The energy-critical nonlinear wave equation with an inverse-square potential

Miao, Changxing; Murphy, Jason; Zheng, Jiqiang

Annales de l'I.H.P. Analyse non linéaire, Tome 37 (2020) no. 2, pp. 417-456.

We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussing case, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.

[1] Aronszajn, N.; Krzywicki, A.; Szarski, J. A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. Mat., Volume 4 (1962), pp. 417–453 | DOI | MR | Zbl

[2] Bahouri, H.; Gérard, P. High frequency approximation of solutions to critical nonlinear wave equations, Am. J. Math., Volume 121 (1999) no. 1, pp. 131–175 | DOI | MR | Zbl

[3] Brézis, H.; Lieb, E. A relation between pointwise convergence of functions and convergence of functionals, Proc. Am. Math. Soc., Volume 88 (1983), pp. 486–490 | DOI | MR | Zbl

[4] Burq, N.; Planchon, F.; Stalker, J.; Tahvildar-Zadeh, A. Strichartz estimates for the wave and Schrödinger equations with the inverse-square potential, J. Funct. Anal., Volume 203 (2003) no. 2, pp. 519–549 | DOI | MR | Zbl

[5] Duyckaerts, T.; Jia, H.; Kenig, C.; Merle, F. Soliton resolution along a sequence of times for the focusing energy critical wave equation, Geom. Funct. Anal., Volume 27 (2017) no. 4, pp. 798–862 | DOI | MR

[6] Grillakis, M. Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. Math. (2), Volume 132 (1990), pp. 485–509 | DOI | MR | Zbl

[7] Grillakis, M. Regularity for the wave equation with a critical nonlinearity, Commun. Pure Appl. Math., Volume 45 (1992), pp. 749–774 | DOI | MR | Zbl

[8] Grillakis, M. Energy estimates and the wave map problem, Commun. Partial Differ. Equ., Volume 23 (1998) no. 5–6, pp. 887–911 | MR | Zbl

[9] Hong, Y. Scattering for a nonlinear Schrödinger equation with a potential, Commun. Pure Appl. Anal., Volume 15 (2016) no. 5, pp. 1571–1601 | DOI | MR

[10] Ionescu, A.D.; Pausader, B. Global well-posedness of the energy-critical defocusing NLS on $\mathbb{R}\times {\mathbb{T}}^3$ , Commun. Math. Phys., Volume 312 (2012) no. 3, pp. 781–831 | DOI | MR | Zbl

[11] Ionescu, A.D.; Pausader, B. The energy-critical defocusing NLS on ${\mathbb{T}}^3$ , Duke Math. J., Volume 161 (2012) no. 8, pp. 1581–1612 | DOI | MR | Zbl

[12] Ionescu, A.D.; Pausader, B.; Staffilani, G. On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE, Volume 5 (2012) no. 4, pp. 705–746 | DOI | MR | Zbl

[13] Jao, C. The energy-critical quantum harmonic oscillator, Commun. Partial Differ. Equ., Volume 41 (2016) no. 1, pp. 79–133 | MR

[14] Jao, C. Energy-critical NLS with potentials of quadratic growth, Discrete Contin. Dyn. Syst., Volume 38 (2018) no. 2, pp. 563–587 | MR

[15] Jao, C. The quintic NLS on perturbations of ${\mathbb{R}}^3$ , Am. J. Math., Volume 141 (2019) no. 4, pp. 981–1035 | MR

[16] Kalf, H.; Schmincke, U.W.; Walter, J.; Wüst, R. On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Spectral Theory and Differential Equations, Lect. Notes in Math., vol. 448, Springer, Berlin, 1975, pp. 182–226 | MR | Zbl

[17] Kapitanski, L. Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., Volume 1 (1994), pp. 211–223 | DOI | MR | Zbl

[18] Kenig, C.; Merle, F. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., Volume 201 (2008) no. 2, pp. 147–212 | DOI | MR | Zbl

[19] Killip, R.; Kwon, S.; Shao, S.; Visan, M. On the mass-critical generalized KdV equation, Discrete Contin. Dyn. Syst., Ser. A, Volume 32 (2012), pp. 191–221 | DOI | MR | Zbl

[20] Killip, R.; Miao, C.; Visan, M.; Zhang, J.; Zheng, J. Sobolev spaces adapted to the Schrödinger operator with inverse-square potential, Math. Z., Volume 288 (2018) no. 3–4, pp. 1273–1298 | MR

[21] Killip, R.; Miao, C.; Visan, M.; Zhang, J.; Zheng, J. The energy-critical NLS with inverse-square potential, Discrete Contin. Dyn. Syst., Volume 37 (2017) no. 7, pp. 3831–3866 | DOI | MR

[22] Killip, R.; Murphy, J.; Visan, M.; Zheng, J. The focusing cubic NLS with inverse-square potential in three space dimensions, Differ. Integral Equ., Volume 30 (2017) no. 3–4, pp. 161–206 | MR

[23] Killip, R.; Oh, T.; Pocovnicu, O.; Visan, M. Solitons and scattering for the cubic-quintic nonlinear Schrödinger equation on ${\mathbb{R}}^3$ , Arch. Ration. Mech. Anal., Volume 225 (2017) no. 1, pp. 469–548 | DOI | MR

[24] Killip, R.; Stovall, B.; Visan, M. Scattering for the cubic Klein–Gordon equation in two space dimensions, Trans. Am. Math. Soc., Volume 364 (2012), pp. 1571–1631 | DOI | MR | Zbl

[25] Killip, R.; Visan, M. Nonlinear Schrödinger equations at critical regularity, Evolution Equations, Clay Math. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 2013, pp. 325–437 | MR | Zbl

[26] Killip, R.; Visan, M. The defocusing energy-supercritical nonlinear wave equation in three space dimensions, Trans. Am. Math. Soc., Volume 363 (2011) no. 7, pp. 3893–3934 | DOI | MR | Zbl

[27] Killip, R.; Visan, M.; Zhang, X. Quintic NLS in the exterior of a strictly convex obstacle, Am. J. Math., Volume 138 (2016) no. 5, pp. 1193–1346 | DOI | MR

[28] Killip, R.; Visan, M.; Zhang, X. The focusing cubic NLS on exterior domains in three dimensions, Appl. Math. Res. Express, Volume 1 (2016), pp. 146–180 | MR

[29] Li, D.; Zhang, X. Dynamics for the energy critical nonlinear wave equation in high dimensions, Trans. Am. Math. Soc., Volume 363 (2011), pp. 1137–1160 | MR | Zbl

[30] Liskevich, V.; Sobol, Z. Estimates of integral kernels for semigroups associated with second order elliptic operators with singular coefficients, Potential Anal., Volume 18 (2003), pp. 359–390 | DOI | MR | Zbl

[31] Lu, J.; Miao, C.; Murphy, J. Scattering in $H^1$ for the intercritical NLS with an inverse-square potential, J. Differ. Equ., Volume 264 (2018) no. 5, pp. 3174–3211 | MR

[32] Milman, P.D.; Semenov, Yu.A. Global heat kernel bounds via desingularizing weights, J. Funct. Anal., Volume 212 (2004), pp. 373–398 | DOI | MR | Zbl

[33] Murphy, J. The nonlinear Schrödinger equation with an inverse-square potential, Contemp. Math., Volume 725 (2019), pp. 215–225 | DOI | MR

[34] Nakanishi, K. Unique global existence and asymptotic behaviour of solutions for wave equations with non-coercive critical nonlinearity, Commun. Partial Differ. Equ., Volume 24 (1999), pp. 185–221 | DOI | MR | Zbl

[35] Pausader, B.; Tzvetkov, N.; Wang, X. Global regularity for the energy-critical NLS on ${\mathbb{S}}^3$ , Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 31 (2014) no. 2, pp. 315–338 | Numdam | MR | Zbl

[36] Shatah, J.; Struwe, M. Well posedness in the energy space for semilinear wave equations with critical growth, Int. Math. Res. Not., Volume 7 (1994), pp. 303–309 | MR | Zbl

[37] Smets, D. Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Am. Math. Soc., Volume 357 (2005) no. 7, pp. 2909–2938 | DOI | MR | Zbl

[38] Struwe, M. Globally regular solutions to the $u^5$ Klein–Gordon equation, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 15 (1989), pp. 495–513 | Numdam | MR | Zbl

[39] Sterbenz, J.; Tataru, D. Regularity of wave-maps in dimension $2+1$ , Commun. Math. Phys., Volume 298 (2010) no. 1, pp. 231–264 | DOI | MR | Zbl

[40] Tao, T. Global regularity of wave maps III. Large energy from ${\mathbb{R}}^{1+2}$ to hyperbolic spaces (Preprint) | arXiv

[41] Vasquez, J.L.; Zuazua, E. The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., Volume 173 (2000), pp. 103–153 | MR | Zbl

[42] Zhang, J.; Zheng, J. Scattering theory for nonlinear Schrödinger with inverse-square potential, J. Funct. Anal., Volume 267 (2014), pp. 2907–2932 | DOI | MR | Zbl