Abstract
In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in \(H^{s}\) with spatial dimension \(n \le 5\). We show this equation, with power \(2\le p\le 1+4/(n-1)\), is (strongly) ill-posed in \(H^{s}\) with \(s = (n+5)/4\) in general. Moreover, when the nonlinearity is quadratic we establish a characterization of the structure of nonlinear terms in terms of the regularity. As a byproduct, we give an alternative proof of the failure of the local in time endpoint scale-invariant \(L_{t}^{4/(n-1)}L_{x}^{\infty }\) Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
When \(n=1\) and \(1/2<s<1\), with \(C^{\alpha \beta }=cm^{\alpha \beta }\), it is understood that the Eq. (1.6) is satisfied in the divergence form \(m^{\alpha \beta }\partial _\beta (e^{-cu}\partial _\alpha u)=0\).
Let \(w(t,x_{1}) = (\partial _{t}-\partial _{x_{1}})u\), then (3.8) becomes an ordinary differential inequality along characteristics
$$\begin{aligned}\frac{dw(t,t+x_{1})}{dt} = w^{p}, \ w(0,x_{1}) = -2\varepsilon \chi '(x_{1})>0, x_1\in (2^{-j}, 2-2^{-j}-2t).\end{aligned}$$It is easy to obtain for \(x_1\in (2^{-j}, 2-2^{-j}-2t)\)
$$\begin{aligned}&w(t, t+x_{1})= \frac{w(0,x_1)}{(1-(p-1)tw^{p-1}(0,x_1))^{1/(p-1)}} \\&\quad = \frac{2\varepsilon |\chi '(x_{1})|}{(1-(2\varepsilon )^{p-1}(p-1)t|\chi '(x_{1})|^{p-1})^{1/(p-1)}}, \end{aligned}$$as long as \(1-(p-1)tw^{p-1}(0,x_1) >0\).
When \(t+\mu (t)< x_{1} < 2-t\), \(t<1\), we have
$$\begin{aligned}&\partial _{x_{1}}h(t,x_{1})= (2\varepsilon )^{p-1}(p-1)^{2}t\alpha \left( \ln \frac{6}{x_{1}-t}\right) ^{\alpha (p-1)-1}\frac{1}{x_{1}-t} > 0, \\&\partial _{x_{1}}^{2}h(t, x_{1})=-\frac{(2\varepsilon )^{p-1}(p-1)^{2}t}{(x_{1}-t)^{2}}\left( \ln \frac{6}{x_{1}-t}\right) ^{\alpha (p-1)-2} \left( \left( \ln \frac{6}{x_{1}-t}\right) -(1-\alpha (p-1))\right) \le 0. \end{aligned}$$.
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics (Amsterdam), vol. 140. Elsevier/Academic Press, Amsterdam (2003)
Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343. Springer, Heidelberg (2011)
Dinh, V.D.: On the Cauchy problem for the nonlinear semi-relativistic equation in Sobolev spaces. Discrete Contin. Dyn. Syst. 38, 1127–1143 (2018)
Ettinger, B., Lindblad, H.: A sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates. Ann. Math. 2, 311–330 (2017)
Fang, D., Wang, C.: Some remarks on Strichartz estimates for homogeneous wave equation. Nonlinear Anal. 65, 697–706 (2006)
Fang, D., Wang, C.: Local well-posedness and ill-posedness on the equation of type \(\Box u = u^{k}(\partial u)^{\alpha }\). Chin. Ann. Math. 3, 361–378 (2005)
Fang, D., Wang, C.: Almost global existence for some semilinear wave equations with almost critical regularity. Commun. Partial Differ. Equ. 38, 1467–1491 (2013)
Fujiwara, K., Georgiev, V., Ozawa, T.: On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases. J. Math. Pures Appl. 136, 239–256 (2020)
Guo, Z., Li, J., Nakanishi, K., Yan, L.: On the boundary Strichartz estimates for wave and Schrödinger equations. J. Differ. Equ. 265, 5656–5675 (2018)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 2424. Pitman (Advanced Publishing Program), Boston (1985)
Hidano, K., Wang, C.: Fractional Derivatives of Composite Functions and the Cauchy Problem for the Nonlinear Half Wave Equation, Selecta Math. (N.S.), vol. 25, No. 1, Paper No. 2 (2019)
Hidano, K., Jiang, J., Lee, S., Wang, C.: Weighted fractional chain rule and nonlinear wave equations with minimal regularity. Rev. Mat. Iberoam. 36, 341–356 (2020)
Klainerman, S.: Global existence for nonlinear wave equations. Commun. Pure Appl. Math. 33, 43–101 (1980)
Klainerman, S., Machedon, M.: Estimates for null forms and the spaces \(H_{s,\delta }\). Int. Math. Res. Notices 1996(17), 853–865 (1996)
Klainerman, S., Machedon, M.: On the algebraic properties of the \(H_{n/2,1/2}\) spaces. Int. Math. Res. Notices 1998(15), 765–774 (1998)
Klainerman, S., Selberg, S.: Bilinear estimates and applications to nonlinear wave equations. Commun. Contemp. Math. 2, 223–295 (2002)
Lindblad, H.: A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations. Duke Math. J. 72, 503–539 (1993)
Lindblad, H.: Counterexamples to local existence for semi-linear wave equations. Am. J. Math. 118, 1–16 (1996)
Lindblad, H.: Counterexamples to local existence for quasi-linear wave equations. Math. Res. Lett. 5, 605–622 (1998)
Machihara, S., Nakamura, M., Nakanishi, K., Ozawa, T.: Endpoint Strichartz estimates and global solutions for the nonlinear Dirac equation. J. Funct. Anal. 219, 1–20 (2005)
Ponce, G., Sideris, T.C.: Local regularity of nonlinear wave equations in three space dimensions. Commun. Partial Differ. Equ. 18, 169–177 (1993)
Sterbenz, J.: Global regularity and scattering for general non-linear wave equations. II. (4+1) dimensional Yang–Mills equations in the Lorentz gauge. Am. J. Math. 129, 611–664 (2007)
Tao, T.: Nonlinear dispersive equations, local and global analysis, CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society, Providence (2006)
Tataru, D.: On the equation \(\Box u = |\nabla u|^{2}\) in 5+1 dimensions. Math. Res. Lett. 6, 469–485 (1999)
Zhou, Y.: On the equation \(\Box u = |\nabla u|^{2}\) in four space dimensions. Chin. Ann. Math. Ser. 24, 293–302 (2003)
Acknowledgements
The authors were supported in part by NSFC 11971428. The second author would like to thank Professor Gang Xu for helpful discussion on the extension of initial data in low dimensions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by F. H. Lin.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
