This paper analyzes a nonlinear second-order reaction–diffusion problem subject to inhomogeneous mixed boundary conditions. The model's novelty lies in the simultaneous incorporation of Cauchy–Neumann and Cauchy–Stefan–Boltzmann conditions, allowing a realistic representation of conduction, convection, and radiation effects. From a theoretical perspective, existence, uniqueness, regularity, and a priori estimates of the solution are established in the functional framework W^1, 2ₚ (Q) via the Leray–Schauder principle and Lᵖ theory for quasi-linear parabolic equations. These findings ensure well-posedness and provide connections with related models in the literature. The model is then applied to laser welding of Al–Si 5% alloy plates using realistic physical parameters. Numerical simulations are carried out with a Galerkin finite element method considering three heat source formulations: Gaussian, conical, and ellipsoidal. The computed temperature distributions and molten pool geometries qualitatively agree with experimentally-observed weld morphologies. In particular, the Gaussian source predicts peak temperatures consistent with industrial data, achieving full penetration and a stable, regular weld bead. The conical source produces excessive thermal peaks and irregular pools, whereas the ellipsoidal source leads to shallow melting. The results support the Gaussian formulation for industrial use and motivate further developments toward quantitative validation and multi-physics extensions.
Miranville et al. (Thu,) studied this question.