A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems

Abstract

We prove a Liouville-type theorem for semilinear parabolic systems of the form

$$\begin{aligned} {\partial _t u_i}-\Delta u_i =\sum _{j=1}^{m}\beta _{ij} u_i^ru_j^{r+1}, \quad i=1,2,\ldots ,m \end{aligned}$$

in the whole space ${\mathbb R}^N\times {\mathbb R}$. Very recently, Quittner (Math Ann. 364, 269–292, 2016) has established an optimal result for $m=2$ in dimension $N\le 2$, and partial results in higher dimensions in the range $p< N/(N-2)$. By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions $N\ge 3$. In particular, our results solve the important case of the parabolic Gross–Pitaevskii system—i.e. the cubic case $r=1$—in space dimension $N=3$, for any symmetric (m, m)-matrix $(\beta _{ij})$ with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space ${\mathbb R}^N_+\times {\mathbb R}$. As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.

Appendix

We give the following version of the maximum principle for cooperative systems, which is suitable to our needs. Related results are given in [26, Section 3.8] or [11, Theorem 3.2], but do not quite satisfy our requirements (unbounded domain, parabolic inequalities assumed on the positivity set only). Here, for given vector $W:=(w_i)_{1\le i\le m}$ and real number M, the inequality $W\le M$ is understood as $w_i\le M$ for all $i=1,\dots ,m$.

Proposition 6.1

Let $m\ge 2$, $N\ge 1$, $T>0$, let $\Omega$ be an arbitrary domain of ${\mathbb R}^N$ (bounded or unbounded, not necessarily smooth). We denote $Q_T=\Omega \times (0,T)$ and $\partial _P Q_T=(\overline{\Omega }\times \{0\})\cup (\partial \Omega \times (0,T))$ its parabolic boundary. Let $W=(w_i)\in C(\overline{\Omega }\times [0,T);{\mathbb R}^m)\cap C^{2,1}(Q_T;{\mathbb R}^m)$ and denote

$$\begin{aligned} D_i=\left\{ (x,t)\in Q_T:\ w_i(x,t)>0\right\} . \end{aligned}$$

Assume that W is a bounded, classical solution of the system

$$\begin{aligned} \partial _t w_i-\Delta w_i-K|\nabla w_i|\le \sum _{j=1}^m c_{ij}(x,t)w_j\quad \hbox { in} D_i,\quad i=1,\dots ,m, \end{aligned}$$

(49)

where $K>0$ is a constant and the coefficients $c_{ij}$ are measurable, bounded and satisfy

$$\begin{aligned} c_{ij}\ge 0 \quad {\mathrm{for}\,\mathrm{all}}\quad i\ne j. \end{aligned}$$

(50)

• (i) If $W\le 0$ on $\partial _PQ_T$, then $W\le 0$ in $Q_T$.

• (ii) Let $M>0$ and assume in addition that

  $$\begin{aligned} \sum _{j=1}^m c_{ij}\le 0,\quad i=1,\ldots ,m. \end{aligned}$$

  (51)

If $W\le M$ on $\partial _PQ_T$, then $W\le M$ in $Q_T$.

Proof of Proposition 6.1

(i) It follows by the Stampacchia method, e.g. along the lines of [29, Proposition 52.21] and [29, Remark 52.11(a)]. We give the proof for the convenience of the reader and for completeness.

First consider the case when $\Omega$ is bounded. Let $i\in \{1,\dots ,m\}$. By (51), we have

$$\begin{aligned} \left[ \partial _t w_i-\Delta w_i-K|\nabla w_i|\right] (w_i)_+\le \sum _{j=1}^m c_{ij}(x,t)(w_j)(w_i)_+ \quad \hbox {in} Q_T. \end{aligned}$$

For $t\in (0,T)$, since $(w_i)_+(\cdot ,t)\in H^1_0(\Omega )$ by our assumption, we may integrate by parts, to obtain

$$\begin{aligned} \frac{1}{2} \frac{d}{dt}\int _\Omega (w_i)_+^2&=\int _\Omega (\partial _t w_i) (w_i)_+ \\&\le -\int _\Omega \nabla w_i\cdot \nabla (w_i)_+ +K\int _\Omega |\nabla w_i| (w_i)_+ + \sum _{j=1}^m \int _\Omega c_{ij}(w_j)(w_i)_+\\&\le -\int _\Omega |\nabla (w_i)_+|^2 +\int _\Omega |\nabla (w_i)_+|^2\\&\quad +\frac{K^2}{4}\int _\Omega (w_i)_+^2 + \sum _{j=1}^m \int _\Omega c_{ij}(w_j)(w_i)_+. \end{aligned}$$

By assumption (52), it follows that

$$\begin{aligned} \frac{1}{2} \frac{d}{dt}\int _\Omega (w_i)_+^2&\le \frac{K^2}{4}\int _\Omega (w_i)_+^2 + \int _\Omega c_{ii}(w_i)_+^2 + \sum _{\begin{array}{c} j=1 \\ j\ne i \end{array}}^m \int _\Omega c_{ij}(w_j)_+(w_i)_+ \\&\le \left( \frac{K^2}{4}+\Vert c_{ii}\Vert _\infty \right) \int _\Omega (w_i)_+^2 + \frac{1}{2}\sum _{\begin{array}{c} j=1\\ j\ne i \end{array}}^m \int _\Omega c_{ij}((w_j)_+^2+(w_i)_+^2). \end{aligned}$$

Adding up for $i=1,\dots ,m$, we get

$$\begin{aligned} \frac{d}{dt}\sum _{j=1}^m \int _\Omega (w_i)_+^2\le L\sum _{j=1}^m \int _\Omega (w_i)_+^2 \end{aligned}$$

for some constant $L>0$. Since $W\le 0$ at $t=0$, it follows by integration that $\sum _{j=1}^m \int _\Omega (w_i)_+^2\le 0$, hence $W\le 0$ in $Q_T$.

Now, in the case of an unbounded domain, we fix $\varepsilon >0$ and consider the modified functions

$$\begin{aligned} \tilde{w}_i=e^{-\lambda t}w_i-\varepsilon \psi , \qquad \psi =(N+K)t +(1+|x|^2)^{1/2}>0 \end{aligned}$$

with $\lambda >0$ to be chosen. We also set

$$\begin{aligned} \tilde{D}_i=\left\{ (x,t)\in Q_T;\ \tilde{w}_i(x,t)>0\right\} \!. \end{aligned}$$

Since $\psi _t-\Delta \psi -K|\nabla \psi |\ge 0$, we have, in $\tilde{D}_i\subset D_i$,

$$\begin{aligned} \partial _t \tilde{w}_i-\Delta \tilde{w}_i-K|\nabla \tilde{w}_i|&\le e^{-\lambda t}\left[ \partial _t w_i-\Delta w_i-K|\nabla w_i|-\lambda w_i\right] \\&\quad -\varepsilon \left[ \psi _t-\Delta \psi -K|\nabla \psi |\right] \\&\le e^{-\lambda t}\left[ -\lambda w_i+\sum _{j=1}^m c_{ij}(x,t)w_j\right] \\&= \left[ -\lambda \tilde{w}_i+\sum _{j=1}^m c_{ij}(x,t)\tilde{w}_j\right] +\varepsilon \left[ -\lambda +\sum _{j=1}^m c_{ij}(x,t)\right] \psi \\&\le (c_{ii}(x,t)-\lambda ) \tilde{w}_i+\sum _{{\begin{array}{c} j=1 \\ j\ne i \end{array}}}s^m c_{ij}(x,t)\tilde{w}_j, \end{aligned}$$

by choosing $\lambda =\max _i\sum _{j=1}^m \Vert c_{ij}\Vert _\infty$. Noting that $\tilde{w}_i<0$ for |x| large, we may then apply the previous argument to get $\tilde{W}\le 0$ in $Q_T$. The conclusion follows upon letting $\varepsilon \rightarrow 0$.

(ii) Set $\hat{w}_i:=w_i-M$. Note that, in view of assumption (53), we have, for each $i=1,\dots ,m$,

$$\begin{aligned} \partial _t \hat{w}_i-\Delta \hat{w}_i- K |\nabla \hat{w}_i|&=\partial _t w_i-\Delta w_i- K |\nabla w_i| \\&\le \sum _{j=1}^m c_{ij}(x,t)(\hat{w}_j+M)\le \sum _{j=1}^m c_{ij}(x,t)\hat{w}_j \end{aligned}$$

in $D_i\supset \left\{ (x,t)\in Q_T:\ \hat{w}_i(x,t)>0\right\}$. Applying assertion (i) to the functions $\hat{w}_i$, we have the conclusion. $\square$