arXiv subjects
Publications and source records attributed to Hamid El Bahja.
Following Dibenedetto's intrinsic scaling method, we prove local Hölder continuity of weak solutions to obstacle problems related to some anisotropic parabolic equations under the condition for which only Hölder's continuity of the obstacle is known.
We study a two-species chemotaxis system that exhibits two qualitatively different regimes: subcritical dynamics, in which solutions remain smooth, and supercritical dynamics, in which strong aggregation may lead to finite-time blow-up. In the subcritical regime, we use a standard continuous-time physics-informed neural network (PINN) with alternating training for the coupled fields and show that it provides accurate and efficient approximations. In the supercritical regime, however, this formulation is not sufficiently robust to resolve the highly localized structures and steep gradients associated with blow-up. To address this limitation, we introduce a discrete-time PINN based on backward Euler time stepping, combined with a logarithmic transformation to stabilize large solution values and a residual-adaptive collocation strategy that concentrates training points near regions of high activity. This hybrid framework allows us to use a simple and computationally inexpensive PINN when the solution is smooth, while switching to a more robust discrete formulation when singular dynamics emerge. Numerical experiments confirm that the continuous-time formulation accurately resolves subcritical dynamics, while the discrete-time formulation reliably captures singular aggregation and finite-time blow-up in the supercritical regime.
This paper investigates the local boundedness of weak solutions to a direction-dependent double-phase nonlocal elliptic equation. By employing refined energy estimates and De Giorgi-type techniques, we establish the local boundedness of these solutions.
This paper establishes a probabilistic representation for the solution of the parabolic obstacle problem associated with the normalized $p$-Laplacian. We introduce a zero-sum stochastic tug-of-war game with noise in a space-time cylinder, where one player has the option to stop the game at any time to collect a payoff given by an obstacle function. We prove that the value functions of this game exist, satisfy a dynamic programming principle, and converge uniformly to the unique viscosity solution of the continuous obstacle problem as the step size $\varepsilon$ tends to zero.
Deep learning has been highly successful in some applications. Nevertheless, its use for solving partial differential equations (PDEs) has only been of recent interest with current state-of-the-art machine learning libraries, e.g., TensorFlow or PyTorch. Physics-informed neural networks (PINNs) are an attractive tool for solving partial differential equations based on sparse and noisy data. Here extend PINNs to solve obstacle-related PDEs which present a great computational challenge because they necessitate numerical methods that can yield an accurate approximation of the solution that lies above a given obstacle. The performance of the proposed PINNs is demonstrated in multiple scenarios for linear and nonlinear PDEs subject to regular and irregular obstacles.