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- W4288346048 abstract "It is known that the De Giorgi’s conjecture does not hold in two dimensions for semilinear elliptic equations with a nonzero drift, in general, $$begin{aligned} Delta u+ qcdot nabla u+f(u)=0 text {in } {mathbb {R}}^2, end{aligned}$$ when $$q=(0,-c)$$ for $$cne 0$$ . This equation arises in the modeling of Bunsen burner flames. Bunsen flames are usually made of two flames: a diffusion flame and a premixed flame. In this article, we prove De Giorgi type results, and stability conjecture, for the following local-nonlocal counterpart of the above equation (with a nonlocal premixed flame) in two dimensions, $$begin{aligned} Delta u + c L[u] + f(u)=0 quad text {in} {{mathbb {R}}}^n, end{aligned}$$ when L is a nonlocal operator, $$fin C^1({mathbb {R}})$$ and $$cin {mathbb {R}}^+$$ . In addition, we provide a priori estimates for the above equation, when $$nge 1$$ , with various jumping kernels. The operator $$Delta +cL$$ is an infinitesimal generator of jump-diffusion processes in the context of probability theory." @default.
- W4288346048 created "2022-07-29" @default.
- W4288346048 creator A5065904836 @default.
- W4288346048 date "2022-08-10" @default.
- W4288346048 modified "2023-10-07" @default.
- W4288346048 title "De Giorgi type results for equations with nonlocal lower-order terms" @default.
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- W4288346048 doi "https://doi.org/10.1007/s10231-022-01251-5" @default.
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