This paper is devoted to the study of the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in R N. We first prove a general Freidlin-G{a}rtner type formula for the spreading speeds of the solutions in any direction. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of independent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of traveling fronts with positive speed. Furthermore, we show some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we prove some asymptotic one-dimensional symmetry properties for the elements of the $Omega$-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we show some logarithmicin-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The paper also contains some related conjectures and open problems.
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