A fractional Laplacian problem with mixed singular nonlinearities and nonregular data

In this note, we study the existence and uniqueness of a positive solution to a doubly singular fractional problem with nonregular data. Besides, for some cases, we will show the existence and uniqueness of another notion of a solution, so-called entropy solution. Also, with suitable assumptions on data, we will discuss the uniqueness. Finally, we will have some relaxation on the assumption to prove the existence results.Comment: We are grateful for any feedback or comments. arXiv admin note: text overlap with arXiv:1910.0471

A note on the existence of a unique positive entropy solution to a fractional Laplacian with singular nonlinearities

In this paper, we prove the existence of a unique positive entropy solution to a fractional Laplacian problem involving nonlinear singular terms and also a non-negative bounded Radon measure as a source term.Comment: We are grateful for any feedback or comment

Nonlocal Lazer-McKenna type problem perturbed by the Hardy's potential and its parabolic equivalence

In this paper, we study the effect of Hardy potential on the existence or non-existence of solutions to a fractional Laplacian problem involving a singular nonlinearity. Also, we mention a stability result.Comment: arXiv admin note: text overlap with arXiv:1412.8159 by other author

On the structure of ionizing shock waves in magnetofluiddynamics

Ionizing shock waves in magnetofluiddynamics occur when the coefficient of electrical conductivity is very small ahead of the shock and very large behind it. For planner motion of plasma, the structure of such shock waves are stated in terms of a system of four-dimensional equations. In this paper, we show that for the above electrical conductivity as well as for limiting cases, that is, when this coefficient is zero ahead of the shock and/or is infinity behind it, ionizing fast, slow, switch-on and switch-off shocks admit structure. This means that physically these shocks occur

On the One-dimensional Stability of Viscous Strong Detonation Waves

Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting Navier-Stokes equations. The primary result, following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting Navier-Stokes equations are includedComment: 66 pages, 7 figure