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In this article, we consider the projective bundle $\mathbb{P}_X(E)$ over asmooth complex projective variety $X$, where $E$ is a semistable bundle on $X$with $c_2(End(E)) =0$. We give a necessary and sufficient condition to get theequality $ Nef^1\bigl(\mathbb{P}_X(E)\bigr) =\overline{Eff}^1\bigl(\mathbb{P}_X(E)\bigr)$ of nef cone and pseudoeffectivecone of divisors in $\mathbb{P}_X(E)$. As an application of our result, we showthe equality of nef and pseudoeffective cones of divisors of projective bundlesover some special varieties. In particular, we show that weak Zariskidecomposition exists on these projective bundles. We also show that asemistable bundle $E$ of rank $r \geq 2$ with $c_2\bigl(End(E)\bigr) = 0$ on asmooth complex projective variety of Picard number 1 is $k$-homogeneous i.e.$\overline{Eff}^k\bigl(\mathbb{P}_X(E)\bigr) =Nef^k\bigl(\mathbb{P}_{X}(E)\bigr)$ for all $1 \leq k < r$. Finally, we showthat weak Zariski decomposition exists for a fibre product$\mathbb{P}_C(E)\times_C\mathbb{P}(E')$ over a smooth projective curve $C$.

Pseudo-effective cones of projective bundles and weak Zariski decomposition