Abstract
The formation of singularities in finite time in nonlocal Burgers’ equations, with time-fractional derivative, is studied in detail. The occurrence of finite-time singularity is proved, revealing the underlying mechanism, and precise estimates on the blowup time are provided. The employment of the present equation to model a problem arising in job market is also analyzed.
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Notes
The notion of solution in Theorem 1.2 is such that for all \(x\in {\mathbb {R}}\), the map \([0,T_\star )\ni t\mapsto u(x,t)\) is continuous, and it is in \(C^{0,\alpha }((0,T_\star ))\), being the latter the space defined, e.g., in formula (3.1) of Kilbas and Marzan (2004). In particular, the Caputo derivative of u is well defined for all \(t\in (0,T_\star )\). Furthermore, for all \(t\in (0,T_\star )\), the map \({\mathbb {R}}\ni x\mapsto u(x,t)\) is smooth, making the classical derivative in space well defined too.
We observe that we cannot use here the Comparison Principle in Lemma 2.6 and Remark 2.1 on pages 219-220 in Vergara and Zacher (2015), since the monotonicity of the nonlinearity goes in the opposite direction.
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Communicated by Dr. Alain Goriely.
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The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). SD and EV are supported by the Australian Research Council Discovery Project grant “Nonlocal Equations at Work” (NEW). SV is supported by the DECRA Project “Partial differential equations, free boundaries and applications”.
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Coclite, G.M., Dipierro, S., Maddalena, F. et al. Singularity Formation in Fractional Burgers’ Equations. J Nonlinear Sci 30, 1285–1305 (2020). https://doi.org/10.1007/s00332-020-09608-x
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DOI: https://doi.org/10.1007/s00332-020-09608-x