Asymptotic behaviour of bigraded components of local cohomology modules

Let $C$ be a commutative Noetherian ring containing a field $K$ of characteristic zero. Let $R=C[X_1,\dots ,X_n,Y_1,\dots ,Y_m]$ be a polynomial ring over $C$ with $\mathrm{b}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{g}c=(0,0)$ for all $c\in C$, $\mathrm{b}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{g}{X}_i=(1,0)$ and $\mathrm{b}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{g}{Y}_j=(0,1)$ for $i=1,\dots ,n$ and $j=1,\dots ,m$. Let $I$ be a bihomogeneous ideal in $R$. In this article, we study asymptotic behaviour of bigraded pieces of the local cohomology module $H_I^i(R)$. Moreover, under the extra assumption that $C$ is regular, we investigate the asymptotic stability of invariants associated to its bigraded components. Consequently, we obtain certain properties of components of the bigraded local cohomology module $H_I^i(R)$, where $C=K$ is a field and $I$ is a binomial edge ideal.