights which compensate for the singularities.

math.PR↗

arXiv2025

Stability of travelling wave solutions to reaction-diffusion equations driven by additive noise with H\"older continuous paths

In this paper we investigate stability of travelling wave solutions to a class of reaction-diffusion equations perturbed by infinite-dimensional additive noise with H\"older continuous paths, covering in particular fractional Brownian motion with general Hurst index. We obtain long- and short time asymptotic error bounds on the maximal distance from the solution of the stochastic reaction-diffusion equation to the orbit of travelling wave fronts. These bounds, in terms of Hurst index and H\"older exponent, apply to a large class of infinite-dimensional self-similar drivers with H\"older continuous paths, such as linear fractional stable motion. We find that for short times, higher Hurst indices imply higher stability, while for large times, a smaller gap between Hurst index and H\"older exponent implies stability for larger noise amplitudes.

math.PR↗