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Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus. (English) Zbl 1492.35100

Summary: In this paper we study the initial value problem of fully nonlinear third-order equations on the torus, that is \(\partial_t u = F \left( \partial_x^3 u, \partial_x^2 u, \partial_x u, u, x, t\right)\) with \(F\) a smooth function depending on the space variable \(x\), the time variable \(t\), the first three derivatives of \(u\) with respect to \(x\), and \(u\). In particular we find conditions on \(u(0)\) and \(F\) for which one can construct a local and unique solution \(u\). In particular if \(F\) and \(u(0)\) satisfy some conditions then the equation behaves like a diffusive one and it has a parabolic smoothing property: the solution is infinitely smooth in one direction of time and the problem is ill-posed in the other direction of time. If \(F\) and \(u(0)\) satisfy some other conditions then the equation behaves like a dispersive one. We also prove continuous dependence with respect to the data. The proof relies upon energy estimates combined with a gauge transformation [N. Hayashi, Nonlinear Anal., Theory Methods Appl. 20, No. 7, 823–833 (1993; Zbl 0787.35099); N. Hayashi and T. Ozawa, Physica D 55, No. 1–2, 14–36 (1992; Zbl 0741.35081); N. Hayashi and T. Ozawa, Differ. Integral Equ. 7, No. 2, 453–461 (1994; Zbl 0803.35137)] and the Bona-Smith argument [J. L. Bona and R. Smith, Philos. Trans. R. Soc. Lond., Ser. A 278, 555–601 (1975; Zbl 0306.35027)].

MSC:

35G31 Initial-boundary value problems for nonlinear higher-order PDEs
35B10 Periodic solutions to PDEs
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References:

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