## Abstract

We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second-order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial results are given at the origin, where the degeneracy occurs. Some nontrivial obstacles, due to the nonconstant speed of propagation, have to be surmounted.

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## Acknowledgements

The authors would like to thank Professor Mohamed Ali Hamza, for his helpful advices during the preparation of this paper, which greatly improved the presentation of the results. This material is based upon work supported by Tamkeen under the NYU Abu Dhabi Research Institute grant CG002.

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Communicated by Nader Masmoudi.

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## A \(L^2_{loc,u}\) for Radial Functions

### A \(L^2_{loc,u}\) for Radial Functions

Note that we handle only \(L^2\)-type spaces, since the extension to \(H^1\)-type spaces is natural. Consider *u* a radial solution in \(L_{loc,u}^2\) in \({\mathbb {R}}^d\) and introduce \(\tilde{u}\) such that \(u(x)={{\tilde{u}}}(r)\) with \(r=|x|,\, \forall x\in {\mathbb {R}}^d\).

Let \(A=\displaystyle \sup \limits _{x_0\in {\mathbb {R}}^d} \int _{B(x_0,1)}|u(x)|^2 dx \) the square of the \(L_{loc,u}^2\) norm in \({\mathbb {R}}^d\) and \(B=\displaystyle \sup \limits _{r_0\ge 1}\frac{1}{r_0^{d-1}} \int _{r_0-1}^{r_0+1}|{{\tilde{u}}}(r)|^2 r^{d-1}dr .\) We also define for the crown \({\mathcal {C}}(r_0,1)\) by

We aim at proving that the square root of *B* is an equivalent norm to the \(L_{loc,u}^2\) in the radial setting. More precisely, we have the following:

### Lemma A.1

*i*):-
\(\exists {{\bar{\alpha }}} (d)>0\) such that \(A\le {{\bar{\alpha }}} (d) B\).

*ii*):-
\(\exists {{\bar{\beta }}} (d)>0\) such that \(B\le {{\bar{\beta }}} (d) A\).

### Proof

*i*):-
It is enough to show that for any \(x_0\in {\mathbb {R}}^d\),

$$\begin{aligned} \int _{B(0,2)}|u(x)|^2 dx \le {{\bar{\alpha }}} (d) B, \text{ for } \text{ some } {{\bar{\alpha }}} (d) >0 . \end{aligned}$$Consider \(x_0\in {\mathbb {R}}^d\). If \(|x_0|<1\) and \(x\in B(x_0,1)\) then \(|x|<|x_0|+1<2\). Consequently,

$$\begin{aligned} \int _{B(x_0,1)}|u(x)|^2 dx\le \int _{B(0,2)}|u(x)|^2 dx= \omega _{d-1}\int _{0}^2 |{{\tilde{u}}}(r)|^2 r^{d-1}dr \le \omega _{d-1}B, \end{aligned}$$where \(\omega _{d-1}\) is the volume of the sphere \(S^{d-1}\). Now, if \(|x_0|\ge 1\), then we have \( B(x_0,1) \subset {\mathcal {C}}(|x_0|,1) \). Furthermore, for geometric considerations, we know that there exists \(\alpha (d, |x_0|)>0\) such that the crown \({\mathcal {C}}(|x_0|,1)\) contains \(\alpha (d,|x_0|) r_0^{d-1}>0\) disjoint copies of \(B(x_0,1)\), with

$$\begin{aligned} \alpha (d, |x_0|)\equiv \alpha _0 (d) r_0^{d-1} \text{ as } r_0\rightarrow +\infty \text{ for } \text{ some } \alpha _0 (d)>0.\end{aligned}$$(A.1)If we denote by \(x_i \) for \(i\in \{0, \ldots ,\alpha -1\}\) the centers of those balls, then we have

$$\begin{aligned}{} & {} \int _{\bigcup \limits _{i=0}^{\alpha -1} B(x_i,1) }|u(x)|^2 dx\le \int _{{\mathcal {C}}(|x_0|,1)}u(r)^2 r^{d-1}dr \nonumber \\{} & {} = \omega _{d-1}\int _{r_0-1}^{r_0+1} |{{\tilde{u}}}(r)|^2 r^{d-1}dx \le \omega _{d-1} Br_0^{d-1}, \end{aligned}$$(A.2)on the one hand. On the other hand, since the difference between the two crown’s radii is 2 and the balls are of radius 1, it follows that

$$\begin{aligned} |x_i|=|x_0| ,\, \forall i\in \{0, \ldots \alpha -1\} \end{aligned}$$(A.3)Since

*u*is radial and the balls \(B(x_i,1) \) are disjoint, using (A.3) we see that$$\begin{aligned} \int _{\bigcup \limits _{i=0}^{\alpha -1} B(x_i,1) }|u(x)|^2 dx=\alpha (d,r_0)\int _{B(x_0,1) }|u(x)|^2 dx. \end{aligned}$$Combining this with (A.2) and (A.1), we conclude the proof of item

*i*). *ii*):-
Consider \(r_0\ge 1\). From geometric considerations, there exists \(\beta (d,r_0)>0\) such that the crown \({\mathcal {C}}(r_0,1)\) is contained in \(\beta (d, r_0)\) copies of

*B*(0, 1), with$$\begin{aligned} \beta (d, r_0)\equiv \beta _0 (d) r_0^{d-1} \text{ as } r_0\rightarrow +\infty \text{ for } \text{ some } \beta _0 (d)>0. \end{aligned}$$(A.4)Denoting by \(y_i \) for \(i\in \{0, \ldots ,\beta -1\}\) the centers of those balls, we have

$$\begin{aligned} \frac{1}{r_0^{d-1}} \int _{r_0-1}^{r_0+1}|{{\tilde{u}}}(r)|^2 r^{d-1}dr&=\frac{1}{\omega _{d-1}r_0^{d-1}} \int _{{\mathcal {C}}(|x_0|,1)}|u(x)|^2 dx\\&\le \frac{1}{\omega _{d-1}r_0^{d-1}} \sum _{i=0}^{\beta -1} \int _{B(y_i,1) }|u(x)|^2 dx \le \frac{\beta (d,r_0) }{\omega _{d-1}r_0^{d-1}} A. \end{aligned}$$Using (A.4), we conclude the proof of item

*ii*). \(\square \)

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Azaiez, A., Zaag, H. Classification of the Blow-Up Behavior for a Semilinear Wave Equation with Nonconstant Degenerate Coefficients.
*Ann. Henri Poincaré* **24**, 1417–1437 (2023). https://doi.org/10.1007/s00023-022-01247-0

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DOI: https://doi.org/10.1007/s00023-022-01247-0