  The purpose of this article (composed of two parts) is the study of the
generalized dispersal operator of a reaction-diffusion equation in $L^p$-spaces
set in the finite conical domain $S_{\omega,\rho}$ of angle $\omega>0$ and
radius $\rho>0$ in $\mathbb{R}^2$. This first part is devoted to the behaviour
of the solution near the top of the cone which is completely described in the
weighted Sobolev space $W^{4,p}_{3-\frac{1}{p}}(S_{\omega,\rho})$, see Theorem
2.2.
