Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

Abstract

AbstractThe ordinary diffusion models in continuous media are derived from basic principles: Conservation of energy and momentum. The classical telegraph equation is one of the earliest and most commonly used equation to describe the transmission of electromagnetic waves. In fractal and disordered media, to capture the complex diffusion feature, we consider rather fractional models. Indeed, the mean square displacement in anomalous media is no longer proportional to the time but rather proportional to a power of time whose exponent may be between zero and one or between one and two. This is the case for the present fractional telegraph equation in the presence of non‐negligible voltage wave. Here, the well‐posedness and stability of a telegraph problem with two time‐fractional derivatives is discussed. In addition to being a noninteger problem, the equation in the model is subject to a nonlinear source as well as a nonlinear lower order fractional derivative term. First, the well‐posedness in an appropriate underlying space is established by the use of resolvent operators. Then, it is proved that, despite the presence of these nonlinearities, solutions are Mittag–Leffler stable. This confirms (and extends) the fact that the lower order term plays the role of a damper in the fractional case. To this end, we combine the energy method with some properties of the Caputo fractional derivative.