Global existence and asymptotic behavior of solutions to a semilinear parabolic equation on Carnot groups

Zixia Yuan

doi:10.1186/s13661-015-0383-8

Abstract

In this paper we consider the semilinear parabolic equation $\sum_{i,j = 1}^{m} {a_{ij} X_{i} X_{j} u} - \partial_{t} u + Vu^{p} = 0$ with a general class of potentials $V=V(\xi,t)$ , where $A = \{a_{ij} \}_{i,j} $ is a positive definite symmetric matrix and the $X_{i}$ ’s denotes a system of left-invariant vector fields on a Carnot group G. Based on a fixed point argument and by establishing some new estimates involving the heat kernel, we study the existence and large-time behavior of global positive solutions to the preceding equation.

Highlights

Global existence and asymptotic behavior of solutions to nonlinear parabolic equations have been followed with interest over the past years [ – ].In this paper we are concerned with the existence and asymptotic behavior of global positive solutions for the semilinear parabolic equation Hu = m i,j= aijXiXju ∂ ∂t uV (ξ, t)up (ξ, t) ∈ G × (, +∞), ( . )u(ξ, ) = u (ξ ), ξ ∈ G.Here p >, X, . . . , Xm are left-invariant vector fields on a Carnot group G, and the matrix A = {aij}i,j is symmetric and positive definite, that is,– |Z| ≤ AZ, Z ≤ |Z|
Based on a fixed point argument and by establishing some new estimates involving the heat kernel, we study the existence and large-time behavior of global positive solutions to the preceding equation
1 Introduction Global existence and asymptotic behavior of solutions to nonlinear parabolic equations have been followed with interest over the past years [ – ]

Summary

Introduction

In this paper we are concerned with the existence and asymptotic behavior of global positive solutions for the semilinear parabolic equation The asymptotic behavior of the global solutions are obtained by establishing global Gaussian upper bounds for the fundamental solution of certain linear degenerate parabolic operators on Carnot groups. Following [ ], we briefly recall some well-known results on the fundamental solution for the operator LA =

Results

Conclusion