Asymptotics for nonlinear damped wave equations with large initial data

N. Hayashi ; E. I. Kaikina ; P. I. Naumkin

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 4 (2007), pp. 249-277.

Résumé

We study the one dimensional nonlinear damped wave equation \begin{equation} \begin{cases} u_{tt}+u_{t}-u_{xx}=\lambda|u|^{\sigma}u,\in\mathbf{R},\quad t>0,\\ u(0,x)=u_0(x), x\in\mathbf{R},\\ u_t(0,x)=u_1(x), x\in\mathbf{R}, \end{cases} \tag{0.1} \end{equation} where $\sigma>0$, $\lambda\in\mathbf R$. Our aim is to prove the large time asymptotic formulas for solutions of the Cauchy problem (0.1) without any restriction on the size of the initial data.

Bibliographie

[1] A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations”, Acta Mathematica, 181:2 (1998), 229–243 | DOI | MR | Zbl

[2] M. Escobedo and O. Kavian, “Asymptotic behavior of positive solutions of a non-linear heat equation”, Houston Journal of Mathematics, 13:4 (1987), 39–50 | MR

[3] M. Escobedo, O. Kavian and H. Matano, “Large time behavior of solutions of a dissipative nonlinear heat equation”, Comm. Partial Diff. Eqs., 20 (1995), 1427–1452 | DOI | MR | Zbl

[4] M. V. Fedoryuk, Asymptotics: integrals and series, Mathematical Reference Library, Nauka, Moscow, 1987, 544 pp. | MR

[5] H. Fujita, “On the blowing-up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$”, J. Fac. Sci. Univ. of Tokyo, Sect. I, 13 (1966), 109–124 | MR | Zbl

[6] V. A. Galaktionov, S. P. Kurdyumov and A. A. Samarskii, “On asymptotic eigenfunctions of the Cauchy problem for a nonlinear parabolic equation”, Math. USSR Sbornik, 54 (1986), 421–455 | DOI | MR | Zbl

[7] K. Hayakawa, “On non-existence of global solutions of some semi-linear parabolic equations”, Proc. Japan Acad., 49 (1973), 503–505 | DOI | MR | Zbl

[8] A. Gmira and L. Veron, “Large time behavior of the solutions of a semilinear parabolic equation in $R^1$”, J. Diff. Eqs., 53 (1984), 258–276 | DOI | MR | Zbl

[9] N. Hayashi, N. Ito, E. I. Kaikina and P. I. Naumkin, “On some nonlinear dissipative equations with sub-critical nonlinearities”, Taiwanese Journal of Math., 8 (2004), 135–154 | MR | Zbl

[10] N. Hayashi, E. I. Kaikina and P. I. Naumkin, “Large time behavior of solutions to the dissipative nonlinear Schrödinger equation”, Proceedings of the Royal Soc. Edingburgh A, 130 (2000), 1029–1043 | DOI | MR | Zbl

[11] N. Hayashi, E. I. Kaikina and P. I. Naumkin, “Global existence and time decay of small solutions to the Landau–Ginzburg type equations”, Journal d'Analyse Mathematique, 90 (2003), 141–173 | DOI | MR | Zbl

[12] N. Hayashi, E. I. Kaikina and P. I. Naumkin, “Damped wave equation with a critical nonlinearity”, Trans. Amer. Math. Soc., 358:3 (2006), 1165–1185 | DOI | MR | Zbl

[13] N. Hayashi, E. I. Kaikina and P. I. Naumkin, “Damped wave equation with supercritical nonlinearities”, Differential Integral Equations, 17:5–6 (2004), 637–652 | MR | Zbl

[14] N. Hayashi, E. I. Kaikina and P. I. Naumkin, “Damped wave equation in the subcritical case”, J. Differential Equations, 207:1 (2004), 161–194 | DOI | MR | Zbl

[15] N. Hayashi, E. I. Kaikina and P. I. Naumkin, Subcritical nonlinear heat equation, Preprint, 2005 | MR

[16] N. Hayashi and P. I. Naumkin, “Asymptotics for the Korteweg–de Vries–Burgers equation”, Acta Mathematica Sinica, English Series, 22 (2006), 1441–1456 | DOI | MR | Zbl

[17] G. Karch, “Selfsimilar profiles in large time asymptotics of solutions to damped wave equations”, Studia Math., 143:2 (2000), 175–197 | MR | Zbl

[18] O. Kavian, “Remarks on the large time behavior of a nonlinear diffusion equation”, Ann. Inst. Henri Poincaré, Analyse non linéaire, 4:5 (1987), 423–452 | MR | Zbl

[19] S. Kawashima, M. Nakao, K. Ono, “On the decay property of solutions to the Cauchy problem of the semilinear wave equation with a dissipative term”, J. Math. Soc. Japan, 47:4 (1995), 617–653 | DOI | MR | Zbl

[20] T. Komatsu, “On the martingale problem for generators of stable processes with perturbations”, Osaka J. Math., 21 (1984), 113–132 | MR | Zbl

[21] T. T. Li and Y. Zhou, “Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha}$”, Discrete Conti. Dynam. Systems, 1:4 (1995), 503–520 | DOI | MR | Zbl

[22] A. Matsumura, “On the asymptotic behavior of solutions of semi-linear wave equations”, Publ. Res. Inst. Math. Sci., Kyoto Univ., 12 (1976), 169–189 | DOI | MR | Zbl

[23] K. Nishihara, “$\mathbf{L}^p-\mathbf{L}^q$ estimates of solutions to the damped wave equation in 3-dimensional space and their application”, Math. Z, 244 (2003), 631–649 | MR | Zbl

[24] K. Nishihara and H. Zhao, “Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption”, J. Math. Anal. Appl., 313:2 (2006), 598–610 | DOI | MR | Zbl

[25] K. Ono, “Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations”, Discrete and Continuous Dynamical. Systems, 9 (2003), 651–662 | DOI | MR | Zbl

[26] M. E. Schonbek, “The Fourier splitting method”, Advances in geometric analysis and continuum mechanics (Stanford, CA, 1993), Internat. Press, Cambridge, MA, 1995, 269–274 | MR | Zbl

[27] G. Todorova and B. Yordanov, “Critical exponent for a nonlinear wave equation with damping”, J. Diff. Equs., 174 (2001), 464–489 | DOI | MR | Zbl

[28] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944, 804 pp. | MR | Zbl

[29] E. Zuazua, “Some recent results on the large time behavior for scalar parabolic conservation laws”, Elliptic and parabolic poblems, Proc. 2nd European Conference, Pitman Res. Notes Math. Ser., 325, 1995, 251–263 | MR | Zbl