Classification of the Blow-Up Behavior for a Semilinear Wave Equation with Nonconstant Degenerate Coefficients

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.

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

$$\begin{aligned} \forall r_0\ge 1,\,{\mathcal {C}}(r_0,1)=\{ x\in {\mathbb {R}}^d, |\,\, r_0-1\le |x| < r_0+1\}. \end{aligned}$$

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). $\square$