2009
DOI: 10.1016/j.na.2009.01.156
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Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

Abstract: Abstract. In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.

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Cited by 12 publications
(34 citation statements)
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“…In [5], Barles, Cardaliaguet, Ley and the author proved a general result of existence of weak solutions for these nonlocal equations. In the framework described above, the essential assumptions under which existence is known are the following; they concern the local equation (1.5), where the nonlocal dependence is frozen, that is to say,…”
Section: H[χ](x T P A) ≤ H[χ](x T P B) If a ≤ Bmentioning
confidence: 99%
See 3 more Smart Citations
“…In [5], Barles, Cardaliaguet, Ley and the author proved a general result of existence of weak solutions for these nonlocal equations. In the framework described above, the essential assumptions under which existence is known are the following; they concern the local equation (1.5), where the nonlocal dependence is frozen, that is to say,…”
Section: H[χ](x T P A) ≤ H[χ](x T P B) If a ≤ Bmentioning
confidence: 99%
“…Finally, for the same computational reasons, we point out that even though existence of solutions to (1.1) is known in a more general setting (see [5]), in this article we consider equations depending on the past, which means that…”
Section: H[χ](x T P A) ≤ H[χ](x T P B) If a ≤ Bmentioning
confidence: 99%
See 2 more Smart Citations
“…In particular, papers [24,70] contain higher order regularity results for viscosity solutions. Papers [13,14] treat evolution of interfaces moving with non-local velocity. Papers and books [1,5,6,7,20,21,39,51,59,62] are motivated by applications to finance and control problems.…”
Section: Introductionmentioning
confidence: 99%