Concerning ill-posedness for semilinear wave equations

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.

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Notes

  1. 1.

    In fact, we can show \(\Box u = (u_{t}-u_{x_{1}})^{p}\) is strongly ill-posed or \(\Box u=u^{2}_{t}\) (\(n\le 4\)) is ill-posed in \(H^{s_{l}}\), see Theorems 3.4 and 5.1.

  2. 2.

    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 \(m^{\alpha \beta }\partial _\beta (e^{-cu}\partial _\alpha u)=0\).

  3. 3.

    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 \(1-(p-1)tw^{p-1}(0,x_1) >0\).

  4. 4.

    When \(t+\mu (t)< x_{1} < 2-t\), \(t<1\), we have

    $$\begin{aligned}&\partial _{x_{1}}h(t,x_{1})= (2\varepsilon )^{p-1}(p-1)^{2}t\alpha \left( \ln \frac{6}{x_{1}-t}\right) ^{\alpha (p-1)-1}\frac{1}{x_{1}-t} > 0, \\&\partial _{x_{1}}^{2}h(t, x_{1})=-\frac{(2\varepsilon )^{p-1}(p-1)^{2}t}{(x_{1}-t)^{2}}\left( \ln \frac{6}{x_{1}-t}\right) ^{\alpha (p-1)-2} \left( \left( \ln \frac{6}{x_{1}-t}\right) -(1-\alpha (p-1))\right) \le 0. \end{aligned}$$

    .

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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.

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Correspondence to Chengbo Wang.

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Liu, M., Wang, C. Concerning ill-posedness for semilinear wave equations. Calc. Var. 60, 19 (2021). https://doi.org/10.1007/s00526-020-01899-y

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Mathematics Subject Classification

  • 35L05
  • 35L15
  • 35L67
  • 35L71
  • 35B33