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We investigated implications of recently released `Joint Light-curveAnalysis' (JLA) supernova Ia (SNe Ia) data for dark energy models with timevarying equation of state of dark energy, usually expressed as $w(z)$ in termsof variation with corresponding redshift $z$. From a comprehensive analysis ofthe JLA data, we obtain the observational constraints on the differentfunctional forms of $w(z)$, corresponding to different varying dark energymodels often considered in literature, \textit{viz.} CPL, JBP, BA andLogarithmic models. The constraints are expressed in terms of parameters ($w_a,w_b$) appearing in the chosen functional form for $w(z)$, corresponding to eachof the above mentioned models. Realising dark energy with varying equation ofstate in terms of a homogeneous scalar field $\phi$, with its dynamics drivenby a $k-$essence Lagrangian $L=VF(X)$ with a constant potential $V$ and adynamical term $F(X)$ with $X=(1/2)\nabla^\mu\phi\nabla_\mu\phi$ wereconstructed form of the function $F(X)$. This reconstruction has beenperformed for different varying dark energy models at best-fit values ofparameters ($w_a, w_b$) obtained from analysis of JLA data. In the context of$k-$essence model, we also investigate the variation of adiabatic sound speedsquared, $c_s^2(z)$, and obtained the domains in ($w_a, w_b$) parameter spacecorresponding to the physical bound $ c_s^2>0$ implying stability of densityperturbations.

Implications of JLA data for $k-$essence model of dark energy with given equation of state