Classification of the blow-up behavior for a semilinear wave equation with nonconstant degenerate coefficients

TL;DR

This paper investigates the blow-up behavior and regularity of solutions for a nonlinear wave equation with nonconstant, degenerate coefficients, addressing challenges posed by variable wave speeds.

Contribution

It provides new insights into blow-up phenomena and regularity for wave equations with degenerate, nonconstant coefficients, including partial results at degeneracy points.

Findings

Characterization of blow-up behavior

Analysis of blow-up set regularity

Partial results at degeneracy points

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.

Equations

$$\left\{\begin{array}[]{l}\displaystyle\partial^{2}_{t}u=a(x)\left(\partial^{2}_{x}u+\frac{N-1}{x}\partial_{x}u\right)+b(x)|u|^{p-1}u+f(u)+g(x,t,\partial_{x}u,\partial_{t}u),\\ \partial_{x}u(x,t)\sqrt{a(x)}\rightarrow 0\mbox{ at }x=0,\\ u(0)=u_{0}\mbox{ and }u_{t}(0)=u_{1},\end{array}\right.$$

$$\;\;\;\left\{\begin{array}[]{l}\;a(x)>0,b(x)>0,\\ \;\int_{0}^{1}\frac{dx}{\sqrt{a(x)}}<+\infty,\\ \;\;|(N-1)\frac{\sqrt{a(x)}}{x}-\frac{1}{2}\frac{a^{\prime}(x)}{\sqrt{a(x)}}-\frac{d-1}{\phi(x)}|\leq M,\end{array}\right.$$

$$d\in {\mathbb{N}}^{\ast },$$

$$\varphi (x)={\int }_0^x\frac{dy}{\sqrt{a(y)}}.$$

$$p>1\text{mbox}andp<\frac{d+3}{d-1}\text{mbox}ifd\ge 2.$$

$$\left\{\begin{array}[]{l}|f(u)|\leq M(1+|u|^{q}),\mbox{ for all }u\in\mathbb{R}\mbox{ with }(q<p,M>0),\\ |g(x,t,v,z)|\leq M(1+|v|\sqrt{a(x)}+|z|),\mbox{ for all }x,t,v,z\in\mathbb{R}.\end{array}\right.$$

$$a(x)=|x{|}^{\alpha }\text{mbox}with\alpha <2.$$

$$H_0=\{v\in L_{loc}^2({\mathbb{R}}^{+})|V\in L_{loc,u,rad}^2({\mathbb{R}}^{+})\},$$

$$H_1=\{v\in L_{loc}^2({\mathbb{R}}^{+})|V\in H_{loc,u,rad}^1({\mathbb{R}}^{+})\},$$

$$V(X)=v(x),X=\varphi (x),$$

$$L_{loc,u,rad}^2({\mathbb{R}}^{+})=\{V\in L_{loc}^2({\mathbb{R}}^{+})|\underset{r_0\ge 1}{\sup }\frac{1}{r_0^{d-1}}{\int }_{r_0-1}^{r_0+1}v(r{)}^2{r}^{d-1}dr<+\infty \},$$

$$H_{loc,u,rad}^1({\mathbb{R}}^{+})=\{V\in L_{loc,u,rad}^2({\mathbb{R}}^{+})|{\partial }_{r}V\in L_{loc,u,rad}^2({\mathbb{R}}^{+})\}.$$

$$||v|{|}_{L_{loc,u}^2}^2=\underset{a\in {\mathbb{R}}^d}{\sup }{\int }_{|x-a|<1}|v(x){|}^{2}dx\text{mbox}and||v|{|}_{H_{loc,u}^1}^2=||v|{|}_{L_{loc,u}^2}^2+||\nabla v|{|}_{L_{loc,u}^2}^2,$$

$${\partial }_t^{2}u={\partial }_x^{2}u+|u{|}^{p-1}u$$

$$U(X,t)=u(x,t),X=\varphi (x)$$

$${\partial }_t^{2}U={\partial }_X^{2}U+\left((N-1)\frac{\sqrt{a(x)}}{x}-\frac{1}{2}\frac{a^{\prime }(x)}{\sqrt{a(x)}}\right){\partial }_{X}U+\beta (X)|U{|}^{p-1}U+f(U)+g(x,t,\frac{{\partial }_{X}U}{\sqrt{a(x)}},{\partial }_{t}U)$$

$${\partial }_t^{2}U={\partial }_X^{2}U+\frac{d-1}{X}{\partial }_{X}U+\beta (X)|U{|}^{p-1}U+f(U)+G(X,t,{\partial }_{X}U,{\partial }_{t}U)$$

$$G(X,t,{\partial }_{X}U,{\partial }_{t}U)=g(x,t,\frac{{\partial }_{x}u}{\sqrt{a(x)}},{\partial }_{t}u)+\left((N-1)\frac{\sqrt{a(x)}}{x}-\frac{1}{2}\frac{a^{\prime }(x)}{\sqrt{a(x)}}-\frac{d-1}{X}\right){\partial }_{X}U.$$

$$|f(U)|$$

$$|G(X,t,{\partial }_{X}U,{\partial }_{t}U)|$$

$${\partial }_{X}U(0,t)=0$$

$$D_U=\{(X,t);0\le t<T_U(X)\},$$

$$D_u=\{(x,t)|0\le t<T(x)\},$$

$$T(x)=T_U(\varphi (x)).$$

$${\mathcal{C}}_{x,t,\delta }=\{(\xi ,\tau )\ne (x,t)|0\le \tau \le t-\delta |\varphi (\xi )-\varphi (x)|\},$$

$$\exists \delta =\delta (x_0)\in (0,1)\text{mbox}suchthatu\text{mbox}isdefinedon{\mathcal{C}}_{x_0,T(x_0),{\delta }_0}.$$

$${\partial }_t^{2}u=a(x){\partial }_x^{2}u.$$

$$w_{x_0}(y,s)=(T(x_0)-t{)}^{\frac{2}{p-1}}u(x,t),y=\frac{\varphi (x)-\varphi (x_0)}{T(x_0)-t},s=-\log (T(x_0)-t).$$

$${\partial }_s^{2}w$$

$$\kappa (\hat{d},y)={\kappa }_0\frac{(1-{\hat{d}}^2{)}^{\frac{1}{p-1}}}{(1+\hat{d}y{)}^{\frac{2}{p-1}}}\text{mbox}with{\kappa }_0={\left(\frac{2(p+1)}{(p-1{)}^2}\right)}^{\frac{1}{p-1}},(\hat{d},y)\in (-1,1{)}^2.$$

$${\bar{\xi }}_i(s)=\left(i-\frac{k+1}{2}\right)\frac{p-1}{2}\log s+{\bar{\alpha }}_i(p,k)$$

Full text

1. 1 Introduction

2. 1.1 Blow-up results

3. 1.1.1 Behavior outside the origin

4. 1.1.2 Behavior at the origin

5. 1.2 Strategy of the proof of the results

6. 2 Proof of the results

7. 2.1 A Lyapunov functional in similarity variables outside the origin

8. 2.2 Blow-up results outside the origin

9. 2.3 Blow-up results at the origin

10. A $\displaystyle L^{2}_{loc,u}$ for radial functions

1 Introduction

We consider the following nonlinear wave equation with nonconstant coefficients in the radial case :

[TABLE]

where $\displaystyle u(t):x\in\mathbb{R}^{+}\to u(x,t)\in\mathbb{R}$, and $\displaystyle N$ is the dimension of the physical space.

We assume that $\displaystyle a\in C^{1}(\mathbb{R}_{+}^{*})$ and $\displaystyle b\in C^{1}(\mathbb{R}_{+})$ satisfy the following conditions for all $\displaystyle x>0$,

[TABLE]

for some $\displaystyle\epsilon_{0}>0$, $\displaystyle M>0$ and

[TABLE]

where $\displaystyle\phi$ is defined by

[TABLE]

Note that the third estimate in (1.2) is useful only near the origin. However, introducing such a change will induce a lot of pure technical and trivial complications in the proof. For that reason, we don’t make this change, and leave it to the interested reader to check by himself that it works straightforwardly.

The exponent $\displaystyle p$ is superlinear and subcritical (in relation to $\displaystyle d$) , in the sense that

[TABLE]

Conditions (1.3) and (1.5) will prove to be meaningful after a change of variables we perform below in (1.11).

We assume in addition that $\displaystyle f$ and $\displaystyle g$ are $\displaystyle C^{1}$ functions, where $\displaystyle f:\mathbb{R}\to\mathbb{R}$ and $\displaystyle g:\mathbb{R}^{4}\to\mathbb{R}$ satisfy

[TABLE]

A typical example that satisfies (1.2) and which will be discussed in this paper is the following :

[TABLE]

The example (1.7) shows a degeneracy at $\displaystyle x=0$ when $\displaystyle\alpha\neq 0$. Note that for $\displaystyle\alpha<0$, the wave speed goes to infinity and for $\displaystyle\alpha\in(0,2)$ it goes to zero. For this case, conditions (1.2), (1.3) and (1.6) are fulfilled for $\displaystyle N\geq 2$, $\displaystyle|g(x,t,v,z)|\leq M(1+|v||x|^{\frac{\alpha}{2}}+|z|)$, and $\displaystyle d=\frac{2(N-\alpha)}{2-\alpha}$.

Note in particular that example (1.7) is not relevant for $\displaystyle N=1$, since the third condition of (1.2) is never fulfilled, for any $\displaystyle\alpha<2$. On the contrary, when $\displaystyle N=2$, example (1.7) is valid for any value $\displaystyle\alpha<2$, and we always have $\displaystyle d=2$. When $\displaystyle N\geq 3$, we need to take $\displaystyle\alpha$ of the form $\displaystyle\alpha_{k}=\frac{2(k-N)}{k-2}$ where $\displaystyle k\geq 3$ is an integer, and in this case, $\displaystyle d=k$.

Initial data $\displaystyle(u_{0},u_{1})$ will be considered in the space $\displaystyle H_{1}\times H_{0}$ defined by

[TABLE]

[TABLE]

where

[TABLE]

$\displaystyle\phi$ was given in (1.4),

[TABLE]

and

[TABLE]

We recall the spaces $\displaystyle L_{loc,u}^{2}(\mathbb{R}^{d})$ and $\displaystyle H_{loc,u}^{1}(\mathbb{R}^{d})$ introduced by Antonini and Merle in [2] by the following norms :

[TABLE]

we show in Appendix A that the spaces $\displaystyle L^{2}_{loc,u,rad}$ and $\displaystyle H^{1}_{loc,u,rad}$ are simply the radial versions of the $\displaystyle L^{2}_{loc,u}$ and $\displaystyle H^{1}_{loc,u}$ spaces.

Equation (1.1) corresponds to physical situations where the wave propagates in non-homogeneous media (see for example [15]). It appears in models of traveling waves in a non-homogeneous gas with damping that changes with the position. The unknown $\displaystyle u$ denotes the displacement, the coefficient $\displaystyle a$, called the bulk modulus, accounts for changes of the temperature depending on the location.

When $\displaystyle a(x)\equiv 1$, this equation was considered by Hamza and Zaag in [5] (see also [4] for some related results). Basically, the authors showed that the results previously proved by Merle and Zaag in [10], [11], [13] and [14] for the unperturbed semilinear wave equation

[TABLE]

do extend to the perturbed case. We also mention the work of Alexakis and Shao [1] who study the energy concentration in backward light cones near blow-up points.

In this paper, we want to explore the case where $\displaystyle a(x)\not\equiv 1$. When $\displaystyle a$ is space dependent, we find that although the blow-up results of [5] remain valid, some nontrivial obstacles have to be surmounted, in particular, at the origin where the degeneracy may occur (see for instance the typical example (1.7)). Since the problem does not have a constant speed of propagation, we have to apply an appropriate transformation to obtain the desired estimates. In fact, we remark that we can reduce to the case $\displaystyle a(x)\equiv 1$ thanks to the following change of variables:

[TABLE]

where $\displaystyle\phi$ is given in (1.4).

Applying this transformation to (1.1), we see that $\displaystyle U$ satisfies:

[TABLE]

where $\displaystyle\beta(X)=b(x)$ and $\displaystyle U(t):X\in\mathbb{R}^{+}\to U(X,t)\in\mathbb{R}$.

We rewrite this equation as follows

[TABLE]

with

[TABLE]

We see from (1.2) and (1.6) that we have

[TABLE]

Note that we have,

[TABLE]

thanks to the condition on the space derivative in (1.1).

As for the Cauchy problem for equation (1.1), we remark that thanks to the change of variables (1.11), we reduce to the formalism of Hamza and Zaag in [6]. Indeed, recalling that $\displaystyle(u_{0},u_{1})\in H_{0}\times H_{1}$, we derive by definition that $\displaystyle(U(X,0),\partial_{t}U(X,0))\in L^{2}_{loc,u,rad}\times H^{1}_{loc,u,rad}$ defined in (1.8) and (1.9).

Therefore, as mentioned in [6] we use a modification of the argument by Georgiev and Todorova [16] to derive a solution $\displaystyle(U,\partial_{t}U)\in C([0,T_{0}),H^{1}_{loc,u,rad}\times L^{2}_{loc,u,rad})$ for some $\displaystyle T_{0}>0$. Thanks to the finite speed of propagation, we extend the definition of $\displaystyle U(X,t)$ to the following domain

[TABLE]

for some $\displaystyle 1-$Lipschitz function $\displaystyle T_{U}$.

Going back to problem (1.1), we see that we have a unique solution $\displaystyle(u,\partial_{t}u)\in C([0,T_{0}),H_{0}\times H_{1})$ which is defined on a larger domain

[TABLE]

where

[TABLE]

Since $\displaystyle T^{\prime}(x)=\frac{T^{\prime}_{U}(\phi(x))}{\sqrt{a(x)}}$, it follows that $\displaystyle T$ is a Lipschitz function, with $\displaystyle\frac{1}{\sqrt{a(x)}}$ as local Lipschitz constant for $\displaystyle x\neq 0$. Note that $\displaystyle T(x)$ and $\displaystyle\Gamma$ will be referred as the blow-up time and the blow-up curve in the following.

Proceeding as in the case $\displaystyle a(x)\equiv 1$, we introduce the following non-degeneracy condition for $\displaystyle\Gamma$. If we introduce for all $\displaystyle x\in\mathbb{R},$ $\displaystyle t\leq T(x)$ and $\displaystyle\delta>0$, the generalized cone

[TABLE]

then our non-degeneracy condition is the following: $\displaystyle x_{0}$ is a non-characteristic point if

[TABLE]

If condition (1.17) is not true, then we call $\displaystyle x_{0}$ a characteristic point.

We denote by $\displaystyle\mathcal{R}$ the set of non characteristic points and $\displaystyle\mathcal{S}$ the set of characteristic points.

Note that the set $\displaystyle\mathcal{C}_{x,t,\delta}$ defined in (1.16) is a cone in the variables $\displaystyle(X,t)$ (1.11). In the $\displaystyle(x,t)$ variables, its boundary is given by the characteristics associated to the linear problem

[TABLE]

In order to state our results, we will use similarity variables associated to $\displaystyle U(X,t)$ defined in (1.11), and which turn out to be a nonlinear version of the standard similarity variables, when related directly to $\displaystyle u(x,t)$:

[TABLE]

Applying this transformation to (1.12), we see that $\displaystyle w_{x_{0}}(y,s)$ satisfies the following equation for all $\displaystyle|y|<1$ and $\displaystyle s\geq-\log T(x_{0})$ :

[TABLE]

Let us introduce the solitons

[TABLE]

We also introduce

[TABLE]

where the sequence $\displaystyle(\bar{\alpha}_{i})_{i=1,...,k}$ is uniquely determined by the fact that $\displaystyle(\bar{\xi}_{i}(s))_{i=1,...,k}$ is an explicit solution with zero center of mass for this ODE system:

[TABLE]

where $\displaystyle\xi_{0}(s)\equiv\xi_{k+1}(s)\equiv 0$ and $\displaystyle c_{1}=c_{1}(p)>0$ appeared for the first time in Proposition $\displaystyle 3.2$ page $\displaystyle 590$ of Merle and Zaag [13].

1.1 Blow-up results

We dissociate two cases in this subsection. In fact, equation (1) has a different structure according to the position of $\displaystyle x_{0}$.

1.1.1 Behavior outside the origin

When $\displaystyle x_{0}\neq 0$, by (1.4) we have $\displaystyle\phi(x_{0})\neq 0$, hence the term $\displaystyle\frac{e^{-s}}{\phi(x_{0})+ye^{-s}}\partial_{y}w_{x_{0}}$ in (1) is a lower order term bounded by $\displaystyle\frac{2e^{-s}}{|\phi(x_{0})|}|\partial_{y}w_{x_{0}}|$ for $\displaystyle s$ large and will be treated as a perturbation, as in Hamza and Zaag [5].

Accordingly, we may write the second and first order space derivatives in equation (1) in the following divergence form:

[TABLE]

where $\displaystyle\rho(y)=(1-y^{2})^{\frac{2}{p-1}}$ exactly as in the one dimensional case of the standard semilinear wave equation (1.10).

We recall that for the unperturbed case (ignoring line 2 and 3 in (1)), the Lyapunov functional is given by

[TABLE]

where $\displaystyle(w,\partial_{s}w)\in H^{1}_{\rho}\times L^{2}_{\rho}$, with

[TABLE]

and

[TABLE]

We see that $\displaystyle E$ is well defined from the fact that the three first terms of its expression in (1.21) are in $\displaystyle L^{1}_{\rho}$ ; for the last term we need to use the Hardy-Sobolev inequality given by Merle and Zaag in Appendix B page $\displaystyle 1163$ of [8]:

[TABLE]

Now, if $\displaystyle u$ is a solution of (1), with blow-up surface $\displaystyle\Gamma\,:\,\{x\rightarrow T(x)\},$ and if $\displaystyle x_{0}\neq 0$, then we have the following:

Theorem 1**.**

(Bound in similarity variables outside the origin)**

$\displaystyle i)$(Non-characteristic case):

If $\displaystyle x_{0}\neq 0$ is a non-characteristic point, then, for all $\displaystyle s$ large enough:

[TABLE]

$\displaystyle ii)$(Characteristic case)*:

If $\displaystyle x_{0}\neq 0$ is a characteristic point, then, for all $\displaystyle s$ large enough:*

[TABLE]

Using the bound in Theorem 1, together with the compactness procedure based on the existence of a Lyapunov for equation (1) (which is a perturbation of the functional $\displaystyle E(w,\partial_{s}w)$ defined in (1.21)), we derive the following:

Theorem 2**.**

(Blow-up behavior in similarity variables outside the origin)*

$\displaystyle i)$ (Non-characteristic case) The set $\displaystyle\mathcal{R}\cap\mathbb{R}_{+}^{*}$ is open, and $\displaystyle T$ is of class $\displaystyle C^{1}$ on that set. Moreover, there exist $\displaystyle\mu_{0}>0$ and $\displaystyle C_{0}>0$ such that for all $\displaystyle x_{0}\in\mathcal{R}\cap\mathbb{R}_{+}^{*}$, there exist $\displaystyle\theta(x_{0})=\pm 1$ and $\displaystyle s_{0}(x_{0})\geq-\log T(x_{0})$ such that for all $\displaystyle s\geq s_{0}$:*

[TABLE]

*where $\displaystyle w=w_{x_{0}}$. Moreover, $\displaystyle E(w,\partial_{s}w)\rightarrow E(\kappa_{0},0)$ as $\displaystyle s\rightarrow\infty.$

$\displaystyle ii)$ (Characteristic case) If $\displaystyle x_{0}\in\mathcal{S}\cap\mathbb{R}_{+}^{*}$, there is $\displaystyle\xi_{0}(x_{0})\in\mathbb{R}$ such that:*

[TABLE]

where $\displaystyle w=w_{x_{0}}$ and $\displaystyle E(w,\partial_{s}w)\rightarrow k(x_{0})E(\kappa_{0},0)$ as $\displaystyle s\rightarrow\infty$, for some $\displaystyle k(x_{0})\geq 2$, $\displaystyle\theta_{i}=\theta_{1}(-1)^{i+1}$, $\displaystyle\theta_{1}=\pm 1$, and continuous $\displaystyle\hat{d}_{i}(s)=-\tan\hat{\xi}_{i}(s)$ with

[TABLE]

where $\displaystyle\bar{\xi}_{i}(s)$ is introduced in (1.20).

Remark: Estimate (1.24) holds in $\displaystyle H^{1}\times L^{2}(-1,1)$, thanks to the covering argument introduced by Merle and Zaag in [9]. From the Sobolev embedding, it holds also in $\displaystyle L^{\infty}\times L^{2}$.

Remark: Following the strategy of Côte and Zaag in [3], refined in [7] by Hamza and Zaag, for every $\displaystyle k_{0}\geq 2$ and $\displaystyle\xi_{0}\in\mathbb{R}$, we are able to construct examples of solutions to equation (1.1) showing a characteristic-point and obeying the modality described in item $\displaystyle ii)$ of Theorem 2.

Going back to $\displaystyle u(x,t)$ thanks to (1.18), we have the following corollary:

Corollary 3**.**

**(Blow-up profile for equation (1.1) in the non-characteristic case outside the origin)  

**If $\displaystyle\mathcal{R}\cap\mathbb{R}_{+}^{*}$, then we have

[TABLE]

uniformly for $\displaystyle x$ such that $\displaystyle|\phi(x)-\phi(x_{0})|<T(x_{0})-t.$

We also obtain the regularity of the blow-up set:

Proposition 4**.**

**(Regularity of the blow-up set outside the origin) 

---

$\displaystyle i)$ (Non-characteristic case) It holds that $\displaystyle\mathcal{R}\neq\emptyset$, $\displaystyle\mathcal{R}\backslash\{0\}$ is an open set, and $\displaystyle x\mapsto T(x)$ is of class $\displaystyle C^{1}$ on $\displaystyle\mathcal{R}\backslash\{0\}$ and for all $\displaystyle x\in\mathcal{R}\backslash\{0\}$, $\displaystyle|T^{\prime}(x)|<\frac{1}{\sqrt{a(x)}}$.

$\displaystyle ii)$ (Characteristic case) Any $\displaystyle x_{0}\in\mathcal{S}\backslash\{0\}$ is isolated. In addition, if $\displaystyle x_{0}\in\mathcal{S}\backslash\{0\}$ with $\displaystyle k(x_{0})$ solitons and $\displaystyle\xi_{0}(x_{0})\in\mathbb{R}$ as center of mass of the solitons’ center as shown in (1.25) and (1.26), then*

[TABLE]

as $\displaystyle x\rightarrow x_{0}$, where $\displaystyle\theta(x)=\frac{x-x_{0}}{|x-x_{0}|}$ and $\displaystyle\nu=\nu(p)>0$.

Remark: If $\displaystyle a$ is Holder continuous, then we may replace $\displaystyle\frac{\theta(x)}{\sqrt{a(x)}}$ by $\displaystyle\frac{\theta(x)}{\sqrt{a(x_{0})}}$ in (1.27), and replace (1.28) by

[TABLE]

1.1.2 Behavior at the origin

When $\displaystyle x_{0}=0$, we have $\displaystyle\phi(x_{0})=0$, hence the term $\displaystyle\frac{e^{-s}(d-1)}{\phi(x_{0})+ye^{-s}}\partial_{y}w_{{}_{0}}=\frac{d-1}{y}\partial_{y}w_{{}_{0}}$ in equation (1) and can no longer be treated as a perturbation.

Accordingly, we may write the second and first order space derivatives in the following divergence form:

[TABLE]

where

[TABLE]

For the case where $\displaystyle(f,g)\equiv(0,0)$, in one space dimension, we introduce the functional

[TABLE]

Note first that $\displaystyle E_{{}_{0}}$ is defined if $\displaystyle(w,\partial_{s}w)\in H^{1}_{\rho_{{}_{0}}}\times L^{2}_{\rho_{{}_{0}}}$, where the norms $\displaystyle L^{2}_{\rho_{{}_{0}}}$ and $\displaystyle H^{1}_{\rho_{{}_{0}}}$ are defined by the same way as in (1.22) and (1.23), but only on the domain $\displaystyle(0,1)$ and with weight $\displaystyle\rho_{{}_{0}}$ given in (1.30).

Adapting the techniques introduced by Antonini and Merle (See Section $\displaystyle 2$ page $\displaystyle 1144$ in [2]) to our case where $\displaystyle\rho_{{}_{0}}$ is given by (1.30), we see that

[TABLE]

as $\displaystyle d$ satisfies (1.3)-(1.5), $\displaystyle E_{{}_{0}}$ (1.31) is decreasing and is a Lyapunov functional. Another way to justify this: the functional in (1.31) is simply the radial version of the functional of [2] considered in the space $\displaystyle\mathbb{R}^{d}$ (which is not the physical space $\displaystyle\mathbb{R}^{N}$).

Considering $\displaystyle w(y,s)$ as a (non necessarily radial) function defined in $\displaystyle\mathbb{R}^{d}$, we may use the perturbative techniques of Hamza and Zaag in [5] to derive the following:

Theorem 5**.**

(Bound in similarity variables at the origin in the non-characteristic case)* If $\displaystyle u$ is a solution of (1) with blow-up surface $\displaystyle\Gamma\,:\,\{x\rightarrow T(x)\},$ and if $\displaystyle 0$ is a non-characteristic point, then, for $\displaystyle s$ large enough:*

[TABLE]

[TABLE]

1.2 Strategy of the proof of the results

Thanks to the transformation (1.11), we reduce to the case where $\displaystyle a(x)\equiv 1$ in the remaining part of the paper. In comparison with the paper by Hamza and Zaag [6], our equation allows a non-constant term in front of the reaction-term $\displaystyle|u|^{p-1}u$, namely $\displaystyle\beta(x)\not\equiv 1$. As in [6], the most delicate point is to obtain a Lyapunov functional in similarity variables defined in (1.18). Thus, in the following section, we focus on the proof of the existence of a Lyapunov functional for equation (1) in the first subsection, then we give some hints on how to adapt the strategy of [6] to derive the blow-up behavior outside and at the origin in the second and third subsections.

**Acknowledgments

**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.

2 Proof of the results

We prove the blow-up results for (1.12) which we recall in the following:

[TABLE]

with

[TABLE]

In fact, this is almost the same equation as in [6] except for the coefficient $\displaystyle\beta(X)$ in front of $\displaystyle|U|^{p-1}U$ which was taken identically equal to $\displaystyle 1$ in [6]. For that reason, we follow the strategy of [6] and focus mainly on the treatment of the term $\displaystyle\beta(X)|U|^{p-1}U$. Given some $\displaystyle X_{0}=\phi(x_{0})\in\mathbb{R}_{+},$ where $\displaystyle\phi$ was defined in (1.4), we introduce the following self-similar change of variables, as in (1.18):

[TABLE]

Note that the curve $\displaystyle T_{U}$ of $\displaystyle U$ is given by the curve $\displaystyle T$ of $\displaystyle u$, in fact :

[TABLE]

This change of variables transforms the backward light cone with vertex $\displaystyle(X_{0},T_{U}(X_{0}))$ into the infinite cylinder $\displaystyle(y,s)\in(-1,1)\times[-\log T_{U}(X_{0}),+\infty).$ The function $\displaystyle w_{X_{0}}$ (we write $\displaystyle w$ for simplicity) satisfies the following equation for all $\displaystyle|y|<1$ and $\displaystyle s\geq-\log T_{U}(X_{0})$:

[TABLE]

In the whole paper, we use the notation

[TABLE]

2.1 A Lyapunov functional in similarity variables outside the origin

In this subsection we prove the existence of a Lyapunov functional and the novelty lays in the new coefficient $\displaystyle\beta(X)\not\equiv 1$. We recall that for the case $\displaystyle(f,G)\equiv(0,0)$ with a constant $\displaystyle\beta$, the Lyapunov functional in one space dimension is

[TABLE]

In order to find a Lyapunov functional for our equation (2), we introduce

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

with

[TABLE]

Then, we claim the following:

Proposition 2.1**.**

**(Energy estimates outside the origin)

$\displaystyle(i)$*** There exist $\displaystyle C=C(p,M)>0$ and $\displaystyle S_{0}\in\mathbb{R}$ such that for all $\displaystyle X_{0}>0$ and for all $\displaystyle s\geq\max(-\log T_{U}(X_{0}),S_{0},-4\log X_{0},-\log\frac{X_{0}}{2})$,*

[TABLE]

$\displaystyle(ii)$* There exists $\displaystyle S_{1}(p,N,M,q)\in\mathbb{R}$ such that, for all $\displaystyle s\geq\max(s_{0},S_{1})$, we have $\displaystyle H(w(s),s)\geq 0$.*

Remark: From $\displaystyle(i)$, we see that $\displaystyle H$ given by

[TABLE]

is a Lyapunov functional for equation (2).

Proof of Proposition 2.1.

In this proof we use the notation, $\displaystyle x_{+}=\max(0,x)$.

$\displaystyle(i)$ We proceed like Hamza and Zaag in [6] (See page $\displaystyle 1092$) and we deal with the new term coming from (2). For that reason, we give the equations, recall the estimates already proved in [6] and focus only on the new term.

We multiply equation (2) by $\displaystyle\partial_{s}w\rho$ and integrate for $\displaystyle y\in(-1,1)$, using (2.7) and (2.8), we have for $\displaystyle X=X_{0}+ye^{-s}$ :

[TABLE]

The terms $\displaystyle I_{1}$, $\displaystyle I_{2}$ and $\displaystyle I_{3}$ can be controlled exactly as in page $\displaystyle 1092$ in [6]. For $\displaystyle I_{5}$, comparing to the previous work, we see that $\displaystyle G$ involves new terms, but as it satisfies condition (1.14), we can also adapt the study of Hamza and Zaag in [6] to get :

[TABLE]

For the new term $\displaystyle I_{4}$, we use the fact that $\displaystyle\beta$ is of class $\displaystyle C^{1}$, we get:

[TABLE]

Using (2.1), (2.12), (2.13), (2.14) and (2.15), we have

[TABLE]

Now, we consider $\displaystyle(K(w(s),s))$ (2.9). Using equation (2) and integration by parts, we write:

[TABLE]

Using (2.7) and (2.8), we get:

[TABLE]

Note that all the terms $\displaystyle K_{1}$, $\displaystyle K_{2}$, $\displaystyle K_{3}$, $\displaystyle K_{4}$, $\displaystyle K_{5}$, $\displaystyle K_{6}$ and $\displaystyle K_{7}$ have been studied in [6] (for details see page $\displaystyle 1094$ in [6]). For the reader’s convenience, we recall the following estimates:

[TABLE]

For the new term $\displaystyle K_{8}$, using the fact that $\displaystyle\beta$ is of class $\displaystyle C^{1}$ we see that:

[TABLE]

By the same way, using the fact that $\displaystyle\beta$ is of class $\displaystyle C^{1}$, we prove that $\displaystyle J$ (2.8) satisfies:

[TABLE]

Using the definitions of $\displaystyle I$ (2.7), $\displaystyle F$(2.4) and the condition (1.6) we see that:

[TABLE]

Using (2.17)-(2.1) and the definition of $\displaystyle\gamma$ (2.10) we deduce that

[TABLE]

Using the definition (2.6) of $\displaystyle E$, (2.16), (2.1) we get (remember from (2.10) that $\displaystyle\gamma\leq\frac{1}{2}$)

[TABLE]

Then, for $\displaystyle S_{0}$ well chosen large enough so that $\displaystyle s\geq\max(-\log T(X_{0}),S_{0},-4\log X_{0},-\log\frac{X_{0}}{2})$, we write

[TABLE]

This yields item $\displaystyle(i)$ of Proposition 2.1.

$\displaystyle(ii)$ This follows from the blow-up criterion proved by Antonini and Merle in [2]. In fact, we need to follow the perturbative argument of Hamza and Zaag [6]. As in [6], it is easy to prove the following identity for large $\displaystyle s$ and for any $\displaystyle w\in\mathcal{H}$:

[TABLE]

For more details see $\displaystyle ii)$ page $\displaystyle 1096$ in [6] and see page $\displaystyle 1147$ in [2]. This concludes the proof of Proposition 2.1. ∎

2.2 Blow-up results outside the origin

In this subsection, we give the main ideas of the proofs of our blow-up results outside the origin (Theorem 1, Theorem 2, Corollary 3 and Proposition 4). However, we will not give the details. In fact, thanks to the transformation (1.11), we obtain the equation (1.12) which is almost the same equation already studied by Hamza and Zaag in [6]. In addition, in Subsection 2.1, we see that a Lyapunov functional is available (see the remark following Proposition 2.1), so with this informations, the reader can easily see that the strategy adapted in [6] from the strategy developed by Merle and Zaag in [8], [9], [10], [11], [13] and [14] together with Côte and Zaag [3] holds with very minor adaptations (See also [12]). For that reason, we will sketch the main steps in the following and explicit only the delicate estimates :

• In Step 1, we show that the solution is bounded in self-similar variables in the energy norm. In particular, we will prove Theorem 1.

• In Step 2, we find the asymptotic behavior and derive the regularity of the blow-up curve. In particular, we will prove Theorem 2, Corollary 3 and Proposition 4.

Step 1 : Boundedness of the solution in similarity variables

We derive with no difficulty the following:

Proposition 2.2**.**

For all $\displaystyle X_{0}>0$, there is a $\displaystyle C_{2}(X_{0})>0$ and $\displaystyle S_{2}(X_{0})\in\mathbb{R}$ such that for all $\displaystyle X\in[\frac{X_{0}}{2},\frac{3X_{0}}{2}]$ and $\displaystyle s\geq S_{2}(X_{0})$,

[TABLE]

Proof of Proposition 2.2.

The adaptation by Hamza and Zaag in page 1091 in [6] to the perturbed case works in our case ($\displaystyle\beta(X_{0}+ye^{-s})\not\equiv 1$) with no difficulty. As in [6], the adaptation is straightforward from [8] and Proposition $\displaystyle 3.5$ page $\displaystyle 66$ in [10]. ∎

Proof of Theorem 1.

$\displaystyle i)$ Consider $\displaystyle x_{0}>0$ and $\displaystyle X_{0}=\phi(x_{0})>0$. Let us start with the upper bound. Since $\displaystyle b$ is continuous, $\displaystyle\beta$ is continuous too and as $\displaystyle\beta(X_{0}+ye^{-s})>0$ for $\displaystyle y\in(-1,1)$ and $\displaystyle s$ large enough., we derive from Proposition 2.2 that

[TABLE]

Using the covering method of Proposition 3.4 in [9], we recover the desired upper bound. As for the lower bound, it follows exactly as in the unperturbed case in Lemma 3.1 in [9], simply because equation (1.12) is well posed in $\displaystyle H^{1}\times L^{2}$, from [16] as we explained in the introduction.

$\displaystyle ii)$ It is a direct consequence of Proposition 2.2. ∎

Step 2 : Dynamics of the solution and properties of the blow-up curve

We recall form the definition of $\displaystyle T$ (1.15) that

[TABLE]

Proof of Theorem 2.

i) Non-characteristic case : As we said before, our equation (1.12) is the same as in [6], except for the coefficient $\displaystyle\beta(X)\not\equiv 1$. Thanks to the Proposition 2.2 above, the adaptation of Hamza and Zaag of the analysis of [10] and [11] works ; in particular, this is the case for Theorem $\displaystyle 1$ page $\displaystyle 1097$ which gives the following profile of $\displaystyle w$

[TABLE]

where $\displaystyle X_{0}=\phi(x_{0})$, in $\displaystyle H^{1}(-1,1)$. Using (2.28), we get our statement.

ii) Characteristic case : The approach of Côte and Zaag in [3], adapted to the perturbed case by Hamza and Zaag in [6] stays valid in our case (for more details see page 1105 in [6]). ∎

Proof of Corollary 3.

Applying the transformation (1.18) to the profile given in (2.29) and using the sobolev embedding, we see that for $\displaystyle X_{0}\in\mathcal{R}\cap\mathbb{R}_{+}^{*}$, we have

[TABLE]

uniformly for $\displaystyle X$ such that $\displaystyle|X-X_{0}|<T_{U}(X_{0})-t$. Applying (2.28), we get the result. ∎

Proof of Proposition 4.

$\displaystyle i)$ We can easily see that the strategy developed in the non-perturbed case in [11], and then adapted to the perturbed case in [6] works in the present case ($\displaystyle\beta(X)\not\equiv 1$), with minor adaptations.

$\displaystyle ii)$ Let $\displaystyle x_{0}\in\mathcal{S}\backslash\{0\}$ with $\displaystyle k(x_{0})$ solitons and $\displaystyle\xi_{0}(x_{0})\in\mathbb{R}$ as center of mass of the solitons as shown in (1.25) and (1.26). Proceeding as in the adaptation by Hamza and Zaag in Theorem 5 in [6] to the perturbed case (see also Theorem 1 and 2 in [14], where this statement was proved with no perturbation), so we get

[TABLE]

as $\displaystyle X\rightarrow X_{0}$, where $\displaystyle\theta_{U}(X)=\frac{X-X_{0}}{|X-X_{0}|}$ and $\displaystyle\nu=\nu(p)>0$. Using the correspondance between $\displaystyle x$ and $\displaystyle X$ and also $\displaystyle T_{U}$ and $\displaystyle T$ shown in (1.11) and (1.15), we recover our conclusion. ∎

2.3 Blow-up results at the origin

Proof of Theorem 5.

The proof is done in the framework of similarity variables (1.18), with $\displaystyle x_{0}=0$. Since $\displaystyle d$ is an integer, one clearly sees that the equation satisfied by $\displaystyle w_{0}$ in (1) is simply the radial version of the multi-dimensional equation considered in $\displaystyle\mathbb{R}^{d}$. Since $\displaystyle p$ is subconformal in relation to $\displaystyle d$, as shown in (1.5), we are in the setting considered by Hamza and Zaag in [6] for perturbed equations, with the exceptions that we have a non-constant coefficient in front of the nonlinear term here. As we have already seen while investigating the Lyapunov functional in Subsection 2.1, that is not an issue, and one can adapt the proof of [6] to the present equation, with no difficulties. ∎

Appendix A $\displaystyle L^{2}_{loc,u}$ for radial functions

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

Let $\displaystyle A=\displaystyle\sup\limits_{x_{0}\in\mathbb{R}^{d}}\int_{B(x_{0},1)}|u(x)|^{2}dx$ the square of the $\displaystyle L_{loc,u}^{2}$ norm in $\displaystyle\mathbb{R}^{d}$ and

$\displaystyle B=\displaystyle\sup\limits_{r_{0}\geq 1}\frac{1}{r_{0}^{d-1}}\int_{r_{0}-1}^{r_{0}+1}|\tilde{u}(r)|^{2}r^{d-1}dr.$ We also define for the crown $\displaystyle\mathcal{C}(r_{0},1)$ by

[TABLE]

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

Lemma A.1**.**

$\displaystyle i)$* $\displaystyle\exists\bar{\alpha}(d)>0$ such that $\displaystyle A\leq\bar{\alpha}(d)B$.

$\displaystyle ii)$ $\displaystyle\exists\bar{\beta}(d)>0$ such that $\displaystyle B\leq\bar{\beta}(d)A$.*

Proof.

$\displaystyle i)$ It is enough to show that for any $\displaystyle x_{0}\in\mathbb{R}^{d}$,

[TABLE]

Consider $\displaystyle x_{0}\in\mathbb{R}^{d}$. If $\displaystyle|x_{0}|<1$ and $\displaystyle x\in B(x_{0},1)$ then $\displaystyle|x|<|x_{0}|+1<2$. Consequently,

[TABLE]

where $\displaystyle\omega_{d-1}$ is the volume of the sphere $\displaystyle S^{d-1}$.

Now, if $\displaystyle|x_{0}|\geq 1$, then we have $\displaystyle B(x_{0},1)\subset\mathcal{C}(|x_{0}|,1)$. Furthermore, for geometric considerations, we know that there exists $\displaystyle\alpha(d,|x_{0}|)>0$ such that the crown $\displaystyle\mathcal{C}(|x_{0}|,1)$ contains $\displaystyle\alpha(d,|x_{0}|)r_{0}^{d-1}>0$ disjoint copies of $\displaystyle B(x_{0},1)$, with

[TABLE]

If we denote by $\displaystyle x_{i}$ for $\displaystyle i\in\{0,...,\alpha-1\}$ the centers of those balls, then we have

[TABLE]

on the one hand. On the other hand, since the difference between the two crown’s radius is $\displaystyle 2$ and the balls are of radius $\displaystyle 1$, it follows that

[TABLE]

Since $\displaystyle u$ is radial and the balls $\displaystyle B(x_{i},1)$ are disjoint, using (A.3) we see that

[TABLE]

Combining this with (A.2) and (A.1), we conclude the proof of item $\displaystyle i)$.

$\displaystyle ii)$ Consider $\displaystyle r_{0}\geq 1$. From geometric considerations, there exists $\displaystyle\beta(d,r_{0})>0$ such that the crown $\displaystyle\mathcal{C}(r_{0},1)$ is contained in $\displaystyle\beta(d,r_{0})$ copies of $\displaystyle B(0,1)$, with

[TABLE]

Denoting by $\displaystyle y_{i}$ for $\displaystyle i\in\{0,...,\beta-1\}$ the centers of those balls, we have

[TABLE]

Using (A.4), we conclude the proof of item $\displaystyle ii)$.

∎