The search for heavy Higgs bosons is an essential step in the exploration of the Higgs sector and in probing the Supersymmetric parameter space. This paper discusses the constraints on the M(A) and tan beta parameters derived from the bounds on the different decay channels of the neutral H and A bosons accessible at the LHC, in the framework of the phenomenological MSSM. The implications from the present LHC results and the expected sensitivity of the 14 TeV data are discussed in terms of the coverage of the [M(A) - tan beta] plane. New channels becoming important at 13 and 14 TeV for low values of tan beta are characterised in terms of their kinematics and the reconstruction strategies. The effect of QCD systematics, SUSY loop effects and decays into pairs of SUSY particles on these constraints are discussed in details.