Well-posedness of a conformable parabolic equation with nonhomogeneous Neumann boundary

Abstract

We study a one dimensional parabolic equation involving a conformable time derivative and nonhomogeneous Neumann boundary data. A weak formulation is introduced in weighted Bochner spaces associated with the natural measure \(d_\beta t=t^{\beta-1}\,dt\), and the conformable derivative is interpreted in a weak sense through a suitable transformation of the time variable. A priori energy estimates are derived for the solution in terms of the initial datum, the source term, and the prescribed boundary flux. Using the change of variables \(\tau=t^\beta/\beta\), the problem is transformed into a classical parabolic problem, which yields the existence of weak solutions. Uniqueness and continuous dependence on the data are then established by energy methods, leading to well-posedness in the sense of Hadamard. Under additional regularity assumptions on the initial and boundary data, higher spatial and temporal regularity of the solution is also obtained.