  We study the renormalization group evolution of the Higgs quartic coupling
$\lambda_{H}$ and the Higgs mass $m_{H}$ in the Standard Model. The one loop
equation for $\lambda_{H}$ is non linear and it is of the Riccati type which we
numerically and analytically solve in the energy range $[m_{t},E_{GU}]$ where
$m_{t}$ is the mass of the top quark and $E_{GU}=10^{14}$ GeV. We find that
depending on the value of $\lambda_{H}(m_{t})$ the solution for
$\lambda_{H}(E)$ may have singularities or zeros and become negative in the
former energy range so the ultra violet cut off of the standard model should be
below the energy where the zero or singularity of $\lambda_{H}$ occurs. We find
that for $0.369\leq\lambda_{H}(m_{t})\leq0.613$ the Standard Model is valid in
the whole range $[m_{t},E_{GU}]$. We consider two cases of the Higgs mass
relation to the parameters of the standard model: (a) the effective potential
method and (b) the tree level mass relations. The limits for
$\lambda_{H}(m_{t})$ correspond to the following Higgs mass relation $150\leq
m_{H}\lessapprox 193$ GeV. We also plot the dependence of the ultra violet cut
off on the value of the Higgs mass. We analyze the evolution of the vacuum
expectation value of the Higgs field and show that it depends on the value of
the Higgs mass. The pattern of the energy behavior of the VEV is different for
the cases (a) and (b). The behavior of $\lambda_{H}(E)$, $m_{H}(E)$ and $v(E)$
indicates the existence of a phase transition in the standard model. For the
effective potential this phase transition occurs at the mass range
$m_{H}\approx 180$ GeV and for the tree level mass relations at $m_{H}\approx
168$ GeV.
