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The present paper is devoted to the investigation of various properties oftransition fronts in nonlocal equations in heterogeneous media of ignitiontype, whose existence has been established by the authors of the present paperin a previous work. It is first shown that the transition front is continuouslydifferentiable in space with uniformly bounded and uniformly Lipschitzcontinuous space partial derivative. This is the first time that regularity oftransition fronts in nonlocal equations is ever studied. It is then shown thatthe transition front is uniformly steep. Finally, asymptotic stability, in thesense of exponentially attracting front like initial data, of the transitionfront is studied.

Regularity and stability of transition fronts in nonlocal equations with time heterogeneous ignition nonlinearity