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We show that all extremal elliptic surfaces in characteristic 2 and 3 are obtained from rational extremal elliptic surfaces as purely inseparable base extensions. As a corollary, we can show that the automorphism group of every supersingular elliptic $K3$ surface has an element of infinite order which acts trivially on the global sections of the sheaf of differential forms of degree 2. We also determine the structures of Mordell- Weil groups for extremal rational elliptic surfaces in these characteristics.
We consider the asymptotic behavior of solutions to the initial value problem for a certain class of hyperbolic-elliptic coupled systems. It will be proved that the solution is time asymptotically approximated by the superposition of diffusion waves constructed in terms of the self-similar solutions of generalized Burgers equations. We will give space-time decay estimates for the residual term through a pointwise estimate for the Green’s function of the linearized system.