A~boundary-value problem for a~non-linear equation with a~fractional derivative

E. I. Kaikina ; P. I. Naumkin ; I. A. Shishmarev

Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1173-1196.

Résumé

We consider a problem on a half-line with a Neumann-type boundary condition. We prove the global existence of solutions and find the leading term of their large-time asymptotic expansion. The case of large initial data and critical non-linearity is considered in the case when the non-linear term of the equation decays in time as rapidly as the linear terms.

Détail

Keywords: Neumann-type boundary-value problem, non-linear equation, fractional derivative, asymptotic behaviour.

Bibliographie

[1] V. A. Zverev, L. A. Ostrovskii, Nelineinaya akustika. Teoreticheskie i eksperimentalnye issledovaniya, In-t prikl. fiz. AN SSSR, Gorkii, 1980, 143–160

[2] L. A. Ostrovsky, “Short-wave asymptotics for weak shock waves and solitons in mechanics”, Internat. J. Non-Linear Mech., 11:6 (1976), 401–416 | DOI | MR | Zbl

[3] E. Ott, R. N. Sudan, “Nonlinear theory of ion acoustic waves with Landau damping”, Phys. Fluids, 12:11 (1969), 2388–2394 | DOI | MR | Zbl

[4] C. J. Amick, J. L. Bona, M. E. Schonbek, “Decay of solutions of some nonlinear wave equations”, J. Differential Equations, 81:1 (1989), 1–49 | DOI | MR | Zbl

[5] P. Biler, “Asymptotic behaviour in time of solutions to some equations generalizing the Korteweg-de Vries–Burgers equation”, Bull. Polish Acad. Sci. Math., 32:5–6 (1984), 275–282 | MR | Zbl

[6] P. Biler, T. Funaki, W. A. Woyczynski, “Fractal Burgers equations”, J. Differential Equations, 148:1 (1998), 9–46 | DOI | MR | Zbl

[7] P. Biler, G. Karch, W. A. Woyczyński, “Multifractal and Lévy conservation laws”, C. R. Acad. Sci. Paris Sér. I Math., 330:5 (2000), 343–348 | DOI | MR | Zbl

[8] J. L. Bona, F. Demengel, K. Promislow, “Fourier splitting and dissipation of nonlinear dispersive waves”, Proc. Roy. Soc. Edinburgh Sect. A, 129:3 (1999), 477–502 | MR | Zbl

[9] J. L. Bona, L. Luo, “Asymptotic decomposition of nonlinear, dispersive wave equations with dissipation”, Phys. D, 152–153 (2001), 363–383 | DOI | MR | Zbl

[10] H. Brezis, F. Browder, “Partial differential equations in the 20th century”, Adv. Math., 135:1 (1998), 76–144 | DOI | MR | Zbl

[11] D. B. Dix, “Temporal asymptotic behavior of solutions of the Benjamin–Ono–Burgers equation”, J. Differential Equations, 90:2 (1991), 238–287 | DOI | MR | Zbl

[12] D. B. Dix, “The dissipation of nonlinear dispersive waves: the case of asymptotically weak nonlinearity”, Comm. Partial Differential Equations, 17:9–10 (1992), 1665–1693 | DOI | MR | Zbl

[13] M. Escobedo, O. Kavian, H. Matano, “Large time behavior of solutions of a dissipative semilnear heat equation”, Comm. Partial Differential Equations, 20:7–8 (1995), 1427–1452 | DOI | MR | Zbl

[14] M. Escobedo, J. L. Vazquez, E. Zuazua, “Asymptotic behaviour and source-type solutions for a diffusion-convection equation”, Arch. Rational Mech. Anal., 124:1 (1993), 43–65 | DOI | MR | Zbl

[15] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov, Blow-up in quasilinear parabolic equations, de Gruyter Exp. Math., 19, de Gruyter, Berlin, 1995 | MR | MR | Zbl | Zbl

[16] Y. Giga, T. Kambe, “Large time behavior of the vorticity of two-dimensional viscous flow and its application to vortex formation”, Comm. Math. Phys., 117:4 (1988), 549–568 | DOI | MR | Zbl

[17] A. Gmira, L. Veron, “Large time behaviour of the solutions of a semilinear parabolic equation in $\mathbb R^{n}$”, J. Differential Equations, 53:2 (1984), 258–276 | DOI | MR | Zbl

[18] N. Hayashi, E. I. Kaikina, P. I. Naumkin, “Large-time behaviour of solutions to the dissipative nonlinear Schrödinger equation”, Proc. Roy. Soc. Edinburgh Sect. A, 130:5 (2000), 1029–1043 | DOI | MR | Zbl

[19] N. Hayashi, E. I. Kaikina, P. I. Naumkin, “Large time behavior of solutions to the Landau–Ginzburg type equations”, Funkcial. Ekvac., 44:1 (2001), 171–200 | MR | Zbl

[20] G. Karch, “Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics”, Math. Methods Appl. Sci., 22:18 (1999), 1671–1697 | 3.0.CO;2-Q class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[21] O. Kavian, “Remarks on the large time behaviour of a nonlinear diffusion equation”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4:5 (1987), 423–452 | MR | Zbl

[22] P. I. Naumkin, I. A. Shishmarëv, Nonlinear nonlocal equations in the theory of waves, Transl. Math. Monogr., 133, Amer. Math. Soc., Providence, RI, 1994 | MR | Zbl

[23] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970 | MR | Zbl

[24] M. E. Schonbek, “Uniform decay rates for parabolic conservation laws”, Nonlinear Anal., 10:9 (1986), 943–956 | DOI | MR | Zbl

[25] M. E. Schonbek, “Lower bounds of rates of decay for solutions to the Navier–Stokes equations”, J. Amer. Math. Soc., 4:3 (1991), 423–449 | DOI | MR | Zbl

[26] W. A. Strauss, “Nonlinear scattering at low energy”, J. Funct. Anal., 41:1 (1981), 110–133 | DOI | MR | Zbl

[27] Q. S. Zhang, “A blow-up result for a nonlinear wave equation with damping: the critical case”, C. R. Acad. Sci. Paris Sér. I Math., 333:2 (2001), 109–114 | DOI | MR | Zbl

[28] N. Hayashi, E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, Asymptotics for dissipative nonlinear equations, Lecture Notes in Math., 1884, Springer-Verlag, Berlin, 2006 | DOI | MR | Zbl

[29] N. Hayashi, E. Kaikina, Nonlinear theory of pseudodifferential equations on a half-line, North-Holland Math. Stud., 194, Elsevier, Amsterdam, 2004 | MR | Zbl

[30] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Tables of integral transforms, vols. I, II, McGraw-Hill, New York–Toronto–London, 1954 | MR | MR | MR | Zbl | Zbl