We establish Fredholm properties for a class of nonlocal differential operators. Using mild convergence and localization conditions on the nonlocal terms, we also show how to compute Fredholm indices via a generalized spectral flow, using crossing numbers of generalized spatial eigenvalues. We illustrate possible applications of the results in a nonlinear and a linear setting. We first prove the existence of small viscous shock waves in nonlocal conservation laws with small spatially localized source terms. We also show how our results can be used to study edge bifurcations in eigenvalue problems using Lyapunov-Schmidt reduction instead of a Gap Lemma.
Bibliographical noteFunding Information:
We would like to thank Bj?rn Sandstede for pointing out a gap in an earlier version of the proof of Proposition 4.7. The authors express thanks for the partial support provided by the National Science Foundation, through grants NSF-DMS-1311414 (first author) and NSF-DMS-0806614 and DMS-1311740 (second author).
Indiana University Mathematics Journal © 2014.
- Edge bifurcations
- Fredholm index
- Nonlocal conservation law
- Nonlocal operator
- Spectral flow