Blowing Up and Dissipation for a Couple of One-dimensional Non-local Conservation Laws

Abreu E.; Huacasi-Machaca M.; Pérez J.; Valencia-Guevara J.C. · Trends in Mathematics (2025) · DOI: 10.1007/978-3-031-77050-0_20

Resumen

We are interested in studying solutions to the non-local conservation law ∂tu+χ∂x(uΛα−1ℋu)=0 subject to initial data in u0L1∩L∞, where χ=±1. We proved that there is a finite existence time T (maximal time) such that the solution u is finite in [0,T) and blows up at T in the sense of mass concentration (χ=−1). We proved that in the case χ=1 the solution exhibits a spreading of the initial datum, which is verified via numerical insights. We also improved comprehension of the results of Biler and Karch (J. Evol. Equ. 10:247–262, 2010) and Li and Zhang (Commun. Pure Appl. Anal. 9(6):1591–1606, 2010) to simplify the analysis and to circumvent the requirement of initial concentration such that our results hold for initial data in L1∩L∞. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2025.

Instituciones: Universidad Nacional de San Agustín de Arequipa