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Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus (2102.02689v4)
Published 4 Feb 2021 in math.AP
Abstract: We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution in one direction of time that is infinitely smooth and the problem in not well-posed in the other direction. Under other conditions on the nonlinearity and the data, we prove that the equation behaves like a dispersive one: there exists a unique local solution in both directions of time.