Advances in Theoretical and Applied Mathematics
  • Year: 2007
  • Volume: 2
  • Issue: 1

Evolution equation associated with the power of the gross laplacian

  • Author:
  • Soumaya Gheryani1, Hui-Hsiung Kuo2, Habib Ouerdiane3
  • Total Page Count: 10
  • Page Number: 41 to 50

1Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia. E-mail: soumaye gheryani@yahoo.fr

2Department of Mathematics, Luisiana State University, Baton Rouge, LA 70803, USA. E-mail: kuo@math.lsu.edu

3Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia. E-mail: habib.ouerdiane@fst.rnu.tn

Abstract

We study an evolution equation associated with the power of the Gross Laplacian ΔpG and a potential function V on an infinite dimensional space. The initial condition is a generalized function. The main technique we use is the representation of the Gross Laplacian as a convolution operator. This representation enables us to apply the convolution calculus on a suitable distribution space to obtain the explicit solution of the perturbed evolution equation. Our results generalize those previously obtained by Hochberg [7] in the one dimensional case with V =0, as well as by BarhoumiKuoOuerdiane for the case p =1.

Keywords

Evolution equation, Gross Laplacian, potential function, white noise analysis, generalized functions, convolution operator, Laplace transform, duality theorem