Acta Mathematica Vietnamica
Global and Nonglobal Solutions for a Mixed Local-nonlocal Heat Equation
Brandon Carhuas-Torre
,
Ricardo Castillo
,
Ricardo Freire
,
Alex Lira
,
Miguel Loayza
Abstract
In this work, we establish optimal criteria characterizing the global and nonglobal existence of solutions to a semilinear parabolic equation with a general source term $f$. The equation is governed by the mixed operator $$\mathcal {L} = -\Delta + (-\Delta )^s, \quad \text {with } 0< s < 1.$$ In our analysis, we study supersolutions and obtain estimates on the semigroup generated by $\mathcal {L}$. In particular, we recover Fujita’s critical exponent $1 + \tfrac{2s}{N}$, previously established by Biagi, Punzo, and Vecchi [2], and by Del Pezzo and Ferreira [18].
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