Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms

@article{Boutiah2017KernelEF,
  title={Kernel estimates for elliptic operators with unbounded diffusion, drift and potential terms},
  author={S. E. Boutiah and A. Rhandi and C. Tacelli},
  journal={arXiv: Analysis of PDEs},
  year={2017}
}
In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^\alpha)\Delta +b|x|^{\alpha-1}\frac{x}{|x|}\cdot\nabla -|x|^\beta$ satisfies $$ k(t,x,y) \leq c_1e^{\lambda_0 t+ c_2t^{-\gamma}}\left(\frac{1+|y|^\alpha}{1+|x|^\alpha}\right)^{\frac{b}{2\alpha}} \frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(\beta-\alpha)}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}\left(|x|^{\frac{\beta-\alpha+2}{2}}+ |y|^{\frac{\beta-\alpha+2}{2}}\right)} $$ for $t>0,\,|x|,\,|y|\ge 1… Expand
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