## 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

### Citations:

Zbl 0787.35099; Zbl 0741.35081; Zbl 0803.35137; Zbl 0306.35027
Full Text:

### References:

 [1] Akhunov, T.; Ambrose, D.; Wright, J., Well-posedness of fully nonlinear KdV-type evolution equations, Nonlinearity, 32, 2914-2954 (2019) · Zbl 1421.35318 [2] Ambrose, D.; Wright, J., Dispersion vs. anti-diffusion: Well-posedness in variable coefficient and quasilinear equations of KdV type, Indiana Univ. Math. J., 62, 4, 1237-1281 (2019) · Zbl 1293.35274 [3] Bona, J. L.; Smith, R., The initial-value problem for the Korteweg-de Vries equation, Philos. Trans. Roy. Soc. London Ser. A, 278, 1287, 555-601 (1975) · Zbl 0306.35027 [4] Harrop-Griffith, B., Large data local well-podedness for a class of KdV-type equations II, Int. Math. Res. Not. IMRN, 18, 8590-8619 (2015) · Zbl 1326.35314 [5] Harrop-Griffith, B., Large data local well-posedness for a class of KdV-type equations, Trans. Amer. Mat. Soc., 367, 2, 755-773 (2015) · Zbl 1312.35151 [6] Hayashi, N., The initial value problem for the derivative nonlinear Schrödinger equation in the energy space, Nonlinear Anal., 20, 7, 823-833 (1993) · Zbl 0787.35099 [7] Hayashi, N.; Ozawa, T., On the derivative nonlinear Schrödinger equation, Phys. D, 55, 1-2, 14-36 (1992) · Zbl 0741.35081 [8] Hayashi, N.; Ozawa, T., Remarks on nonlinear Schrödinger equations in one space dimension, Differ. Integral Equ., 7, 2, 453-461 (1994) · Zbl 0803.35137 [9] Iorio, R. J.; Iorio, V., Fourier Analysis and Partial Differential Equations (2001), Cambridge University Press [10] Kenig, C. E.; Ponce, G.; Vega, L., Higher-order nonlinear dispersive equations, Proc. Amer. Math. Soc., 122, 1, 157-166 (1994) · Zbl 0810.35122 [11] Kenig, C. E.; Ponce, G.; Vega, L., On the hierarchy of the generalized KdV equations, (Proc. Lyon Workshop on Singular Limits of Dispersive Waves. Proc. Lyon Workshop on Singular Limits of Dispersive Waves, NATO ASI Ser, vol. 320 (1994)), 347-356 · Zbl 0849.35121 [12] Kenig, C.; Staffilani, G., Local well-posedness for higher order nonlinear dispersive systems, J. Fourier Anal. Appl., 3, 4, 417-433 (1997) · Zbl 0884.35116 [13] Pilod, D., On the Cauchy problem for higher-order nonlinear dispersive equations, J. Differ. Integral Equ., 245, 8, 2055-2077 (2008) · Zbl 1152.35017 [14] K. Tsugawa, Parabolic smoothing effect and local well-posedness of fifth order semilinear dispersive equations on the torus, preprint. · Zbl 1375.35463
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