article published yes Rowan Killip author Betsy Stovall author Monica Visan author 056daca0-b8d1-11f0-964f-f91054abf8ca We consider both the defocusing and focusing cubic nonlinear Klein–Gordon equations utt − Δu + u ± u3 =0 in two space dimensions for real-valued initial data u(0) ∈ H1/x and ut(0) ∈ L2/x. We show that in the defocusing case, solutions are global and have finite global L4/t,x spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state. These results rely on analogous statements for the two-dimensional cubic nonlinear Schr¨odinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional. It was previously shown by Nakanishi that spacetime bounds for Klein– Gordon equations imply the same for nonlinear Schr¨odinger equations. American Mathematical Society2012 eng 0002-9947 1088-6850 1008.271210.1090/s0002-9947-2011-05536-4 36431571-1631 yes Killip R, Stovall B, Vişan M. 2012. Scattering for the cubic Klein-Gordon equation in two space dimensions. Transactions of the American Mathematical Society. 364(3), 1571–1631. Killip, R., Stovall, B., &#38; Vişan, M. (2012). Scattering for the cubic Klein-Gordon equation in two space dimensions. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">https://doi.org/10.1090/s0002-9947-2011-05536-4</a> Killip, Rowan, et al. “Scattering for the Cubic Klein-Gordon Equation in Two Space Dimensions.” <i>Transactions of the American Mathematical Society</i>, vol. 364, no. 3, American Mathematical Society, 2012, pp. 1571–631, doi:<a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">10.1090/s0002-9947-2011-05536-4</a>. R. Killip, B. Stovall, and M. Vişan, “Scattering for the cubic Klein-Gordon equation in two space dimensions,” <i>Transactions of the American Mathematical Society</i>, vol. 364, no. 3. American Mathematical Society, pp. 1571–1631, 2012. R. Killip, B. Stovall, M. Vişan, Transactions of the American Mathematical Society 364 (2012) 1571–1631. Killip R, Stovall B, Vişan M. Scattering for the cubic Klein-Gordon equation in two space dimensions. <i>Transactions of the American Mathematical Society</i>. 2012;364(3):1571-1631. doi:<a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">10.1090/s0002-9947-2011-05536-4</a> Killip, Rowan, Betsy Stovall, and Monica Vişan. “Scattering for the Cubic Klein-Gordon Equation in Two Space Dimensions.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2012. <a href="https://doi.org/10.1090/s0002-9947-2011-05536-4">https://doi.org/10.1090/s0002-9947-2011-05536-4</a>. 220222026-06-19T07:31:10Z2026-06-22T09:19:43Z