# American Institute of Mathematical Sciences

March  2015, 4(1): 61-67. doi: 10.3934/eect.2015.4.61

## Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions

 1 Institut Élie Cartan de Lorraine, UMR CNRS 7502, Université de Lorraine-Metz, Ile du Saulcy, F-57045 Metz cedex 1, France

Received  March 2014 Revised  January 2015 Published  February 2015

In the present note, we give a concise proof for the equivalence between the local boundedness property for parabolic Dirichlet BVP's and the gaussian upper bound for their Green functions. The parabolic equations we consider are of general divergence form and our proof is essentially based on the gaussian upper bound by Daners [2] and a Caccioppoli's type inequality. We also show how the same analysis enables us to get a weaker version of the local boundedness property for parabolic Neumann BVP's assuming that the corresponding Green functions satisfy a gaussian upper bound.
Citation: Mourad Choulli. Local boundedness property for parabolic BVP's and the Gaussian upper bound for their Green functions. Evolution Equations and Control Theory, 2015, 4 (1) : 61-67. doi: 10.3934/eect.2015.4.61
##### References:
 [1] J. Choi and S. Kim, Green's function for second order parabolic systems with Neumann boundary condition, J. Diffent. Equat., 254 (2013), 2834-2860. doi: 10.1016/j.jde.2013.01.003. [2] D. Daners, Heat Kernel estimates for operators with boundary conditions, Math Nachr., 217 (2000), 13-41. doi: 10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6. [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [4] S. Hofmann and S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat., 48 (2004), 481-496. doi: 10.5565/PUBLMAT_48204_10.

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##### References:
 [1] J. Choi and S. Kim, Green's function for second order parabolic systems with Neumann boundary condition, J. Diffent. Equat., 254 (2013), 2834-2860. doi: 10.1016/j.jde.2013.01.003. [2] D. Daners, Heat Kernel estimates for operators with boundary conditions, Math Nachr., 217 (2000), 13-41. doi: 10.1002/1522-2616(200009)217:1<13::AID-MANA13>3.0.CO;2-6. [3] R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 5, Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-642-58090-1. [4] S. Hofmann and S. Kim, Gaussian estimates for fundamental solutions to certain parabolic systems, Publ. Mat., 48 (2004), 481-496. doi: 10.5565/PUBLMAT_48204_10.
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