The shape of a free boundary driven by a line of fast diffusion

We would like to dedicate this article to Sandro Salsa. He is a wonderful person and an exceptional, generous mathematician. We have greatly enjoyed his work and his friendship.

1.   Introduction

1.1.   Model and question

The system under study involves an unknown real $c > 0$, an unknown function $u(x, y)$ defined in $\mathbb{R}^2_-: = \mathbb{R}\times \mathbb{R}_-$, and an unknown curve $\Gamma\subset \mathbb{R}^2_-$ satisfying

Note that the convergence of $u$ to 1 to the left, and 0 to the right, are not requested to hold in the same sense. This is not entirely innocent, we will explain why in more detail below. We will also be interested in a (seemingly) more complex version of (1.1). We look for a real $c > 0$, a function $u(x)$, defined for $x\in \mathbb{R}$, a function $v(x, y)$ defined in $\mathbb{R}^2_-: = \mathbb{R}_+\times \mathbb{R}_-$, and a curve $\Gamma\subset \mathbb{R}^2_-$ such that

We ask for the global shape of the free boundary $\Gamma$. Before that, we ask about the existence of a solution $(c, \Gamma, u)$ to (1.1), and of a solution $(c, \Gamma, u, v)$ to (1.2), this indeed deserves some thought, as the condition at $-\infty$ is rather strong.

Systems (1.1) and (1.2) arise from a class of models proposed by H. Berestycki, L. Rossi and the second author to model the speed-up of biological invasions by lines of fast diffusion, see for instance ^[3] or ^[4]. The two-dimensional lower half-plane (that was called "the field" in the afore-mentionned references), in which reaction-diffusion phenomena occur, interacts with the $x$ axis ("the road") which has a much faster diffusion $D$ of its own. In Model (1.2), $u(x)$ the density of individuals on the road, and $v(t, x, y)$ the density of individuals in the field. The road yields the fraction $\mu u$ to the field, and retrieves the fraction $\nu v$ in exchange; the converse process occurs for the field. Model (1.1) is obtained from (1.2) by forcing $\mu u = v$ on the road, so that the sole unknown is $v(x, y)$, and the exchange term is replaced by $\frac{v_y}\mu$ (and $v$ has been renamed $u$). In the sequel, Model (1.2) will be called the "model with two species" (that is, the density on the road and in the field may be different), while Model (1.1) will be called "Model with one species". Also note that, in both models, the unknown functions are travelling waves of an evolution problem where the term $c\partial_x u$ (resp. $c\partial_xv)$ is replaced by $\partial_tu$ (resp. $\partial_tv$). This is explained in more detail in ^[6], where the study of (1.1) and (1.2) was initiated.

1.2.   Results

Theorem 1.1. (Existence for the model with one species) Assume $D\geq d$. System (1.1) has a solution $(c > 0, \Gamma, u)$ with $u$ globally Lipschitz, and we have $\partial_xu < 0$. Moreover

$–$ $\Gamma$ is an analytic curve, as well as a locally Lipschitz graph in the $y$ variable:

$–$ $\Gamma\cap\{y = 0\}$ is nonempty, assume $(0, 0)\in\Gamma$. There is $\varepsilon_0 > 0$ such that

Theorem 1.2. (Existence for the model with two species) System (1.2) has a solution $(c > 0, \Gamma, u, v)$ such that the function $v$ is globally Lipschitz, and we have $\partial_xv < 0$. Moreover

$–$ $\Gamma$ is an analytic curve, as well as a locally Lipschitz graph in the $y$ variable:

$–$ $\Gamma\cap\{y = 0\}$ is nonempty, assume $(0, 0)\in\Gamma$. There is $\lambda > 0$ and $\varepsilon_0 > 0$ such that

The main question of this work, namely, how $\Gamma$ looks like, is addressed in the following theorem. Both models with one species and model with two species are concerned. Let $c_0$ be the speed of the basic travelling wave $\phi_0(x)$:

In what follows, we stress the dependence on $D$ of the velocity, free boundary and solution of the PDE by denoting them $c_C$, $\Gamma_D$, $u_D$, $v_D$.

Theorem 1.3. In both models (1.1) and (1.2), there is $D_0\geq0$ such that, for every $D\geq D_0$ we have $c_D > c_0$. Let $\alpha_D$ be given by $\sin\alpha_D = \frac{c_0}{c_D}$. If $\varphi_D$ is given by (1.4) we have

In other words, $\Gamma_D$ has an asymptotic direction, which is a line making the angle $\alpha_D$ with the horizontal. We have a more precise version of Theorem 1.3:

Theorem 1.4. Assume that $\alpha < \frac\pi2$. For every $\omega\in(0, \frac12c_D\sin\alpha_D)$, we have

Thus there is a straight line making the angle $\alpha_D$ with the horizontal that is asymptotic to $\Gamma_D$ at infinity, and the distance between the two shrinks exponentially fast.

1.3.   Underlying question, comments, organisation of the paper

Let us be more specific about the question that we wish to explore here. We want to account for a loss of monotonicity phenomenon for $u$ (Model (1.1)) or $v$ (Model (1.2)) in the $y$ variable, a study that was initiated in ^[6]. This phenomenon was discovered numerically by A.-C. Coulon in her PhD thesis ^[7]. There, she provided simuations for the evolution system (notice that the propagation takes place for $y > 0$):

The figures below account for some of her results; the parameters are

The top figure represents the levels set 0.5 of $v$ at times 10, 20, 30, 40, while the bottom figure represents the shape of $v(40, x, y)$.

The surprise is the location of the leading edge of the invasion front: rather than being located on the road, as one would have expected (especially for large $D$), it sits a little further in the field. This entails a counter-intuitive loss of monotonicity. As working directly with the reaction-diffusion (1.10) has not resulted in a significant outcome so far, the idea was to replace the reaction-diffusion model by a free boundary problem, that may be seen as an extreme instance of reaction-diffusion. Using this idea, a first explanation of the location of the leading edge is provided in ^[6]. The conclusions are included in Theorems 1.1 and 1.2.

The goal of the paper is, as already announced, to account for the global shape of the free boundary. We claim that it will provide a good theoretical explanation of A.-C. Coulon's numerics. Indeed it may be expected (although it is not a totally trivial statement) that the solution of (1.10) will converge to a travelling wave. As the underlying situation is that of an invasion, it is reasonable to assume that no individuals (if we refer, as was the initial motivation, to a biological invasion) are present ahead of the front. This is why we impose the seemingly stringent, but reasonable from the modelling point of view, condition at $-\infty$. Theorem 1.3 shows that it entails a very specific behaviour.

Let us explain the consequences of our results on the understanding of the model. Theorems 1.1 to 1.3 put together depict a free boundary whose leading edge is in the lower half plane and which, after a possibly nonempty but finite set of turns, becomes asymptotic to a line that goes to the right of the lower half plane. This is in good qualitative agreement with the upper two-thirds of the picture presented here, the lower third accounting for the fact (still to be described in mathematically rigorous terms) that the free boundary bends in order to connect to a front propagating downwards, which is logical as we start from a solution that is compactly supported. Let us, however, point out that the analogy should not be pushed further than what is reasonable, as the logistic nonlinearity displays – and this is especially true for the model (1.10) – more counter-intuitive oddities of its own, see ^[5].

Some additional mathematical comments are in order. The first one is that we have put some effort in proving existence theorems. The reason is that we could not entirely rely on our previous study ^[6] for that: Although the leading edge was analysed, we had chosen to wipe out the additional difficulties coming from the study in the whole lower half-plane, by studying a model in a strip of finite length with Neumann conditions at the bottom. While the study of the leading edge is purely local, and will not need any more development, the condition $\lim\limits_{x\to-\infty}u(x, y) = 1$ (resp. $\lim\limits_{x\to-\infty}u(x, y) = 1$) uniformly in $y\in \mathbb{R}_-$ requires some additional care, that is presented in Section 3.

Theorem 1.3 is hardly incidental. It is in fact a general feature in reaction-diffusion equations in the plane. The heuristic explanation is the following: looking down very far in the lower half plane, we may think that the free boundary $\Gamma_D$ propagates like the 1D wave in its normal direction, that is, $V_n = c_0$. On the other hand, it propagates with speed $c_D$ horizontally: this imposes the angle $\alpha_D$. For a rigorous proof of that, we take the inspiration from previous works on conical-shaped waves for reaction-diffusion equations in the plane. A first systematic study may be found in ^[2], while the stability of these objects is studied in ^[9]. Further qualitative properties are derived in ^[10]. Solutions of the one phase free boundary problem for $c\partial_x u-\Delta u = 0$ are classified by Hamel and Monneau in ^[8]. One of their results will play an important role in the proof of Theorem 1.3, this will be explained in detail in Section 4.

Although this work is clearly aimed at understanding the situation for large $D$ (the case $D\leq d$ poses interesting technical questions in for the one species model) we have not really provided a systematic study of the limit $D\to+\infty$. This will be the object of a forthcoming paper. Another interesting question is whether the free boundary has turning points. While the simulations cleary point at convexity properties of the sub-level sets, we do not, at this stage, have real hints of what may be true.

The paper is organised as follows. In Section 2, we provide some universal bounds on the velocities, in therms of the diffusion on the road $D$. In Section 3, we construct the wave for the one species model and prove Theorem 1.2. In Section 4, we indicate the necessary modifications for the two-species model. In Section 5, we prove the exponential convergence of the level sets.

2.   The one species model: Bounds on the velocity in a truncated problem

Solutions to (1.1) will be constructed through a suitable approximation in a finitely wide cylinder; we set

from (a trivial modification of) ^[6], for $L > 1$, there is a solution $(c_D^L, \Gamma_D^L, u_D^L)$ to the auxiliary problem

In biological terms, this means that the boundary $\{y = -L\}$ is lethal for the individuals. The limit at $x = +\infty$, namely the function $(1-y-L)^+$, is of course not chosen at random, it solves the one-dimensional free boundary problem

To ensure the maximum chance to retrieve, in the end, a solution that converges to 1 uniformly in $y$ as $x\to-\infty$, we have imposed the Dirichlet condition $u = 1$ at the bottom of the cylinder.

2.1.   Exponential solutions

At this point, we need to make a short recollection of what the exponential solutions of the linear problem are. System (2.1), linearised around 0, reads

It has exponential solutions that decay to 0 as $x\to-\infty$, i.e., solutions of the form $\psi_D^L(x, y) = e^{\alpha x}h(y)$ (we have to put here the dependence on $L$ and $D$. If $L = +\infty$ we may choose the function $h$ as an exponential in $y$. We have

so that the exponents $\alpha^L_D$ and $\beta^L_D$ satisfy

Three types of limits will be considered.

Case 1. The limit $D\gg L\gg1$, $c$ bounded. We expect $\beta_D^L$ to go to 0 as $D\to+\infty$, so that

Then (2.3) yields estimates of the form

It is to be noted that these equivalents may be pushed up to $c = o(\sqrt D)$.

Case 2. The limit $D\gg L\gg1$, $c\gg\sqrt D$. This time we expect $\beta_D^L$ to go to infinity as $D\to+\infty$, so that

We have

We have here a first occurrence of the critical order of magnitude $c\propto\sqrt D$.

Case 3. The limit $L\gg D\gg1$, $c\gg1$. We expect that $L\beta_D^L$ will go to infinity, so that ${\mathrm{cotanh}}(\beta_D^L)\to 1.$ In this setting, we have the estimates (2.5).

2.2.   Universal upper bound

In the construction of a travelling wave for (1.1), the first task is to bound the velocity from above. We will prove straight away the upper bound that will serve us in the later section, namely that $c_D^L$ cannot exceed a (possibly large) multiple of $\sqrt D$.

Theorem 2.1. There is $K > 0$, independent of $D$ and $L$, such that

Proof. Assume $c_D^L\gg\sqrt D\gg L\gg1$, so that (2.5) hold for $\alpha_D^L$ and $\beta_D^L$. Set

The function $\underline u_D^L$ vanishes on the curve $\underline\Gamma_D^{2L}$ whose equation is

In particular, for $y = -L$ we have

Translate the solution $u_D^L(x, y)$ of (1.1) so that the leftmost point of $\Gamma_D^L$ is located at the vertical of the origin. Then, from (2.7), $\underline\Gamma_D^{2L}$ lies entirely to the left of $\Gamma_D^L$. By the maximum principle we have

Now, translate $\underline u_D^L$ as much as this allows it to remain under $u_D^L$. If $\lambda_D^L$ is the maximum amount by which one may translate $\underline u_D^L$, there is a contact point $(x_D^L, y_D^L)$ between $u_D^L(x, y)$ and $\underline u_D^L(x-\lambda_D^L, y)$.

Still from (2.7), we have $y_D^L > -L$. So, it now depends on whether we have $y_D^L = 0$ or $y_D^L\in(-L, 0)$. In the first case, let $\Gamma_D^L$ intersect the line $\{y = 0\}$ at the point $(\tilde x_D^L, 0)$. If $x_D^L = \tilde x_D^L$, we have $\partial_x u_D^L(x_D^L, 0) = 0$ but we also have

an impossibility. If not, we have $x_D^L < \tilde x_D^L$ and this time we have

the last inequality because of the Hopf lemma. This prevents the Wentzell relation from holding at $(x_D^L, 0)$. So, the only possibility left is $y_D^L < 0$. By the strong maximum principle, the point $(x_D^L, y_D^L)$ is on the free boundary $\Gamma_D^L$. We have

that is,

because $\underline u_D^L(x_D^L, y_D^L) = 0$. Thus, if $D$ is sufficiently large, (2.5) implies

contradicting the free boundary relation for $u_D^L$. This proves the theorem.

2.3.   Universal lower bound

Theorem 2.2. There is $L_0 > 0$ such that the velocity $c_D^L$ in Model (2.1) satisfies

uniformly with respect to $L\geq L_0$.

The proof of Theorem 2.2 will be by contradiction. From now on, and until this has been proved wrong, we assume that, $c^L_D$ is bounded, both with respect to $L$ and $D$. Recall that $\Gamma^L_D$, the free boundary, is an analytic graph $\{x = \varphi^L_D(y)\}$. Moreover, from ^[6], we may always assume that it intersects the line $\{x = 0\}$, so that we may always assume $\varphi^L_D(0) = 0$. Define $\bar x^L_D$ as the last $x$ such that, for all $y\in(-L, 0)$, then $(x, y)\notin\Gamma^L_D$. Our main step is to prove that the front goes far to the left of the domain, this is expressed by the following lemma.

Lemma 2.3. There is $q > 0$ universal such that

Proof. Recall that we have

For every $x < 0$, let $y^L_D(x)$ be the first $y$ such that $x = \varphi^L_D(y)$. We will prove that

which implies Lemma 2.3. Assume (2.9) does not hold, and consider $\delta_0 > 0$ such that, for a sequence $(D_n, L_n)_n$, going to $+\infty$, and a sequence $(\varepsilon_{n})_{n}$ going to $0$ as $n\to+\infty$, we have

Obviously, we must assume the boundedness from below of the sequence $(\varepsilon_n\sqrt {D_n})_n$. For every $n$ set

For $\varepsilon_0$, to be chosen small in due time, the segment $(\bar X_n, X_{n})$ is at distance at least $\delta_0/2$ from the free boundary. By nondegeneracy (recall the boundedness of $(c_{D_n}^{L_n})_n$), there is $q_0 > 0$ universal such that, for all $n$:

On the other hand, recall that $\partial_yu_{D_n}^{L_n}(x, 0)\leq1$, this follows from ^[6]. So, the equation for $u_{D_n}^{L_n}(x, 0)$ reads, simply

Thus we have

This yields, for $n$ large enough, the existence of $\delta_0'' > 0$, universal, such that

This comes from the Hopf boundary lemma. So now, we now write the equation for $u_{D_n}^{L_n}(x, 0)$ as

Integrating this equation on $(- \varepsilon_0\sqrt{D_n}, 0)$ and invoking (2.11) allows us to find a small constant $\delta_0'' > 0$ such that

a contradiction.

Proof of Theorem 2.2. Recall that we have still assumed the boundedness of $(c^{L_n}_{D_n})_{n}$. In order to allevite the notations a little, we omit the index ${}_n$. Integration of (2.1) over $\Sigma^L$ yields

This expression has to be handled with care, because each integral, taken separately, diverges. We will see, however, that there is much less nonsense in (2.12) than it carries at first sight. The curve $\Gamma^L_D$ has an upper branch, that we call $\Gamma^L_{D, +}$, and that connects $(0, 0)$ to $X^L_D$; the latter point being a turning point. The lower branch, called $\Gamma^L_{D, -}$, connects $X^L_D$ to $(+\infty, -L+1)$, in other words it is asymptotic to the line $\{y = -L+1\}$ as $x$ goes to $+\infty$. We decompose $(2.12)$ into

Let us remark that $\partial_yu_D^L(x, -L)\geq-1$ for all $x\in \mathbb{R}$. Indeed, the function $u_D^L(x, y)$ being decreasing in $x$, it is larger than $(1-L-y)^+$. So, we have $u_D^L(x, -L)\geq-1$. Now, the graph $\Gamma^L_D$ has a discrete set of turning points, due to its analyticity. Away from these turning points, for $x\geq x^L_D$, there is a finite number of $y's$: $y_1 > y_2 > ...y_{n_D(x)}$ such that $(x, y_i)\in\Gamma^L_D$ for $1\leq i\leq n_D$. The point $(x, y_1)$ has already been counted in the integration over $\Gamma^L_{D, +}$, and so does not need to be counted again. Notice then that, if $x$ is not the abscissa of a turning point, then $n_D$ (or $n_D-2$) is even, because of the configuration of $\Gamma^L_D$. So, $u_\nu(x, y_i) = u_\nu(x, y_{i+1}) = 1$ for $1\leq i\leq n_D(x)-2$: it suffices to consider the sole point $(x, y_{n_D(x)})$ in the computation of $I\!I\!I$, however at this point we also have $\partial_\nu u_D^L = 1$. Therefore we end up with $I\!I\!I\geq0$. Of course the integral giving $I\!I\!I$ converges, but we do not even have to bother to prove it.

From Lemma 2.3, we have

for some universal $C > 0$. We already saw that $I\!I\!I$ was nonnegative, so let us deal with $I\!I$. By nondegeneracy, there is $A > 0$ universal such that

Notice that we do not change anything if, in $I\!I$, we integrate up to $x^L_D-1$ instead of $x^L_D$. We deduce, because $\psi_D^L(x, -L) = 0$ and $u_D^L(x, -L) = 1$:

This implies

This yields

for a universal $q$, as soon as we choose $D\gg L\gg1$. This is an obvious contradiction. Now, note that, because of the Dirichlet condition at $y = -L$, the sequence $(c^L_D)^L$, for fixed $D$, is increasing. This ends the proof of the theorem.

3.   The one species model: Construction and properties of the free boundary

3.1.   Global solutions of the free boundary problem in the plane

In this short section we recall a result of Hamel and Monneau that we will use to analyse the behaviour of the free boundary at infinity. Consider a solution $(c > 0, \Gamma, u)$ of the free boundary problem in the whole plane

The result is a classification of the solutions of (3.1) having certain additional properties. We rephrase it here to avoid any confusion, since the function $u$ in ^[8] corresponds to $1-u$ in our notations.

Theorem 3.1. (Hamel-Monneau ^[8], Theorem 1.6) Assume that

1). $\Gamma$ is a $C^{1, 1}$ curve with globally bounded curvature,

2). $\mathbb{R}^2\backslash\Omega$ has no bounded connected component,

3). we have

Then $c\geq c_0$ and, if we set

either $u$ is the tilted one-dimensional solution $\phi_0(y\cos\alpha\pm x\sin\alpha)$, or $u$ is a conical front with angle $\alpha$ to the horizontal, that is, the unique solution $u_\alpha(x, y)$ of (3.1) such that $\Gamma$ is asymptotic to the cone $\partial\mathcal{C_\alpha}$, with

Note that the fact that $u_\alpha$ is unique is not exactly trivial, it is given by Theorem 1.3 of ^[8]. Let us already notice that Properties 1 and 2 of this theorem are satisfied by the solution $(c_D^L, \Gamma_D^L, u_D^L)$ of (2.1): Property 1 is clear, and Property 2 is readily granted by the monotonicity in $x$. Property 3 will be a little more involved to check.

3.2.   Construction of a solution in the whole half-plane

From then on, we fix $D > 0$ large enough so that Theorem 2.2 holds. Pick $L_0$ such that $c_D^{L_0} > c_0$, this implies $c_D^L\geq c_D^{L_0} > c_0$. Notice that we have almost all the elements for the proof of Theorem 1.1, we just need, in addition, to control where the free boundary meets the line $\{y = 0\}$. Since we have this freedom, we assume $\Gamma_D^L$ to intersect the line $\{y = 0\}$ at the origin. Let $\bar X_D^L = (\bar x_D^L, \bar y_D^L)$ the point of $\Gamma_D^L$ that is furthest to the left, our sole real task will be to prove that $X_D^L$ cannot escape too far as $L\to+\infty$. Indeed, we notice that the property

holds easily. Indeed, from the maximum principle we have

for all $\gamma < \frac{c_D^{L_0}}D$, as it is a subsolution to the equation for $u$ in the plane and on the line $\{y = 0\}$, and below $u_D^L$ on the bottom line and also the vertical segment $\{x = 0, -L\leq y\leq0\}$. Hence, at that point, we have almost everything for the construction of the wave in the whole half-plane, except the attachment property.

Lemma 3.2. There is a constant $K_D > 0$ independent of $L$ such that $\vert \bar X_D^L\vert\leq K_D$.

Proof. Let us first assume that

Translate $\Gamma_D^L$ and $u_D^L$ so that $\bar X_D^L$ becomes the new origin, the free boundary $\Gamma_D^L$ meets therefore the horizontal line at the point $(-\bar x_D^L, -\bar y_D^L)$. Up to a subsequence the triple $(c_D^L, \Gamma_D^L, u_D^L)$ converges to a solution $(c_D^\infty, \Gamma_D^\infty, u_D^\infty)$ of the free boundary problem (3.1) in the whole plane. Moreover, the origin is its leftmost point. So, we may slide $\phi_0$ from $x = -\infty$ to the first point where it touches $u_D^\infty$, this can only be at $\Gamma_D^\infty$, thus at the origin. But then we have

a contradiction with the Hopf Lemma. So this scenario is impossible, and the family $(\bar y_D^L)_{L\geq L_0}$ is bounded as $L\to\infty$.

To prove that the family $(\bar x_D^L)_{L\geq L_0}$ is bounded, we consider

and integrate the equation for $u_D^L$ on $\Sigma_{\bar L} = \mathbb{R}\times(-\bar L, 0)$, we obtain

From (3.6) and elliptic estimates, we have

moreover the choice of $\bar L$ implies that the rightmost point of $\{u(x, -\bar L) > 0$ is at the left of the origin. Hence $I\!I$ is uniformly bounded with respect to $L$. On the other hand we have

which, from the uniform boundedness of $c_D^L$ in Theorem 2.1, yields the boundedness of the family $(\bar x_D^L)_{L\geq L_0}$.

Proof of Theorem 1.1. We send $L$ to infinity, a sequence $(c_D^{L_n}, \Gamma_D^{L_n}, u_D^{L_n})_n$ will converge to a solution $(c_D, \Gamma_D, u_D)$ of (1.1). Because of Lemma 3.2, the free boundary $\Gamma_D$ meets the line $\{y = 0\}$ at a point $(-\bar x_D, 0)$ and the expansion (1.4) is granted by Theorem 1.4 of ^[6]. Notice also that the uniform limit at $-\infty$ is also granted because of (3.6). So, to finish the proof of the theorem, it remains to prove that, for all $y < 0$, the positivity set of $u$ only extends to a finite range. Such were it not the case, the limit

would exist and be nonzero. It would solve the free boundary problem

This only allows for $u^\infty_D(y) = (y+a)^-$, a contradiction to the bounedness of $u_D^\infty$.

3.3.   The tail at infinity

Let $(c_D, \Gamma_D, u_D)$ the solution constructed in Theorem 1.1 as one of the limits $L\to+\infty$ of $(c_D^L, \Gamma_D^L, u_D^L)$, with

where $\varphi_D$ is a smooth, locally Lipschitz function. Theorem 2.2 readily implies the first part of Theorem 1.3, that is

simply because $c^L_D\leq c_D$. The only use of this result that we are going to make in this section is that there exists $D_0 > 0$ such that $c_D > c_0$ for all $D\geq D_0$, which is the first part of Theorem 1.3. And so, we drop the indexes ${}_D$ in the rest of this section, for the simple reason that the dependence with respect to $D$ will not appear anymore. To prove the second part of Theorem 1.3, we apply Theorem 3.1 to any sequence of translates of $u$:

to infer that any possible limit of $u_n$ is the one-dimensional wave, tilted in the correct direction. Properties 1 and 2 of the theorem being readily true, we concentrate on Property 3. The main step will be to prove that $\phi$ is in fact globally Lipschitz in (any sub-plane of) the half plane, once this is done a suitably designed Hamel-Monneau type ^[8] subsolution, placed under $u$, will give the property.

Proposition 3.3. The function $\varphi$ is globally Lipschitz in $(-\infty, y_0]$, for any $y_0 < 0$.

Proof. Note that the lemma is trivially false if we inisist in making $y$ vary on the whole half-line. Also, as will be clear from the proof, the value of $y_0$ will play no role as soon as it remains a little away from 0. So, we will assume for definiteness $y_0 = -1$. The main step of the proposition consists in proving that no point of $\Gamma\cap\{y\leq-1\}$ may have a horizontal tangent. Assume that there is such a point $X_0 = (\varphi(y_0), 0)$, $y_0\leq -1$. Translate $u$ and $\Gamma$ so that it becomes the origin, still denoting them by $\Gamma$ and $u$. In a neighbourhood of $0$ of size, say, $\rho > 0$, $\Gamma$ may be written (recall that it is an analytic curve) as

We have $\psi(0) = \psi'(0) = 0$. Two cases have to be distinguished. The first one is $\psi\equiv0$ in $[-\rho, \rho]$. By analyticity and Cauchy-Kovalevskaya's Theorem, this implies that $\psi$ is defined and equal to 0 on the whole line; actually this case may happen only if $X_0$ is a point at infinity, that is, $u$ is a limit of translations of infinite size. But then we have

depending on whether the positivity set of $u$ is above or below $\Gamma$. This is in contradiction with the boundedness of $u$. The other case is $\psi$ nonconstant in $[-\rho, \rho]$, so that $u$ is nonconstant either in $B_\rho$. By analyticity again, $u$ has an expansion of the following type, in a neighbourhood of the origin:

with $n\geq 2$, the functions $P$ and $R$ being smooth in their arguments. Because $\partial_xu$ it maximal at the origin, we have $u_{xy}(0, 0)\neq0$ from the Hopf Lemma. Consider the situation where we have, for instance $u_y(0, 0) = -1,$ $u_{xy}(0, 0) = \beta > 0.$ We have therefore

and

Inside $B_\rho$ we also have

so that

This implies, because $n\geq 2$:

Thus the free boundary relation cannot be satisfied in $B_\rho$, except at the origin.

We note that these situations exhaust what can happen on $\Gamma$. Indeed, if there is a sequence $(X_n)_n$ of $\Gamma$, going to infinity, such that the tangent to $\Gamma$ at $X_n$ makes an angle $\alpha_n$ with the horizontal, with $\lim\limits_{n\to+\infty}\alpha_n = 0$, the usual translation and compactness argument yields a solution of the free boundary problem (3.1), where the tangent at the origin is horizontal. Once again we are in one of he above two cases, that are impossible.

The last step is to check a property that will imply Property 3 for any limiting translation of $(\Gamma, u)$ of the form (3.7). This is the goal of the next proposition.

Proposition 3.4. We have

Proof. We use the notations of Theorem 3.1. Assume for definiteness that the leftmost point of $\Gamma$ is located at $X_l = (0, y_l)$ with $y_l < 0$. Pick $\bar X\in\Omega$, call $M > 0$ its distance to $\Gamma$, and also set $X_0 = (x_0, -N)$, both $M$ and $N$ will be assumed to be large, independently of one another. From Proposition 3.3 we claim the existence of a cone ${\mathcal{C}}_\beta$ (the notation is given by (3.4)), the angle $\beta > 0$ depending on the Lipschitz constant of $\varphi$, but independent of $M$ and $N$, and a point $\bar X_{M, N}\in\Omega$, such that

$1).$ we have $d(\bar X_{M, N}, \Gamma)\geq\sqrt{\inf(M, N)}$,

$2).$ we have $\bar X\in \bar X_{M, N}+{\mathcal{C}}_\beta$,

$3).$ the upper branch of $\bar X_{M, N}+\partial{\mathcal{C}}_\beta$ meets the line of fast diffusion $\{y = 0\}$ at a point $(x_{M, N}, 0)$ with

$\gamma > 0$ independent of $M$ and $N$.

Two cases have to be distinguished, depending on whether $\bar X$ is below (see Figure 1) or above (see Figure 2) its projection onto $\Gamma$ (or, if the projection is not unique, the projection that we have selected). Note that, in the first case, there is $C > 0$, once again independent of $M$ and $N$, such that

in such a case we have $\inf(M, N) = M$. We now translate the origin to $\bar X_{M, N}$. Note that the line of fast diffusion becomes the line $\{y = N\}$. Consider a smooth, nonpositive, concave, even function $\varphi_{M, N}(y)$ such that

$1).$ we have $\varphi_{M, N}(0) = 0$,

$2).$ we have $-\varphi_{M, N}''(y)\leq \frac1{\inf(M, N)}$,

$3).$ we have $\vert\varphi_{M, N}'(y)\vert = 2\mathrm{cotan}\beta$ if $\vert y\vert\geq\sqrt{\inf(M, N)}$.

Let $\Gamma_{M, N}$ be the graph

It meets the line of fast diffusion at a point $\tilde x_{M, N}$ that satisfies an estimate of the form

with a universal $\gamma > 0$, possibly different from that of (3.10).

Pick now any small $\varepsilon > 0$ and consider the function

We claim that $\underline u\leq u$ in ${\Gamma}_{M, N}\cap\{y\leq N\}$. Indeed, recall that we have

this is a simple consequence of the uniform convergence of $u(x, .)$ to 1 as $x\to-\infty$ in the original variables, the fact that $(x_{M, N}, N)$ is at the left of the leftmost point of $\Gamma$, and (3.11), and the fact that $\underline u\leq 1- \varepsilon$. Inside ${\Gamma}_{M, N}\cap\{y\leq N\}$, we have

Therefore, $\underline u\leq u$, so that we have, if we still call $\bar X$ the original point $\bar X$ translated by $\bar X_{M, N}$:

which is the sought for estimate.

Proof of Theorem 1.2. Proposition 3.3 proves that any limit of a sequence of translations $(u_n)_n$ in (3.4) satisfies the assumptions of Theorem 3.1. Therefore it is either a conical front $u_\alpha$ (Case 1), a tilted one-dimensional wave $\phi_0(y\cos\alpha-x\sin\alpha)$ (Case 2), or a tilted wave $\phi_0(y\cos\alpha+x\sin\alpha)$ (Case 3). We wish to prove that only Case 3 survives. Let us consider the set $T$ of turning points of $\Gamma$, that is, the set of all points $(\varphi(y), y)$ such that $\varphi'(y) = 0$. From the analyticity of $\Gamma$, $T$ is discrete:

If we manage to prove that it is finite we are done because this excludes Case 1 trivially, and Case 2 because $u(x, y)\to 1$ as $x\to-\infty$, uniformly in $y$. So, assume that $T$ is infinite. We claim the existence of two sequences of $T$, decreasing to $-\infty$, $(y_n^i)_n$, $i\in\{1, 2\}$, such that

● we have $y_{n+1}^1\leq y_n^2\leq y^1_n$ for all $n$,

● we have $\lim\limits_{n\to+\infty}(y_n^2-y_{n+1}^1) = \lim\limits_{n\to+\infty}(y_n^1-y_{n}^2) = +\infty$,

such that, if we set $x_n^i = \varphi(y_n^i)$ we have

uniformly on compact sets in $(x, y)$. Indeed, there is $N_0\in \mathbb{N}$ such that $T$ may be organised in disjoint clusters

each $\Lambda_n$ having at most $N_0$ elements, and

This is because of the fact that $u_\alpha$ has a finite number of turning points and that the convergence of $u$ to its limits implies the convergence of the free boundaries in $C^1$ norms. We claim that $(x_n^1, y_n^1)$ and $(x_n^2, y_n^2)$ cannot be in two consecutive clusters, because of the orientation of $\mathcal{C}_\alpha$. So, if $\Lambda_n^i$ is the cluster of $(x_n^i, y_n^i)$, let $\Lambda^3_n$ be a cluster in-between. Let $(x_n^3, y_n^3)$ be the leftmost point of $\Lambda_n^3$. From the definition of $\Lambda_n^3$, there is $R_n\to+\infty$ such that $(x_n^3, y_n^3)$ is the leftmost point of $\Gamma$ in $B_{R_n}(x_n^3, y_n^3)$. Let $u_\infty(x, y)$ be a limit of translations of $u$ with the sequence of points $(x_n^3, y_n^3)$, it does not converge to any of the functions prescribed by Theorem 3.1, which is a contradiction. This finishes the proof of Theorem 1.2.

4.   The model with two species

The first thing one must understand is (1.2) in the truncated cylinder $\Sigma^L$, namely

4.1.   Exponential solutions

System (4.1), linearised around 0, reads

The solutions that decay to 0 as $x\to-\infty$ are looked for under the form

so that the exponents $\alpha^L_D$ and $\beta^L_D$ satisfy

Once again we consider types of limits.

Case 1. The limit $D\gg L\gg1$, $c$ bounded. We expect $\beta_D^L$ to go to 0 as $D\to+\infty$, so that

Then (4.3) yields estimates of the form

And, once again, the estimate may be pushed up to $c = o(\sqrt D)$.

Case 2. The limit $D\gg L\gg1$, $c\gg\sqrt D$. This time we expect $\beta_D^L$ to go to infinity as $D\to+\infty$, so that

We have

Case 3. The limit $L\gg D\gg1$, $c\gg1$. We expect that $L\beta_D^L$ will go to infinity, so that $\frac{d\beta\mu\mathrm{ch}(\beta L)}{\mathrm{sh}(\beta L)+d\beta\mathrm{ch}(\beta L)}\to 1.$ In this setting, we have the estimates (4.5).

4.2.   Estimates on the velocity

Let $(c_D^L, \Gamma_D^L, v_D^L)$ a solution to (4.1), we may infer its existence from a once again slight modification of Theorem 1.1 of ^[6]. We assume that it meets the line $\{y = 0\}$ at the point $(0, 0)$.

Theorem 4.1. There is a constant $K > 0$, independent of $L$ and $D$, such that

Moreover there is $L_0 > 0$ such that we have

uniformly in $L\geq L_0$.

Proof. That of (4.6) is similar to that of Theorems 2.1, one compares $(u_D^L, v_D^L)$ to

As for (4.7), the only point to prove is that, under the assumption that $(c_D^{L_0})_D$ is bounded, the width $L_0$ being large but fixed, then the leftmost point of $\Gamma_D^L$, denoted by $(\bar x_D^{L_0}, \bar y_D^{L_0})$ satisfies

Once this is proved, the proof of the theorem proceeds much as that of Theorem 2.2, using in particular estimates (4.4) for the linear exponentials. So, assume (4.8) to be false, that. is, there are two diverging positive sequences $(D_n)_n$ and $(x_n)_n$, and a positive constant $d_0$ such that

By nondegeneracy, there is $\gamma > 0$ such that

Choose another sequence $(x_n')_n$ such that

Then we have

just by integrating the ODE for $u_{D_n}^{L_0}$. But then, the Robin condition $\partial_y v_{D_n}^{L_0}+v_{D_n}^{L_0} = o(1)$ on $(-x_n', -x_n)\times\{0\}$, together with (4.9) and the Hopf Lemma, yields the existence of $\gamma' > 0$ such that

The equation for $u_{D_n}^{L_0}$ becomes

as soon as $n$ is large enough. This implies $u_{D_n}{L_0}(-x_n') < 0,$ contradiction.

From then on, the rest of the study of the two species model parallels exactly that of the one species model.

5.   Exponential convergence

In this section, we assume that all the requirements on the coefficients are fulfilled, and we drop the index ${}_D$ for the velocity $c$, the free boundary $\Gamma$ and the solution $u$. Theorem 1.4 is proved by deriving a differential ineqaulity for $\varphi$, exploiting Theorem 1.3 and the fact that $\varphi'$ as a limit at infinity. This allows indeed to write the free boundary problem for $u$ in a suitable perturbative form, and translate the double Dirichlet and Neumann boundary condition into the sought for differential inequality.

In the whole section, the considerations will be rigorously identical for the one species model or the two species model, except the global estimate in Proposition 5.2 below, where the computations are slightly different - but left to the reader. Thus we will concentrate on the one species model (1.1).

Recall from ^[1] that $\Gamma$ is an analytic graph. Theorem 1.3 readily implies

Proposition 5.1. We have

This entails the following improvement of Proposition 3.4.

Proposition 5.2. There is $\rho > 0$ such that, if $u(x, y) > 0$ we have

Proof. For $\varepsilon > 0$ consider $\psi_ \varepsilon(y)$ smooth whose derivative $\psi_ \varepsilon'$ satisfies

Note that this function exists due to Theorem 1.3 and Proposition 5.1. For $\rho > 0$ consider

it can be estimated by $e^{-\delta_ \varepsilon\rho\mathrm{dist}((x, y), \Gamma)}$ for a suitable $\delta_ \varepsilon > 0$, because of Theorem 1.3 again. In the lower half plane we have

On the line we have, on the same pattern:

And so, as soon as $\rho\in(0, \frac{c}D)$ and $\varepsilon > 0$ is small enough, the function $\underline u: = (1-\bar u)^+$ is a sub-solution to the equations for $u$ in the region $\{u > 0\}$, moreover $\partial\{\bar u > 0\}$ coincides $\Gamma$ sufficiently far in the lower half plane. The maximum principle implies the proposition.

Proof of Theorem 1.4. From now on, translate the origin so that we are in the following situation: $\Gamma\cap \mathbb{R}^2_-$ is the graph

with $\phi'(x) < 0$ and $\lim\limits_{x\to+\infty}\phi'(x) = -\mathrm{tan}\alpha.$ Actually, the translation may be adjusted so that $\phi'(x)$ is as colse as we wish to $-\mathrm{tan}\alpha$, this will be quantified later. The following chain of transformations is then made.

$1).$ Rotate the coordinates by the angle $\alpha$, so as to obtain the new set $(x, Y)$ given by

In this new system the free boundary may be written as $Y = \psi(X),$ with, by Proposition 5.1:

$2).$ Straighten the free boundary by setting

In this new coordinate system, $u$ solves the over-determined problem

$3).$ A standard compactness/uniqueness argument shows that

uniformly in $Y' < 0$. As we will not need any additional change of coordinates, let us, for notational simplicity, revert to the initial notation

The function $u(x, y)$ is this looked for under a perturbation of $\phi_0(y)$: $u(x, y) = \phi_0(y)+\tilde u(x, y)$. The free boundary condition writes

The full PDE for $\tilde u$ will not be written, as we need another change of unknowns.

$4).$ In order to transform (5.1) into an over-determined problem with fixed boundary, we look for $\tilde u$ under the form

with $\gamma_0$ smooth, compactly supported, $\gamma(0) = 0$, $\gamma'(0) = 1$. As $\psi$ appears in the equations only in the form of $\psi'$, we set $h(x) = \psi'(x)$.

The system for $(h, v)$ is thus

the function ${\mathcal R}[h](x, y)$ being:

While the expression of ${\mathcal R}[h](x, y)$ is utterly unpleasant, its structure is quite simple: It is quadratic in $h$ and its derivatives up to order 2, which are known to vanish at infinity. Finally, set

By elliptic regularity, all derivatives of $v$ go to 0 as $x\to+\infty$, uniformly in $y$. Problem (5.3) now reads

We will use the estimate

with $\theta(x, y)$ a positive function that tends to 0, uniformly in $y$, as $x\to+\infty$. It only remains to apply Proposition A1 of the appendix together with (5.4)), with the data

From Proposition 5.2, estimate (A1) holds, so that the function $h$ solves the differential inequality

the function $\tilde \theta(x)$ going to 0 as $x\to+\infty$. This implies the exponential estimate.

Acknowledgments

L. A. Caffarelli is supported by NSF grant DMS-1160802. The research of J.-M. Roquejoffre has received funding from the ERC under the European Union's Seventh Frame work Programme (FP/2007-2013) / ERC Grant Agreement 321186 - ReaDi. He also acknowledges a long term visit at UT Austin during the academic year 2018-19, made possible by a delegation within the CNRS for that year, and a J. T. Oden fellowship.

Conflict of interest

All authors declare no conflicts of interest in this paper.

Appendix: Compatibility relations for over-determined equations with right handside

The goal of this section is to prove that the right handside of the solution of an elliptic equation in the lower half plane, both with Dirichlet and Neumann condition, has to satisfy an integral equation.

Proposition A1. Consider $f(x)$ and $g(x, y)$ two smooth, bounded functions defined on $\mathbb{R}_+$ (resp. $\mathbb{R}_+\times \mathbb{R}_-$). Set $e_0(y) = e^{c_0y}$. Assume the existence of $\rho > 0$ such that

Let $u(x, y)$ solve

in the classical sense. Also assume that $u$ and its derivatives satisfy the estimate (A1). There is a continuous function $K(x, y)$, defined on $[2, +\infty)\times \mathbb{R}_-$, such that, for all $\varepsilon > 0$ we have

and such that

Proof. First, transform (A2) into a problem on the whole line by setting

where $\gamma$ is a smooth function, supported on $(1, +\infty)$, and equal to 1 on $[2, +\infty)$. the equation for $v$ is thus

with

In other words, the right handside of (A3) equals that of (A2) as soon as $x\geq2$.

A second observation is that the general Dirichlet problem

is well-posed as soon as the Dirichlet datum $u_0(y)$ and the right handside $F(x, y)$ satisfy the estimate (A1), with any $\rho < c_0$. Indeed, the classical change of unknown $u(x, y) = e^{\rho y}v(x, y)$ changes (A3) into an elliptic equation the the zero order term $\rho c_0-\rho^2$. Thus it is enough to assume tht $g$ and $k$ are compactly supported in $y$, in order to perform Fourier transforms. An easy density argument then allows to treat general data satisfying (A1).

Set $w(x, y) = e^{(c\cos\alpha x+c_0y)/2}v(x, y)$, the equation for $w$ is now

The new data are

and we have set

Thanks to both Dirichlet and Neumann conditions, the function $w(x, y)$ may be extended evenly over the whole plane $\mathbb{R}^2$, where it solves the same equation as (A4), with $G$ and $E_0$ also extended evenly. The new unknown and data are still denoted by $w$, $E_0$ and $G$. Let $\hat w(\xi, \zeta)$ be the Fourier transform of $w$, as well as $\hat F(\xi)$, $\hat E_0(\zeta)$ and $\hat G(\xi, \zeta)$ the Fourier transforms of the data. Thus we have

Now, the Dirichlet condition for $w$ entails $\int_{ \mathbb{R}}\hat w(\xi, \zeta)d\zeta = 0$, that is

From Plancherel's Theorem, we have

and, if we still denote by $\hat G(\xi, y)$ the partial Fourier transform of $G$ with respect to $x$, we have, again by Plancherel's theorem

Inverting the Fourier transform in $x$ yields, by the residue theorem:

Shifting the integration line in $\xi$ from $\mathbb{R}$ to $\mathbb{R}+i\rho$ yields, for $x > 0$:

and $\sqrt{\beta^2+\xi^2-\rho^2+2i\rho\xi}$ has nonzero real part. This entails the proposition.