Bubbling Blow-Up in Critical Elliptic and Parabolic Problems
Monica Musso
Communicated by Notices Associate Editor Daniela De Silva
Mathematical models are often expressed by nonlinear partial differential equations. Solutions of a given partial differential equation can be interpreted as attainable states for the underlying model. In steady as well as in time-dependent problems, a central issue is to determine the behaviour of solutions or the presence of blow-up. Blow-up takes place in regions or instants where solutions, or some quantities depending on them, become unbounded or exhibit irregular behaviour. This usually means that the original model loses validity near these regions and space-time scaling is required to make an accurate description. A particular type of blow-up are the ones that are triggered by bubbling. We will briefly discuss bubbling blow-up in two classical critical elliptic and parabolic problems.
Bubbling in critical elliptic problems
Many problems of physical and geometrical interest have a variational structure. For such problems, the failure of compactness at certain energy levels reflects highly interesting phenomena related to internal symmetries of the systems under study. In several of these situations, bubbling may occur. The term bubbling refers to the presence of families of solutions that at main order look like scalings of a single profile which in the limit become singular but at the same time have an approximately constant energy level. Such phenomena have been observed for the first time by Sacks-Uhlenbeck (1981) in the context of two-dimensional harmonic maps, and independently by Wente (1980) in the context of surfaces of prescribed constant mean curvature.
Classical models where bubbling occurs are semilinear boundary value problems near criticality in $\mathbb{R}^N$. Consider the problem of finding positive solutions to
$$\begin{equation} \Delta u + u^{q} =0 \quad \text{ in }\Omega ,\quad u=0 \quad \text{ on } \partial \Omega , \cssId{P}{\tag{1}} \end{equation}$$
where $\Omega \subset \mathbb{R}^N$,$N\geq 3$, is a bounded domain with smooth boundary $\partial \Omega$ and $q>1$. This equation in 1 is sometimes called the Lane-Emden-Fowler equation. It was used first in the mid-19th century in the study of internal structure of stars, on the other hand it constitutes a basic model equation for steady states of reaction-diffusion systems, nonlinear Schrodinger equations, fast diffusion equations, and nonlinear dispersive equations. The case $q={N+2\over N-2}$ is especially meaningful. In geometry, it is related to the well-known problem of finding conformal metrics on a given manifold with prescribed scalar curvature, as in the Yamabe problem. In the study of nonlinear dispersive equations, it is related to the soliton resolution conjecture for nonlinear wave equations and nonlinear wave maps 10, 16.
The critical exponent $q={N+2 \over N-2}$ sets a threshold where the structure of the solution set of 1 suffers a dramatic change. If $q < {N+2\over N-2}$ a solution may always be found by minimizing the Rayleigh quotient
In fact, the quantity $S_q (\Omega )\equiv \inf _{ u\in H_0^1(\Omega )\setminus \{0\}} \, Q (u)\,$ is achieved thanks to compactness of Sobolev embeddings $H^1_0 (\Omega ) \hookrightarrow L^{q+1} (\Omega )$ for $q< {N+2\over N-2}$. A suitable scalar multiple of a minimizer turns out to be a solution of 1. The case $q\ge {N+2\over N-2}$ is considerably more delicate: for $q= {N+2\over N-2}$ compactness of the embedding is lost while for $q> {N+2\over N-2}$ there is no such an embedding. This obstruction is not just technical for the solvability question, but essential. If $\Omega$ is strictly star-shaped around a point $x_0 \in \Omega$ and $u$ solves 1 then Pohozaev’s identity (1965) yields
where $\nu$ is the unit outer normal to $\partial \Omega$. Hence necessarily $q <{N+2 \over N-2}$, and thus no solutions at all exist if $q \geq {N+2 \over N-2}$.
Pohozaev’s result puts in evidence the central role of topology or geometry in the domain for solvability. Kazdan and Warner (1975) observed that Problem 1 is actually solvable for any $q>1$ if $\Omega$ is a radial annulus, as compactness in the Rayleigh quotient $Q$ is gained within the class of radially symmetric functions. On the other hand Coron (1984) found via a variational method that 1 is solvable at the critical exponent $q={N+2\over N-2}$ whenever $\Omega$ is a domain exhibiting a small hole. Substantial improvement of this result was found by Bahri and Coron 1, proving that if some homology group of $\Omega$ with coefficients in $\mathbf{Z}_2$ is not trivial, then 1 has at least one solution for $q$ critical, in particular in any three-dimensional domain which is not contractible to a point. Examples showing that this condition is actually not necessary for solvability at the critical exponent were found by Dancer (1988), Ding (1989) and Passaseo (1989, 1998).
The change of structure of the solution set taking place at the critical exponent is strongly linked to the presence of unbounded sequences of solutions or bubbling solutions. By a bubbling solution for 1 near the critical exponent we mean an unbounded sequence of solutions $u_n$ of 1 for $q=q_n \to {N+2\over N-2}$. Setting
in the expanding domain $\Omega _n = M_n^{{(q_n -1)/ 2}}(\Omega - \xi _n)$. Assuming for instance that $\xi _n$ stays away from the boundary of $\Omega$, elliptic regularity implies that locally over compacts around the origin, $v_n$ converges up to subsequences to a positive solution of
$$\begin{equation*} \Delta U + U^{N+2\over N-2} = 0 \end{equation*}$$
in entire space. Positive solutions to this equation are known from the classical works of Rodemich (1966), Aubin (1976), Obata (1972) and Talenti (1976) to be the functions
A natural problem is that of constructing solutions exhibiting this property around one or several points of the domain when the exponent $q$ approaches the critical value ${N+2 \over N-2}$.
For $q$ slightly sub-critical, $q= {N+2 \over N-2}- \varepsilon$,$\varepsilon >0$, a solution $u_\varepsilon$ given by a minimizer of the Rayleigh quotient 2 clearly cannot remain bounded as $\varepsilon \downarrow 0$, since otherwise Sobolev’s constant ${\mathcal{S}}_N$ in 3 would be achieved by a function supported in $\Omega$. In this case, $u_\varepsilon$ has asymptotically just a single maximum point $\xi _\varepsilon$ and the asymptotic 4 holds globally in $\Omega$ with $M_\varepsilon \sim \varepsilon ^{-\frac{1}{2}}$. Moreover, $\xi _\varepsilon$ approaches a critical point of Robin’s function $H(x,x)$. Here $H(x,y)$ is the regular part of Green’s function $G(x,y)$ for the Laplace operator in $\Omega$ under Dirichlet boundary conditions.
This conclusion can be refined to the case of solutions $u_\varepsilon$ exhibiting bubbling at multiple points, for both slightly sub-critical and super-critical exponents $q={N+2 \over N-2}\mp \varepsilon$. The general result can be phrased in the following terms: Given a nondegenerate critical point or a topologically nontrivial critical point of the reduced functional of $(\xi , \lambda ) = (\xi _1 , \ldots , \xi _k , \lambda _1 , \ldots , \lambda _k ) \in \Omega ^k \times \mathbb{R}_+^k$,
to problem 1 with $q={N+2 \over N-2}\mp \varepsilon$. Needless to say, it is a delicate task to find critical points for this reduced functional for a general domain $\Omega$, and they may even not exist.
But what is the origin of the reduced functional $\Psi _{k}^{\mp \varepsilon }$? An alternative way to find a solution to 1 is as a critical point of the energy functional
The scaled bubbles give a precise description of the solution near the blow up points. Far from these points, the bubbles need to be correct to match the zero Dirichlet boundary condition. An efficient way to do this is by using a proper multiple of the regular part $H(x,y)$ of the Green’s function. Hence a better approximate solution is given by
Hence finding a critical point of the reduced functional suggests the existence of a solution $u_\varepsilon$ close (in some topology) to the sum $v_\varepsilon$ of $k$ corrected bubbles. As mentioned before, in the sub-critical setting $q={N+2 \over N-2}-\varepsilon$, the natural topology for this problem is the energy space $H^1_0 (\Omega )$. In this regime, the reported results have been obtained by Brezis-Peletier 3, Han (1991), Rey (1990) and Bahri-Li-Rey 2. In the super-critical regime $q={N+2 \over N-2}+\varepsilon$ the embedding $H^1_0 (\Omega ) \hookrightarrow L^{q+1} (\Omega )$ is not available. New weighted $L^\infty$ spaces were first introduced in 7 to treat the super-critical regime.
A main implication of the result in 7 states that in a domain with a small hole problem 1 with $q= {N+2 \over N-2 } +\varepsilon$ has a two-bubble solution. More generally, if several spherical holes are drilled, a solution obtained by gluing together several two-bubbles can be found. Two-bubble solutions are the simplest to be obtained: single-bubble solutions for Problem 1 with $q= {N+2 \over N-2 } +\varepsilon$do not exist, as shown by M. Ben Ayed, K. El Mehdi, M. Grossi, O. Rey (2003). Solutions with different blow-up orders, known as tower of bubbles were found in 18.
$$\begin{equation} \begin{aligned} \partial _t u & =\Delta u + u^{q} \quad \text{ in }\Omega \times [0,T], \,\quad \\ u&=0 \quad \text{ on } \partial \Omega \times [0,T],\, \end{aligned} \cssId{PP}{\tag{7}} \end{equation}$$
for $0< T \leq \infty$, is a widely studied classical problem, usually referred to as the Fujita problem, after his work in 1969. The heat operator $\partial _t u= \Delta u$ in 7 describes the diffusion of a density-function $u = u(x,t)$, where $x$ is the space variable and $t$ denotes time, and the term $f(u) = u^q$ represents a source. This is the simplest model of semilinear parabolic equations, which are ubiquitous as they can be found in numerous applications ranging from physics and biology to materials and social sciences. We refer the reader to reference 20 for a comprehensive survey on Problem 7 and more general versions of it.
Despite its simple look, Problem 7 encodes the fundamental features of a general semilinear parabolic problem. If the initial condition $u_0 = u_0(x)$ at time $t=0$ is smooth and has value $0$ on the boundary of $\Omega$, Problem 7 has a unique (classical) solution $u= u(x,t)$ defined on some time interval $[0, T )$ with $0 < T \leq \infty$. If we call $T = T (u_0)$ the maximal possible time of existence, the solution cannot be extended beyond $T$. If $T < \infty$, then necessarily the solution blows up at $T$, in the sense that $\| u(\cdot , t) \|_{L^\infty (\Omega ) } \to \infty$ as $t \nearrow T$. If $T = \infty$, we say that the solution is global. In this case, two possibilities can occur: either $u$ remains bounded as $t \to \infty$, or
The latter is sometimes referred to as infinite time blow-up or grow-up of $u$. One of the fundamental problems concerning equation 7 is whether the infinite time blow-up can actually occur for some $u_0$ or not.
When $q$ is the critical Sobolev exponent $q={N+2 \over N-2}$, one expects that blow-up by bubbling for specific situations appears in the form
where now $\lambda _j (t)$ and $\xi _j (t)$ are functions of the time variable $t$, with $\lambda _j(t) \to 0$ as $t \to T$. Those solutions are usually asymptotically not self-similar and, while not generic, their presence is among the most important features of the full dynamics since they correspond to threshold solutions between different generic regimes.
Consider an initial condition of the form $u_0 (x) = \alpha \varphi (x)$, where $\varphi$ is a fixed positive smooth function in $\Omega$ with zero boundary value and $\alpha$ is a positive constant, and denote by $u_\alpha (x,t)$ the unique (local) solution to 7 with this initial condition. For all sufficiently small $\alpha$, it is possible to prove that $u_\alpha (x,t)$ is globally defined and that $u_\alpha (x,t) \to 0$ uniformly for $x \in \Omega$ as $t \to \infty$. To see this, let $\lambda _1$ be the first eigenvalue of $-\Delta$ in $\Omega$ under Dirichlet boundary conditions and $\phi _1$ a positive first eigenfunction:
Let $\delta >0$ and consider the function $\bar{u} (x,t) = \delta e^{-\gamma t} \phi _1 (x)$, where $0<\gamma <\lambda _1$. Then a direct computation gives
provided $\delta >0$ is small. By the maximum principle, $\bar{u}(x,t)$ is a supersolution of 7. Hence any solution to 7, whose initial value at time $t=0$ is bounded above by $\bar{u} (x,0)$, stays bounded by $\bar{u} (x,t)$ at all times. If we take $0<\alpha$ small, then $u_\alpha (x,0) =\alpha \varphi (c) \leq \bar{u} (x,0)$ and hence, for some positive constant $C>0$, and any $x \in \Omega$
$$\begin{equation*} u_\alpha (x,t) \leq C e^{-\gamma t}, \quad {\text{as}} \quad t \to \infty . \end{equation*}$$
On the other hand, if we now take $\alpha$ in the initial condition $u_0 (x) = \alpha \phi (x)$ to be large, then $u_\alpha (x,t)$ blows-up in finite time. To see this, we assume that the solution $u_\alpha (x,t)$ is defined in $\Omega \times [0,T)$, we mutiply the equation against $\phi _1 (x)$ and integrate on $\Omega$. Using the divergence Theorem, we get
is well defined and $0<\alpha _\star <\infty$. The solution $u_{\alpha _\star } (x,t)$ somehow lies in the dynamic threshold between solutions globally defined in time and those that blow-up in finite time. Ni, Sacks, Tavantzis (1984) prove that this solution is a well-defined $L^1$-weak solution of the Fujita problem, but it is not clear whether it will be smooth for all times.
When $1 < q < {N+2 \over N-2}$,$u_{\alpha _\star } (x,t)$ is uniformly bounded and smooth, and up to subsequences it converges to a (positive) solution of the stationary problem 1. When $q > {N+2 \over N-2}$,$\Omega$ is a ball, and $u_{\alpha _\star }$ is radially symmetric then $u_{\alpha _\star } (x,t) \to 0$ as $t \to \infty$. The case $q={N+2 \over N-2}$ is completely different: Galaktionov and Vázquez 13 proved that if $\Omega =B(0,1)$ and if the threshold solution $u_{\alpha *}$ is radially symmetric, then no finite time singularities for $u_{\alpha *}(r,t)$ occur and it must become unbounded as $t\to +\infty$, thus exhibiting infinite-time blow up
Galaktionov and King 12 discovered that this radial blow-up solution to 7 at $q={N+2 \over N-2}$ does have a bubbling asymptotic profile as $t\to +\infty$ of the form 8 with $k=1$ and $\lambda _1(t) \sim t^{-\frac{1}{N-4}}\to 0$ for $N\ge 5$. This critical infinite-time bubbling occurs also in the nonradial setting, as shown in 6: there are global solutions to 7 with $q={N+2 \over N-2}$ which have infinite time blow-up at any collection of points $p=( p_1, \ldots , p_k) \in \Omega ^k$ if $p$ lies in the open region of $\Omega ^k$ where a certain $k \times k$ matrix $\mathcal{G} (q)$ is positive definite. The matrix ${\mathcal{G}}$ is explicitly defined in terms of the Robin’s and the Green’s functions in $\Omega$, introduced in the previous section: