Let $R=k[x_1,\cdots, x_n]$ be a polynomial ring over a field $k$ of characteristic $p>0.$ If $I$ is an ideal of $R,$ we denote $H^i_I(R)$ the $i$-th local cohomology module of $R$ with support in $I.$ After some introductory material on local cohomology, we will give a lower bound of the dimension of associated primes $P$ of $H^i_I(R)$ in terms of the degrees of the generators of $I.$ Let $\m=(x_1,\cdots, x_n)$ be the maximal ideal generated by the variables and let $I_1,\cdots, I_s$ be homogeneous ideals of $R.$ We will describe the grading of $H^i_{\mathfrak{m}}(H^{j_1}_{I_1}\circ\cdots \circ H^{j_s}_{I_s}(R))$ and also give two algorithms to calculate it.