A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton-Jacobi equations

TL;DR

This paper proves a Liouville-type theorem in a half-space and applies it to analyze boundary gradient blow-up in superquadratic diffusive Hamilton-Jacobi equations, revealing universal blow-up profiles and boundary condition loss.

Contribution

It establishes a one-dimensional symmetry result for elliptic solutions in a half-space and applies this to understand boundary blow-up behavior in higher dimensions.

Findings

Solutions near the boundary exhibit a universal blow-up profile in the normal direction.

The tangential derivatives dominate the normal derivatives at blow-up points.

Viscosity solutions generally lose boundary conditions after blow-up.

Abstract

We consider the elliptic and parabolic superquadratic diffusive Hamilton-Jacobi equations with homogeneous Dirichlet conditions. For the elliptic problem in a half-space, we prove a Liouville-type classification, or symmetry result, which asserts that any solution has to be one-dimensional. This turns out to be an efficient tool to study the behavior of boundary gradient blow-up (GBU) for the parabolic problem in general bounded domains. Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior, with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions {\it generically} lose boundary conditions after GBU. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.

Equations

$$\{\begin{array}{ll}-\Delta u=|\nabla u{|}^p,& x\in {\mathbb{R}}_{+}^n,\\ u(\tilde{x},0)=0,& \tilde{x}\in {\mathbb{R}}^{n-1},\end{array}$$

$$\{\begin{array}{ll}{u}_t-\Delta u=|\nabla u{|}^p,& \text{mbox}in\Omega \times (0,\infty ),\\ u=0,& \text{mbox}on\partial \Omega \times (0,\infty ),\\ u(\cdot ,0)=u_0,& \text{mbox}in\Omega ,\end{array}$$

$$\{\begin{array}{ll}-\Delta u=|\nabla u{|}^p+f(x),& \text{mbox}in\Omega ,\\ u=0,& \text{mbox}on\partial \Omega .\end{array}$$

$$U_0(s)=c_p{s}^{1-\beta },s\ge 0,with{c}_p=\frac{p-1}{p-2}(p-1{)}^{-1/(p-1)}=\frac{{\beta }^{\beta }}{1-\beta }.$$

$$\beta =\frac{1}{p-1}.$$

$$U_0^{\prime }(s)=d_p{s}^{-\beta },s>0,with{d}_p={\beta }^{\beta }=(1-\beta )c_p,$$

$$X:=\{\varphi \in C^1(\overline{\Omega });{\varphi }_{|\partial \Omega }=0\}.$$

$$\Vert u(\cdot ,t){\Vert }_{\infty }\le \Vert u_0{\Vert }_{\infty },0<t<T,$$

$$T<\infty ⟹\underset{t\to T^{-}}{\lim }\Vert \nabla u(\cdot ,t){\Vert }_{\infty }=\infty .$$

$$GBUS(u_{0})=\Bigl{\{}x_{0}\in\overline{\Omega};\,\limsup_{t\to T^{-},\,x\to x_{0}}|\nabla u(x,t)|=\infty\Bigr{\}}$$

$$GBUS(u_0)\subset \partial \Omega .$$

$$|\nabla u(x,t)|\le C(n,p){\delta }^{-\beta }(x)+C(u_0)in\; \Omega \times [0,T),$$

$$\delta (x)=\mathrm{dist}(x,\partial \Omega ),x\in \Omega ,$$

$$-C(u_0)\delta (x)\le u(x,t)\le C(n,p){\delta }^{1-\beta }(x)+C(u_0)\delta (x)in\; \Omega \times [0,T)$$

$$|\nabla u(x,t)|\le (1+\epsilon )d_p{\delta }^{-\beta }(x)+Cin\; \Omega \times [0,T],$$

$$u(x,t)\le (1+\epsilon )c_p{\delta }^{1-\beta }(x)+C\delta (x)in\; \Omega \times [0,T],$$

$${\Omega }_{\epsilon }=\{x\in \Omega ;\delta (x)<\epsilon \},\epsilon >0.$$

$$x\in {\Omega }_{{\delta }_0}$$

$$\nu (x)={\nu }_{P(x)},x=P(x)+\delta (x)\nu (x).$$

$$u_{\xi }(x,t)=\nabla u(x,t)\cdot \xi (x),u_{\xi \zeta }(x,t)=\xi (x)D^{2}u(x,t)\zeta (x).$$

$$\bigl{|}u_{\nu\nu}+|u_{\nu}|^{p}\bigr{|}\leq\varepsilon|u_{\nu}|^{p}+C_{\varepsilon}\quad\hbox{ in $\Omega_{\delta_{0}}\times[T/2,T]$}$$

$$|u_{\tau \tau }|+|u_{\nu \tau }|+|u_{\tau }{|}^p\le \epsilon |u_{\nu }{|}^p+C_{\epsilon }in{\Omega }_{{\delta }_0}\times [T/2,T],$$

$$\underset{{\Omega }_{{\delta }_0}\times (0,T)}{\inf }{u}_{\nu }>-\infty .$$

$$\underset{s\to 0}{\lim }{s}^{\beta }\nabla u(a+s{\nu }_a,T)=d_p{\nu }_a.$$

$$\underset{x\to a,x\in \partial \Omega }{\lim }|x-a{|}^{\beta }{u}_{\nu }(x,T)=\infty .$$

$$\underset{t\to T^{-},x\to a}{\lim }{u}_{\nu }=\infty .$$

$$\underset{t\to T^{-},x\to a}{\lim }\frac{u_{\nu \nu }}{u_{\nu }^p}=-1$$

$$\ \Bigl{[}u_{\nu}^{1-p}(P(x),t)+(1+\varepsilon)(p-1)\delta(x)\Bigr{]}^{-\beta}\leq u_{\nu}(x,t)\leq\Bigl{[}u_{\nu}^{1-p}(P(x),t)+(1-\varepsilon)(p-1)\delta(x)\Bigr{]}^{-\beta}$$

$$\bigl{|}u_{\nu}^{1-p}(z,t)-u_{\nu}^{1-p}(a,t)\bigr{|}\leq o(|z-a|)\quad\hbox{ as $t\to T$ and $z\to a$, with $z\in\partial\Omega$.}$$

$$u_{\nu \nu }\sim -|u_{\nu }{|}^p,$$

Full text

Namely, we show that in a neighborhood of the boundary, at leading order, solutions display a global ODE type behavior of the form $u_{\nu\nu}\sim-u_{\nu}^{p}$, with domination of the normal derivatives upon the tangential derivatives. This leads to the existence of a universal, sharp blow-up profile in the normal direction at any GBU point, and moreover implies that the behavior in the tangential direction is more singular. A description of the space-time profile is also obtained. The ODE type behavior and its connection with the Liouville-type theorem can be considered as an analogue of the well-known results of Merle and Zaag [31] for the subcritical semilinear heat equation, with the significant difference that for the latter, the ODE behavior is in the time direction (instead of the normal spatial direction).

On the other hand, it is known that any GBU solution admits a weak continuation, under the form of a global viscosity solution. As another consequence, we show that these viscosity solutions generically lose boundary conditions after GBU. Namely, solutions without loss of boundary conditions after GBU are exceptional and can be characterized as thresholds between global classical solutions and GBU solutions which lose boundary conditions. This result, as well as the above GBU profile, were up to now essentially known only in one space-dimension.

Finally, in the case of elliptic Dirichlet problems, we deduce from our Liouville theorem an optimal Bernstein-type estimate, which gives a partial improvement of a local estimate of P.-L. Lions [28].

Keywords. Diffusive Hamilton-Jacobi equations, Liouville-type theorem, gradient blow-up, final profile, loss of boundary conditions, Bernstein-type estimates.

1. Introduction and main results

In this paper we consider superquadratic diffusive Hamilton-Jacobi equations, in both elliptic and parabolic settings. Namely our goal is two-fold:

• (i)

to establish a Liouville-type classification theorem for the elliptic problem in a half-space:

[TABLE]

where $\mathbb{R}^{n}_{+}=\{(\tilde{x},x_{n})=(x_{1},\dots,x_{n})\in\mathbb{R}^{n},\,\,x_{n}>0\}$ and $p>2$;

• (ii)

to derive a number of applications of our Liouville-type theorem to the initial-boundary value problem:

[TABLE]

as well as for the inhomogeneous Dirichlet problem:

[TABLE]

Throughout this article, $\Omega$ is a bounded domain of $\mathbb{R}^{n}$ with boundary of class $C^{2+\mu}$ for some $\mu\in(0,1)$. Let us denote by $\nu_{x}$ the inward unit normal vector at any $x\in\partial\Omega$.

Let us begin with our Liouville-type classification, or symmetry, result. It asserts that any solution in a half-space is one-dimensional.

Theorem 1.1**.**

Let $p>2$ and let $u\in C^{2}(\mathbb{R}^{n}_{+})\cap C(\overline{\mathbb{R}^{n}_{+}})$ be a solution of (1.1). Then $u$ depends only on the variable $x_{n}$.

As a consequence of Theorem 1.1 and straightforward ODE analysis, any solution of (1.2) is thus given by either $u=0$ or $u=U_{\alpha}(x_{n}):=U_{0}(\alpha+x_{n})-U_{0}(\alpha)$ for some $\alpha\geq 0$, where

[TABLE]

Here and in the rest of the paper we define

[TABLE]

For future reference we also write

[TABLE]

and we note that all solutions $U_{\alpha}$ for $\alpha>0$ are $C^{1}$, whereas $U^{\prime}_{0}$ is singular at $s=0$ and $U_{0}$ displays the key Hölder exponent $1-\beta$. We stress that Theorem 1.1 does not assume $C^{1}$ regularity at the boundary for $u$, and that this feature will be crucial in our applications. We do not make any a priori assumption on the behavior of $u$ at infinity either.

Remark 1.1**.**

(a) For the whole space case, it was proved in [28] that any classical solution of $-\Delta u=|\nabla u|^{p}$ in $\mathbb{R}^{n}$ with $p>1$ has to be constant. For the half-space problem (1.1) in the subquadratic case $p\in(1,2]$, a result similar to Theorem 1.1 was proved in [37]. Our proof is based on a moving planes technique, combined with Bernstein type estimates from [28] and a compactness argument. It is rather different from the proof in [37], which relies on the existence of a finite limit as $x_{n}\to\infty$, a property which does not hold in the superquadratic case.

Let us finally recall that Bernstein type gradient estimates go back to the early work [9] and that the technique was further developed in important papers such as [24], [4], [40], [29], [28].

(b) The Liouville-type theorem in [37] was motivated by the study of the so-called “large solutions” of elliptic equations with gradient terms, initiated in [25] in the framework of stochastic control problems with state constraints (see also, e.g., [6], [26], [27], [2], [16] and the references therein). As for the question of one-dimensional symmetry of solutions of elliptic equations in a half-space, it has also attracted much attention, especially for equations of the form $-\Delta u=f(u)$, see, e.g., [8], [15], [12], [14] and the references therein.**

Let us turn to the applications of Theorem 1.1 to the study of the parabolic problem (1.2). Problem (1.2) is locally well-posed for all $u_{0}\in X$, with

[TABLE]

Denoting by $T=T(u_{0})$ the existence time of the unique maximal classical solution $u$ of (1.2), it is known that

[TABLE]

as a consequence of the maximum principle, and that

[TABLE]

This is called gradient blow-up (GBU) and it is also known that $T<\infty$ whenever $u_{0}$ is suitably large, whereas solutions exist globally and decay to [math] if $\|u_{0}\|_{C^{1}}$ is sufficiently small (see e.g. [3], [1], [41], [23]). The singular set, or GBU set, of $u$ is defined by

[TABLE]

and the elements of $GBUS(u_{0})$ are called GBU points. It is known [42] that

[TABLE]

More precisely, we have the following upper bound of Bernstein type:

[TABLE]

where

[TABLE]

is the distance to the boundary (see [42] and cf. [28] in the elliptic case). This also implies

[TABLE]

(the upper estimate follows by integrating (1.6) in the normal direction; the lower estimate is immediate since $u$ is a supersolution of the heat equation). In view of (1.6) and parabolic estimates, the solution $u$, which primarily belongs to $C^{2,1}(\overline{\Omega}\times(0,T))\cap C^{1,0}(\overline{\Omega}\times[0,T))$, can be extended to a function $u\in C^{2,1}(\Omega\times(0,T])\cap C(\overline{\Omega}\times[0,T])$.

As a first consequence of Theorem 1.1, we have the following optimal Bernstein-type upper estimate, which improves (1.6). Actually, the optimality will follow from Theorem 1.4.

Theorem 1.2**.**

Let $p>2$ and let $u_{0}\in X$ be such that $T(u_{0})<\infty$. For any $\varepsilon>0$ there exists $C=C(\varepsilon,u_{0})>0$ such that

[TABLE]

[TABLE]

where the constants $c_{p},d_{p}$ are given by (1.4), (1.5).

In view of the next statements, let us introduce some notation. Set

[TABLE]

Recall that, thanks to the regularity of $\Omega$, there exists $\delta_{0}>0$ such that

[TABLE]

The point $P(x)$ is the projection of $x$ onto $\partial\Omega$. In this way, we can in particular extend the normal vector field to the neighborhood $\Omega_{\delta_{0}}$ of the boundary, by setting, for all $x\in\Omega_{\delta_{0}}$:

[TABLE]

Also, for any unit vector fields $\xi,\zeta$, we denote the corresponding first and second order derivatives in space of $u$ by

[TABLE]

The following theorem describes a global behavior of the normal derivatives, and their dominance with respect to the tangential derivatives, for any GBU solution of (1.2).

Theorem 1.3**.**

Let $p>2$ and let $u_{0}\in X$ be such that $T(u_{0})<\infty$. For any $\varepsilon>0$ there exists $C_{\varepsilon}>0$ (possibly depending on $u_{0}$) such that

[TABLE]

and

[TABLE]

where $\tau$ is any tangential vector field (i.e. $\tau\perp\nu$ and $|\tau|=1$ in $\Omega_{\delta_{0}}$). Moreover, we have

[TABLE]

Theorem 1.3 has interesting consequences on the behavior near a GBU point.

Theorem 1.4**.**

Let $p>2$ and let $u_{0}\in X$ be such that $T(u_{0})<\infty$. For each GBU point $a\in\partial\Omega$, we have the following properties:

(i) (Universal final blow-up profile in the normal direction)

[TABLE]

(ii) (More singular behavior in the tangential direction)

[TABLE]

(iii) (Continuity of $u_{\nu}$ with values in $[0,\infty]$) As $t\to T$ and $x\to a$, the normal derivative $u_{\nu}$ (and hence $|\nabla u|$) blows up in the strong sense:

[TABLE]

(iv) (Space-time behavior) We have

[TABLE]

and consequently, for each $\varepsilon\in(0,1)$ there exists $\eta\in(0,\delta_{0})$ such that

[TABLE]

for all $(x,t)\in B_{\eta}(a)\times(T-\eta,T)$. Moreover, we have

[TABLE]

Theorems 1.3 and 1.4 show that any GBU solution follows a global ODE-like behavior in the normal direction. More precisely, in the singular region, the dominating terms in the PDE $u_{t}-\Delta u=|\nabla u|^{p}$ are the normal derivatives $u_{\nu}$ and $u_{\nu\nu}$, with

[TABLE]

whereas all other derivatives are of lower order, as illustrated by the following scheme (in two space dimensions):

[TABLE]

For the tangential parts of the gradient and of the Laplacian, this is stated in (1.12). As for the time-derivative, we actually have the well-known uniform bound

[TABLE]

(this follows from the maximum principle applied to $u_{t}$, see e.g. [42]). We stress that the value $u_{\nu}(x,T)=\infty$ in (1.14) is allowed, so that the statement applies to both isolated or non-isolated GBU points.

We next turn to the post blow-up behavior. It is known [7] that problem (1.2) admits a unique, continuous global viscosity solution $\bar{u}\in C(\overline{\Omega}\times[0,\infty))$, which extends the maximal classical solution after $t=T$. It is actually a classical solution in $\Omega\times(0,\infty)$, namely, $\bar{u}\in C^{2,1}(\Omega\times(0,\infty))$, but the homogeneous boundary conditions have to be understood in the generalized viscosity sense (or state constraints) and need not be satisfied in the usual sense. The solution $\bar{u}$ can also be obtained by monotone approximation of problems with truncated nonlinearities (see [38], [36] and the references therein). Moreover, as shown in [35], [39], the global weak solution $\bar{u}$ may lose boundary conditions after gradient blow-up, i.e.

[TABLE]

However, it was shown in [35] that there are also solutions which never lose the boundary conditions after gradient blow-up. In particular, for any nontrivial $\phi\in X_{+}$, where $X_{+}=\{\phi\in X;\,\phi\geq 0\}$, it is shown in [35] that

[TABLE]

that $T(\lambda^{*}\phi)<\infty$ and that the corresponding solution does not lose boundary conditions. In one space dimension, it was moreover proved in [35] that solutions without loss of conditions are exceptional: they constitute thresholds between global classical solutions and GBU solutions with loss of boundary conditions. As a consequence of Theorems 1.2 and 1.4, we can show that this threshold property remains true in any space dimension.

Theorem 1.5**.**

Let $p>2$ and let $u_{0},v_{0}\in X_{+}$ be such that $T(u_{0})<\infty$ and $v_{0}\not\equiv u_{0}$. Denote by $\bar{u},\bar{v}$ the corresponding unique global viscosity solutions of (1.2).

• (i)

If $v_{0}\leq u_{0}$ and no loss of boundary conditions occurs for $\bar{u}$, then $T(v_{0})=\infty$.

• (ii)

If $v_{0}\geq u_{0}$, then $T(v_{0})<T(u_{0})<\infty$ and $\bar{v}$ loses boundary conditions before $T(u_{0})$, i.e., there exists $t\in(T(v_{0}),T(u_{0}))$ such that

[TABLE]

See [38], [35], [36] for further results on the behavior of the viscosity solution $\bar{u}$ for $t\geq T$. We refer to [18], [17] and the references therein for results on the continuation after GBU for some other one-dimensional parabolic problems.

Let us finally briefly consider the inhomogeneous elliptic problem (1.3). It is shown in [28, Théorème IV.1] that any (local) solution $u\in C^{2}(\Omega)$ of

[TABLE]

with $f\in W^{1,\infty}(\Omega)$, satisfies the Bernstein-type estimate

[TABLE]

As a consequence of Theorem 1.1, we obtain the following optimal estimate, which is a partial improvement of the result in [28] for the case when the boundary value problem (1.3) is considered instead of the local equation (1.20).

Theorem 1.6**.**

Let $p>2$ and $f\in W^{1,\infty}(\Omega)$ with $\|f\|_{W^{1,\infty}}\leq M$. Assume that $u\in C^{2}(\Omega)\cap C(\overline{\Omega})$ is a solution of (1.3). Then for every $\varepsilon>0$ there exists $C=C(\varepsilon,M)>0$ such that

[TABLE]

Remark 1.2**.**

As far as we know, this paper provides the first study of the spatial GBU behavior and final profiles valid for general solutions of (1.2) in all space dimensions. Previously, the behavior was known only in the one-dimensional case, see [10], [5], [21], or for domains with some symmetry [34].

Theorems 1.1 and 1.3 can be seen as the analogues of the well-known results of Merle and Zaag [31] (see also [32]) concerning the subcritical nonlinear heat equation

[TABLE]

with $1<p<(n+2)/(n-2)_{+}$. As a key difference, the ODE behavior in (1.22) is in the time direction for $u$, whereas the ODE behavior in (1.2) is in the spatial normal direction for $\nabla u$. Namely, the Liouville-type theorem in [31] states that any ancient solution of (1.22) with self-similar temporal decay at $-\infty$ must depend on the time-variable only. This is then used to show that blow-up solutions of (1.22) satisfy

[TABLE]

(see also [33], [19], [22] for related results based on the Liouville theorem in [31]).

The proof of Theorem 1.3 relies on Theorem 1.1, combined with suitable rescaling and compactness arguments. It follows the general strategy of [31] (see also [22]), but with notable differences. First, whereas the solutions of (1.22) considered in [31] blow up only at interior points of the domain $\Omega$ (as a consequence of a convexity assumption on $\Omega$), GBU for (1.2) occurs at the boundary. Due to this, we have to deal with rather delicate boundary estimates in our rescaling procedures and in the preliminary nondegeneracy properties, relying in particular on flow coordinates (cf. (1.10)). Moreover, the nondegeneracy properties require different arguments from those in [31], due to the lack of variational structure of problem (1.2). Also, instead of using type I temporal estimates from [20] for (1.22) as basic a priori estimates, we rely on the spatial Bernstein type estimate (1.6).

As another qualitative difference with [31], we note that our results on the parabolic problem (1.2) are derived from an elliptic Liouville-type theorem. This is allowed by the above mentioned bound (1.19) on $u_{t}$ in (1.2), so that the time derivative vanishes in rescaling limits. Let us stress that the apparently simplifying mechanisms (1.11)-(1.12) and (1.19) are far from making the dynamics of the equation trivial. Indeed, they are not sufficient to provide complete information on the transition (in space and/or in time) between the singular and regular parts of the solution and on the corresponding transition speeds (time rate of GBU and tangential space profile near an isolated GBU point of the boundary). These questions are delicate; see Remark 1.3(b) for results in that direction.**

Remark 1.3**.**

(a) Ancient solutions of (1.2) in $\mathbb{R}^{n}\times(-\infty,0)$ have been studied in [42]. The half-space case $\mathbb{R}^{n}_{+}\times(-\infty,0)$, which is a topic of possible independent interest, will be studied in a forthcoming paper.

(b) As mentioned above, Theorem 1.3 is not sufficient to determine the sharp final GBU profile in the tangential direction near isolated GBU points and we suspect that the latter is not universal but that various tangential profiles may exist, depending on the solution. However, Theorem 1.4(ii) shows that the profile is always anisotropic: it is more singular in the tangential direction than in the normal one.

In some very special cases, more precise information on the final GBU profile in the tangential direction can be found in [34]. Namely, for $n=2$ and $p\in(2,3]$, under suitable symmetry assumptions on the domain $\Omega$ and initial data $u_{0}$, and assuming that $\Omega$ coincides with the half-plane $\{(x,y);\,y>0\}$ near the origin, we have single-point GBU at the origin, with the final profile

[TABLE]

For results on the GBU set, especially sufficient conditions ensuring single-point GBU, see [30], [13]. As for the time rate $\|\nabla u(\cdot,t)\|_{\infty}$ of GBU, it remains an open problem in general, and so is the time behavior of $u_{\nu}(a,t)$ – cf. (1.18). Nevertheless the rate is known to be always non self-similar, unlike for (1.22), and of type II, with a lower bound $\|\nabla u(\cdot,t)\|_{\infty}\geq C(T-t)^{-1/(p-2)}$; see [10], [21], [43], [36] for results on the GBU rate.

(c) Similar to (1.12), we also have the same bound for the mixed tangential derivative:

[TABLE]

where $\tau,\hat{\tau}$ are any vector fields such that $(\nu,\tau,\hat{\tau})$ is orthonormal in $\Omega_{\delta_{0}}$ (see the proof of Theorem 1.3). **

The outline of the rest of the paper is as follows. In Section 2 we establish Theorem 1.1. In Sections 3 and 4, we use Theorem 1.1 to prove Theorems 1.2 and 1.6, respectively. Theorem 1.3 is next proved in Section 5. In Section 6 we deduce Theorem 1.4 from Theorem 1.3, and Theorem 1.5 from Theorems 1.2 and 1.4. Finally, in the appendix, we provide the proof of Proposition 5.1, a technical result which is used in the proof of Theorem 1.3.

2. Proof of Theorem 1.1

It is based on a moving planes argument combined with Bernstein type estimates.

Proof.

Write $x=(\tilde{x},y)\in\mathbb{R}^{n-1}\times[0,\infty)$ and fix any $h\in\mathbb{R}^{n-1}\setminus\{0\}$. Let

[TABLE]

It suffices to show that $v\equiv 0$. Assume for contradiction that

[TABLE]

(the case $\inf_{\mathbb{R}^{n}_{+}}v<0$ is similar). By the Bernstein estimate in [28], we have

[TABLE]

(more precisely, we apply Case 2 of Theorem IV.1 and Remark p. 250 in [28], with $C_{1}=C_{2}=0$, to the function $-u$). It follows that

[TABLE]

Hence $|v|\leq\sigma/2$ for $y\geq A$ large. Therefore

[TABLE]

On the other hand, $v$ satisfies the equation

[TABLE]

where

[TABLE]

Observe that, as a consequence of (2.2), the function $a$ is bounded for $y$ bounded away from [math], hence in particular on compact subsets of $\mathbb{R}^{n}_{+}$. By the strong maximum principle applied to (2.4), it follows that the solution $v$ cannot achieve any local maximum in $\mathbb{R}^{n}_{+}$. Otherwise $v$ would be constant, and we assumed the contrary in view of (2.1) and the fact that $v(\tilde{x},0)=0$. In particular, $\sigma$ in (2.3) is not attained and there exists a sequence $(\tilde{x}_{j},y_{j})\in\mathbb{R}^{n-1}\times(0,A)$ with $|\tilde{x}_{j}|\to\infty$ such that $v(\tilde{x}_{j},y_{j})\to\sigma$.

Next define

[TABLE]

and note that

[TABLE]

and

[TABLE]

Since $u_{j}$ is a solution of (1.2) in $\mathbb{R}^{n}_{+}$, it satisfies the (uniform) Bernstein estimate (2.2), namely:

[TABLE]

Owing to $u_{j}(\tilde{x},0)=0$, by integration in the $y$ direction, we also have

[TABLE]

and for all $j$. It then follows from interior elliptic estimates that $(u_{j})_{j}$ is relatively compact in $C^{2}_{\mathrm{loc}}(\mathbb{R}^{n}_{+})$. Therefore, some subsequence of $(u_{j})_{j}$ converges in that topology to a solution $U\in C^{2}(\mathbb{R}^{n}_{+})$ of $-\Delta U=|\nabla U|^{p}$. As a consequence of (2.7), we also have $U\in C(\overline{\mathbb{R}^{n}_{+}})$ and $U(\tilde{x},0)=0$. Moreover, we may assume that $y_{j}\to y_{\infty}\in[0,A)$ and we get

[TABLE]

owing to (2.6), which implies $y_{\infty}>0$.

Put now

[TABLE]

It follows from (2.5) and (2.8) that $\sigma=\sup_{\mathbb{R}^{n}_{+}}V=V(0,y_{\infty})$. But $V$ satisfies

[TABLE]

where

[TABLE]

is bounded on compact subsets of $\mathbb{R}^{n}_{+}$. This contradicts the strong maximum principle and completes the proof. ∎

3. Proof of Theorem 1.2

The proof of Theorem 1.2 is based on the Liouville-type Theorem 1.1 and on a suitable rescaling argument. By the same ideas, one also obtains the following proposition, which states that tangential derivatives are of lower order than normal derivatives in terms of the distance to the boundary. This will be an important preliminary step for the proof of Theorem 1.3 in Section 5.

Proposition 3.1**.**

Let $p>2$ and let $u_{0}\in X$ be such that $T(u_{0})<\infty$. For each $\varepsilon>0$ there exists a constant $C_{\varepsilon}>0$ such that, for any vector fields $\tau,\hat{\tau}$ such that $\tau,\hat{\tau}\perp\nu$ and $|\tau|=|\hat{\tau}|=1$ in $\Omega_{\delta_{0}}$, we have

[TABLE]

Proof of Theorem 1.2 and of Proposition 3.1. First recall that $u$ satisfies the following Bernstein-type estimates:

[TABLE]

for some constant $C>0$ depending only on $\Omega,p,u_{0}$; cf. (1.6)-(1.7).

Assume that either (1.8) or (3.1) fails. Then there exist $c>0$, a sequence of points $(x_{j},t_{j})$ in $\Omega_{\delta_{0}}\times[T/2,T)$ and unit vectors $\tau_{j},\hat{\tau}_{j}\perp\nu(x_{j})$ such that either

[TABLE]

or

[TABLE]

In view of (3.2)-(3.3) and of parabolic estimates, we have $\delta(x_{j})\to 0$. Set

[TABLE]

so that

[TABLE]

After extracting a subsequence, we have $z_{j}\to a\in\partial\Omega$, and we may assume without loss of generality that $a=0$ and $\nu_{0}=e_{n}$. Hence $x_{j}\to 0$, and

[TABLE]

We may also assume that

[TABLE]

Next we rescale $v$ by setting:

[TABLE]

where $\Omega_{j}:=\lambda_{j}^{-1}(\Omega-z_{j})$ and $\lambda_{j}=\delta(x_{j})\to 0$. We have

[TABLE]

Since $\beta+1=\beta p$, it follows that $v_{j}$ satisfies

[TABLE]

Also $\Omega_{j}$ converges to the half-space $\{y_{n}>0\}$ and $t_{j}\lambda_{j}^{-2}\geq T/(2\lambda_{j}^{2})\to\infty$, as $j\to\infty$.

We now proceed to prove a suitable local compactness property of the sequence $(v_{j})_{j}$ by making use of the Bernstein estimates (3.2)-(3.3). To this end we need to convert these estimates in terms of the variable $y$. Since $\nu_{0}=e_{n}$, we first note that

[TABLE]

Since $\partial\Omega_{j}=\lambda_{j}^{-1}(\partial\Omega-z_{j})$ and $z_{j}\in\partial\Omega$, $z_{j}\to 0$, $\lambda_{j}\to 0$, it follows from (3.8), applied with $x=z_{j}+\lambda_{j}\xi$ and $x^{\prime}=z_{j}$, that, for any $R>0$,

[TABLE]

Also, there exists $j_{0}(R)$ such that for any $j\geq j_{0}(R)$ and any $y\in\Omega_{j}\cap B_{R}$, the projection of $y$ onto $\partial\Omega_{j}$, denoted by $\bar{y}^{j}$, is well defined and satisfies

[TABLE]

Consequently, for any $y\in\Omega_{j}\cap B_{R}$, since ${\rm dist}(y,\partial\Omega_{j})\leq|y|\leq R$, we have

[TABLE]

Therefore, (3.9) and (3.10) give

[TABLE]

Fix any $R,\eta$, with $0<\eta<1<R$. By (3.8) and (3.11), there exists $j_{1}(R,\eta)$ such that

[TABLE]

and

[TABLE]

Since

[TABLE]

we deduce from (3.2), (3.3) and (3.13) that, for all $j\geq j_{1}(R,\eta)$ and all $(y,s)\in D_{R,\eta}\times(-t_{j}\lambda_{j}^{-2},0]$,

[TABLE]

and

[TABLE]

By interior parabolic estimates, it follows that the sequence $(v_{j})_{j}$ is precompact in $C^{2,1}(Q_{R,\eta})$, where $Q_{R,\eta}=D_{R,\eta}\times[-R,0]$. By a diagonal procedure, we deduce that some subsequence of $(v_{j})_{j}$, not relabeled, converges in each $C^{2,1}(Q_{R,\eta})$ to a classical solution $V(y,s)\in C^{2,1}(\mathbb{R}^{n}_{+}\times(-\infty,0])$ of

[TABLE]

On the other hand, (1.19) yields

[TABLE]

so that we actually have $V_{s}=0$, hence $V=V(y)$. Moreover, (3.16) guarantees that

[TABLE]

Consequently, $V$ extends to a function $V\in C(\overline{\mathbb{R}^{n}_{+}})$, with $V=0$ on $\partial\mathbb{R}^{n}_{+}$. It follows from Theorem 1.1 that either $V=0$ or $V(y)=V_{\alpha}(y_{n})$ for some $\alpha\geq 0$.

Now, in case (3.4) holds, we have

[TABLE]

We then deduce from (3.6) that

[TABLE]

But this contradicts the fact that either $V=0$ or \bigl{|}\nabla V(e_{n})\bigr{|}=V^{\prime}_{\alpha}(1)=d_{p}(\alpha+1)^{-\beta}\leq d_{p}. We thus conclude that (1.8) is true. Estimate (1.9) then follows by integration in the normal direction and this completes the proof of Theorem 1.2.

Finally, in case (3.5) holds, we have

[TABLE]

Hence, using (3.6), (3.7), we get

[TABLE]

This contradicts the fact that $V$ depends only on $y_{n}$ and completes the proof of Proposition 3.1. ∎

Remark 3.1**.**

Let $\bar{u}$ be the global viscosity solution of (1.2) (cf. the paragraph before Theorem 1.5) and assume that, for some $\tau\in(T,\infty]$,

[TABLE]

i.e., $\bar{u}$ does not lose boundary conditions for $t\leq\tau$. Then, for any $\varepsilon>0$, estimates (1.8) and (1.9) in Theorem 1.2 remain valid for $\bar{u}$ in $\Omega\times[0,\tau]$.

Indeed, we know that $\bar{u}$ satisfies $\sup_{\Omega\times[T/2,\infty)}|\bar{u}_{t}|<\infty$ and that the gradient estimate (3.2) remains valid for $\bar{u}$ in $\Omega\times[0,\infty)$. This follows respectively from [36, Lemma 10.1] and from [36, Theorem 3.1], applied with $F(\xi)=|\xi|^{p}$ and $\theta=(p-1)/p$. Now, by integrating (3.2) in the normal direction and using (3.17), we see that estimate (3.3) remains valid for $\bar{u}$ in $\Omega\times[0,\tau)$. The proof of Theorem 1.2 then applies without changes.**

4. Proof of Theorem 1.6

The proof is similar to that of Theorem 1.2, based on a rescaling argument, combined with the elliptic Bernstein estimates [28] and the Liouville-type Theorem 1.1.

Assume for contradiction that there exist $d>d_{p}$ and sequences $\{f_{j}\},\{u_{j}\},\{x_{j}\}$, with $\|f_{j}\|_{W^{1,\infty}}\leq M$, such that

[TABLE]

Let

[TABLE]

By extracting a subsequence, we may assume without loss of generality that $z_{j}\to 0\in\partial\Omega$, hence $x_{j}\to 0$, and

[TABLE]

Set

[TABLE]

We have

[TABLE]

Since $\beta+1=\beta p$, it follows that $v_{j}$ satisfies

[TABLE]

in $\Omega_{j}:=\lambda_{j}^{-1}(\Omega-z_{j})$, hence

[TABLE]

Also $\Omega_{j}$ converges to the half-space $\{y_{n}>0\}$ as $j\to\infty$.

On the other hand, it follows from the elliptic Bernstein estimate (1.21) that

[TABLE]

for some constant $C=C(n,p,M)>0$ independent of $j$. Setting $Q_{R,\varepsilon}=B_{R}\cap\{y_{n}>\varepsilon\}$, arguing as in the proof of Proposition 3.1 and using interior elliptic $L^{q}$ estimates, we can find a subsequence of $v_{j}$, not relabeled, which converges in $W^{2,q}(Q_{R,\varepsilon})$ for each $q,R,\varepsilon>0$ to a strong, hence classical, solution $V(y)\in C^{2}(\mathbb{R}^{n}_{+})\cap C(\overline{\mathbb{R}^{n}_{+}})$ of

[TABLE]

It follows from Theorem 1.1 that either $V=0$ or $V(y)=U_{\alpha}(y_{n})$ for some $\alpha\geq 0$. In particular, $|\nabla V(y)|\leq d_{p}y_{n}^{-\beta}$ by (1.5). Since, by (4.2),

[TABLE]

This contradicts (4.1), being $d>d_{p}$. ∎

5. Proof of Theorem 1.3

The proof of Theorem 1.3 relies on Proposition 3.1, along with two other preliminary results. The first one, Proposition 5.1, which is rather technical, is a nondegeneracy property for GBU points. It states that if the singularity of $u_{\nu}$ is sufficiently weak at small (but positive) distance from a given boundary point and in some time interval, then $|\nabla u|$ satisfies a uniform space-time bound near that point.

Proposition 5.1 is an improvement of [30, Lemma 2.2], where the weak singularity assumption had to be made in a whole space-time neighborhood of the boundary point (and not only at positive distance). This improvement is crucial to our arguments and is made possible by relying on estimate (3.1) from Proposition 3.1. The proof amounts to showing that the weak singularity will propagate from small finite distance up to the boundary. Moreover, in the course of the proof of Theorem 1.3, Proposition 5.1 will be applied to get uniform gradient estimates for a sequence of suitably renormalized versions of $u$ in rescaled domains. Thus we need a local statement in more general domains under precise regularity assumptions on the boundary.

Proposition 5.1**.**

Let $p>2$, let $\omega$ be a bounded domain of class $C^{2}$, $a\in\partial\omega$, $R>0$, and set $D=B_{2R}(a)\cap\omega$. Assume that

[TABLE]

[TABLE]

Set $L:=\|\Delta\delta\|_{L^{\infty}(D)}$. Let $T_{0}\in\mathbb{R}$, $\theta>0$ and let $v\in C^{2,1}(\overline{D}\times[T_{0}-\theta,T_{0}))$ be a solution of

[TABLE]

satisfying

[TABLE]

and the property that for any $\varepsilon\in(0,1)$ there exists a constant $C_{\varepsilon}>0$ such that for any vector field $\tau\perp\nu$,

[TABLE]

For each $k\in(0,d_{p})$ there exists $r_{0}>0$, depending only on $p,k,M,L$ and on the constants $C_{\varepsilon}$, such that, if for some $r\in(0,\min\{r_{0},R\}]$ and some $\sigma\in(0,r]$

[TABLE]

then

[TABLE]

The proof of Proposition 5.1, which is rather long and technical, is postponed to the appendix.

Our last preliminary result, Proposition 5.2, gives an (optimal) lower bound on the final space profile of $|u_{\nu}|$ in the normal direction to a GBU point (however, the absolute value will be eventually removed in Theorem 1.4). It will be proved as a direct consequence of Proposition 5.1.

Proposition 5.2**.**

Let $p>2$ and let $u_{0}\in X$ be such that $T:=T(u_{0})<\infty$. If $a$ is a GBU point of $u$ (i.e., $\limsup_{t\to T^{-},\,x\to a}|\nabla u(x,t)|=\infty$), then

[TABLE]

Proof of Proposition 5.2. Assume for contradiction that (5.7) fails. Hence,

[TABLE]

for some $k\in(0,d_{p})$. Take $R\in(0,\delta_{0}/2)$ such that $\Omega$ satisfies an inner sphere condition of radius $R$ at each point of $\partial\Omega$. In view of (1.19) and Proposition 3.1, we may apply Proposition 5.1 to $v=u$ with $\omega=\Omega$, $T_{0}=T$, $\theta=T/2$. Let $r_{0}$ be given by Proposition 5.1 for the above value of $k$. By (5.8), there exists $r\in(0,\min\{r_{0},\,R\}]$ such that

[TABLE]

Since $u\in C^{2,1}(\Omega\times(0,T])$, by continuity, there exist $\sigma\in(0,r)$ and $\theta\in(0,T/2)$ so small that

[TABLE]

It follows from Proposition 5.1, that $a$ is not a GBU point: a contradiction. ∎

We are now in a position to give the proof of Theorem 1.3, by combining Propositions 3.1-5.2 and an appropriate rescaling argument. As mentioned in Remark 1.2, we shall adapt the strategy in [31] to our problem, also using some simplifications from [22].

Proof of Theorem 1.3. To establish (1.11), it suffices to prove (1.12), since (1.11) then follows in view (1.2) and (1.19). We shall actually prove estimate (1.23) at the same time, as noted in Remark 1.3(c).

Assume that either (1.12) or (1.23) fails. Thus there exist $c_{1}>0$, a sequence of couples $(x_{j},t_{j})\in\Omega_{\delta_{0}}\times[T/2,T)$ and unit vectors $\tau_{j},\hat{\tau}_{j}$ with $\tau_{j}$ and $\hat{\tau}_{j}\perp\nu(x_{j})$, such that

[TABLE]

Set

[TABLE]

so that $\delta(x_{j})=|x_{j}-z_{j}|$. The proof will be done in several steps.

Step 1. Nondegeneracy at points $z_{j}$. First, it follows from Proposition 3.1 and (5.9) that, for all $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that

[TABLE]

Therefore, $t_{j}\to T$, $\delta(x_{j})\to 0$ and

[TABLE]

with $\varepsilon_{j}\to 0$ as $j\to\infty$. After extracting a subsequence, we have $z_{j}\to a\in\partial\Omega$, and we may assume without loss of generality that $a=0$, hence $x_{j}\to 0$, and that

[TABLE]

Note that [math] is in particular a GBU point (i.e., $\limsup_{t\to T^{-},\,x\to 0}|\nabla u(t,x)|=\infty$), since otherwise by parabolic regularity, $K_{j}$ would be bounded.

We claim that there exists a subsequence of $\{(x_{j},t_{j})\}_{j}$, not relabeled, and a sequence $\rho_{j}\to 0$ such that

[TABLE]

To prove the claim, in view of (5.10), by continuity, it suffices to show that, for each $\rho>0$ and each $j_{0}\geq 1$ there exist $j\geq j_{0}$ and $s\in(\delta(x_{j}),\rho)$ such that

[TABLE]

If this were false, then there would exist $\rho>0$ and $j_{0}\geq 1$ such that, for all $j\geq j_{0}$ and $s\in(\delta(x_{j}),\rho)$, $s^{\beta}u_{\nu}(z_{j}+s\nu_{j},t_{j})<d_{p}/2$. Hence, letting $j\to\infty$,

[TABLE]

Applying Proposition 5.2, we would deduce that [math] is not a GBU point, which is a contradiction. This proves the claim.

Step 2. Rescaling and convergence to a one dimensional profile. We rescale similarly as in the proof of Proposition 3.1, but now taking $\rho_{j}$ as rescaling parameter. Namely, we set:

[TABLE]

where $\Omega_{j}:=\rho_{j}^{-1}(\Omega-z_{j})$. The function $v_{j}$ satisfies

[TABLE]

and $\Omega_{j}$ converges to the half-space $\{y_{n}>0\}$ as $j\to\infty$. Let D_{R,\eta}:=\bigl{\{}y\in B_{R};\,y_{n}>\eta\bigr{\}} and $Q_{R,\eta}=D_{R,\eta}\times[-R,0]$ for $0<\eta<1<R$. Arguing exactly as in the proof of Proposition 3.1, we can find a subsequence of $(v_{j})_{j}$, not relabeled, which converges in each $C^{2,1}(Q_{R,\eta})$ to a classical solution $w(y,s)\in C^{2,1}(\mathbb{R}^{n}_{+}\times(-\infty,0])$ of

[TABLE]

Moreover, by (1.19) we have

[TABLE]

Hence $w_{s}\equiv 0$. On the other hand, by (the analogues of) (3.12)-(3.14), we have

[TABLE]

By Proposition 3.1, for any $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that, for any unit vector field $\tau\perp\nu$,

[TABLE]

In view of (5.11), we deduce that $\partial_{y_{1}}w=\dots=\partial_{y_{n-1}}w\equiv 0$. Therefore $w=w(y_{n})$ and the function $w$ solves

[TABLE]

Next, using $\nu_{j}\equiv\nu(z_{j})=\nu(z_{j}+\rho_{j}\nu_{j})$, property (5.12) yields

[TABLE]

so that $|w^{\prime}(1)|=d_{p}/2$. But then necessarily $w^{\prime}(1)=d_{p}/2$, since any solution of (5.15), with $w^{\prime}(1)<0$, ceases to exist after some finite $y_{n}>1$. Integrating (5.15), we thus obtain

[TABLE]

On the other hand, letting

[TABLE]

we deduce from (5.9) that

[TABLE]

Since $w$ depends only on $y_{n}$, we expect to reach a contradiction with (5.16). However, since $r_{j}$ may approach [math] and the convergence obtained so far is only valid locally for $y_{n}>0$ bounded away from [math], we need to extend the convergence near $y_{n}=0$. To this end, in the next step, we shall apply Proposition 5.1 to get a priori estimates of $|\nabla v_{j}|$ near the boundary.

Step 3. Uniform regularity of rescaled solutions and conclusion. Put

[TABLE]

First, since

[TABLE]

we deduce from Step 2 that for any $\eta\in(0,1)$, there exists $j_{1}(\eta)\geq 1$ such that

[TABLE]

We want to show that the assumptions of Proposition 5.1 with $\omega=\Omega_{j}=\rho_{j}^{-1}(\Omega-z_{j})$ are satisfied, uniformly with respect to $j$ large. It is easy to see that (5.1) is satisfied for some $R\in(0,1)$ independent of $j$, and moreover (5.2) is true with $D=D_{j}:=B_{2R}\cap\Omega_{j}$ (see e.g. [11] for regularity properties of the function distance to the boundary). Set

[TABLE]

and observe that the normal vector field to $\partial\Omega_{j}$ is given by

[TABLE]

which is well defined in $D_{j}$. Since $\delta_{j}(y)=\rho_{j}^{-1}\delta(z_{j}+\rho_{j}y)$, we have

[TABLE]

Also, it follows from (1.19) that

[TABLE]

where $M$ is independent of $j$. Moreover, by Proposition 3.1 for any $\varepsilon\in(0,1)$ there exists $C_{\varepsilon}>0$, independent of $j$, such that for any vector field $\tau(y)\perp\nu^{j}(y)=\nu(z_{j}+\rho_{j}y)$

[TABLE]

Now let $r_{0}=r_{0}(p,k,M,L,C_{\varepsilon})$ be given by Proposition 5.1 with $k=d_{p}/2$ , and choose $r\in(0,\min\{r_{0},\,R\}]$ such that

[TABLE]

By the proof of (3.11), we have

[TABLE]

where \sigma_{j}:=\sup\bigl{\{}|\xi_{n}|;\ \xi\in\partial\Omega_{j},\,|\xi|\leq 2\bigr{\}}\to 0 as $j\to\infty$. Therefore,

[TABLE]

for all $j\geq j_{2}(r)$ large enough which, along with (5.17) and (5.18), implies

[TABLE]

We then deduce from Proposition 5.1 that

[TABLE]

Writing $y=(y^{\prime},y_{n})$ and setting $\Sigma_{j}=\Omega_{j}\cap\{y;\ |y^{\prime}|<r/16,\,|y|\leq 3\}$, we infer that, for all $j\geq j_{3}$ large enough,

[TABLE]

(consider the cases $y_{n}\leq r/16$ and $y_{n}>r/16$ and use (5.19) and (5.17), respectively).

Next note that, for $j$ large enough, $\partial\Omega_{j}\cap B_{4}$ can be written as the graph $\{y_{n}=\psi_{j}(y^{\prime})\}$ of a $C^{2}$ function $\psi_{j}$, with a uniform $C^{2}$-bound with respect to $j$. Going back to (5.14), we may thus apply parabolic interior-boundary estimates uniformly in $j$, to deduce from (5.20) that

[TABLE]

for some $\gamma\in(0,1)$. Passing to a subsequence and recalling that $\nu_{j}\to e_{n}$, we may also assume that $\tau_{j}\to\tau_{\infty}$, $\tilde{\tau}_{j}\to\tilde{\tau}_{\infty}$, with $|\tau_{\infty}|=|\tilde{\tau}_{\infty}|=1$, $\tau_{\infty},\tilde{\tau}_{\infty}\perp e_{n}$, and that $r_{j}\to\ell\in[0,1]$. In view of (5.16), this implies

[TABLE]

This contradicts the fact that $w=w(y_{n})$ and the proof of (1.12) and (1.23) is completed.

Step 4. Proof of (1.13). Denote by $\lambda_{1}>0$ the first eigenvalue of $-\Delta$ in $H^{1}_{0}(\Omega)$, and by $\varphi_{1}$ the corresponding positive eigenfunction, normalized by $\max_{\overline{\Omega}}\varphi_{1}=1$. Since $u_{0}\in X$, there exists $M_{1}>0$ such that $u_{0}\geq-M_{1}\varphi_{1}$ in $\overline{\Omega}$. Observing that $\underline{u}(x,t):=-M_{1}e^{-\lambda_{1}t}\varphi_{1}$ is a subsolution of (1.2), if follows from the comparison principle that $u\geq\underline{u}$ in $\Omega\times(0,T)$. In particular,

[TABLE]

But, by (1.11), we have

[TABLE]

For each $t\in[T/2,T)$ and $a\in\partial\Omega$, the function $\phi(s):=-u_{\nu}(a+s\nu_{a},t)$ thus satisfies $\phi^{\prime}\leq 2|\phi|^{p}+C_{1}$ in $[0,\delta_{0}]$, with $\phi(0)\leq M_{2}$. It then follows from standard ODE arguments that

[TABLE]

where $s_{0}\in(0,\delta_{0})$ depends only on $p,M_{2},C_{1}$. But on the other hand, as a consequence of (3.2), we have

[TABLE]

Combining (5.21) and (5.22), we obtain $\displaystyle\inf_{\Omega_{\delta_{0}}\times(T/2,T)}u_{\nu}>-\infty$. This implies (1.13), owing to the $C^{1}$ spatial regularity of $u$ on $\overline{\Omega}\times[0,T/2]$. ∎

6. Proof of Theorems 1.4 and 1.5

Proof of Theorem 1.4. By Proposition 5.2 and (1.13) in Theorem 1.3, we have

[TABLE]

Assertion (i) then follows from (1.8) in Theorem 1.2, being $\varepsilon$ arbitrary.

Let us next prove assertion (iii). For fixed $b\in\partial\Omega$ and $t\in[T/2,T)$, we set

[TABLE]

Since $\phi^{\prime}(r)=u_{\nu\nu}(b+r\nu_{b},t)$, estimate (1.11) guarantees that

[TABLE]

for some constant $C_{1}=C_{1}(u_{0})$. For given $M>0$, take $\varepsilon=\varepsilon(M)\in(0,\delta_{0})$ such that $[2(2M)^{p}+C_{1}]\varepsilon<M$. A standard argument shows that, if $|\phi(r)|\leq M$ at some $r\in(0,\varepsilon)$, then $|\phi(s)|\leq 2M$ for all $s\in(0,r+\varepsilon]$.

Now assume for contradiction that $u_{\nu}(x_{j},t_{j})<M$ for some $M>0$ and some sequences $x_{j}\to a$, $t_{j}\to T$. By (1.13), we thus have $|u_{\nu}(x_{j},t_{j})|<M$ for a possibly larger $M$. We may write $x_{j}=b_{j}+r_{j}\nu_{b_{j}}$ with $b_{j}=P(x_{j})$ and $r_{j}=\delta(x_{j})\to 0$. Since $r_{j}<\varepsilon(M)$ for all $j$ large, we deduce from the previous paragraph that

[TABLE]

Passing to the limit, we get

[TABLE]

But this contradicts Proposition 5.2 and so assertion (iii) follows.

Let us now prove assertion (ii). Fix $\varepsilon\in(0,1)$. By the smoothness of $\partial\Omega$, we may find $\eta_{\varepsilon}>0$ such that, for all $x_{0}\in\partial\Omega\cap B_{\eta_{\varepsilon}}(a)$, with $x_{0}\neq a$, there exists a $C^{1}$ curve $\gamma:[0,1]\to\partial\Omega$ such that $\gamma(0)=a$ and $\gamma(1)=x_{0}$, and there exists a unit vector $\tau\perp\nu_{a}$ such that

[TABLE]

Assertion (iii) gives

[TABLE]

For $t\in(T/2,T)$ and $x_{0}\in\partial\Omega\cap B_{\eta_{\varepsilon}}(a)$, we may then write

[TABLE]

By (6.1), we have

[TABLE]

Since $|u_{\nu\tau}|\leq\varepsilon|u_{\nu}|^{p}+C_{\varepsilon}$ and $|\nabla u_{\nu}|\leq 2|u_{\nu}|^{p}+C$ in $\Omega_{\delta_{0}}\times[T/2,T]$ by Theorem 1.3, it follows that

[TABLE]

Hence, using (6.2),

[TABLE]

for all $(x_{0},t)\in[\partial\Omega\cap B_{\eta_{\varepsilon}}(a)]\times(T-\eta_{\varepsilon},T)$, with $\eta_{\varepsilon}>0$ possibly smaller. Now, if $x_{0}$ is not a GBU point, then $u_{\nu}(x_{0},t)$ has a finite limit as $t\to T$. Letting $t\to T$ and using (6.2) again, we obtain

[TABLE]

If $x_{0}$ is a GBU point, then $\lim_{t\to T^{-}}u_{\nu}(x_{0},t)=\infty$, so that (6.4) remains true as well. Since $\varepsilon$ is arbirarily small, we have thus proved (1.14).

Finally, to check assertion (iv), we observe that (1.16) is a consequence of (1.11) and (1.15). As for (1.17), it follows from (1.16) by integrating in the normal direction, whereas (1.18) is a consequence of (6.3). ∎

Proof of Theorem 1.5. It suffices to prove assertion (ii), since assertion (i) will then clearly follow by exchanging the roles of $u_{0}$ and $v_{0}$. Recall (see e.g. [36]) that $\bar{u},\bar{v}\in C^{2,1}(\Omega\times(0,\infty))\cap C(\overline{\Omega}\times[0,\infty))$ and that, since $v_{0}\geq u_{0}$, we have

[TABLE]

Pick $t_{0}\in(0,T(v_{0}))$. A consequence of Hopf’s Lemma gives $v(\cdot,t_{0})\geq\lambda u(\cdot,t_{0})$ for some $\lambda>1$. Assume for contradiction that

[TABLE]

Then, by Theorem 1.2 and Remark 3.1, we have

[TABLE]

On the other hand, since $w:=\lambda u$ satisfies in $\Omega\times(0,T(u_{0}))$

[TABLE]

and $w=\bar{v}=0$ in $\partial\Omega\times(0,T(u_{0}))$, it follows from the comparison principle (cf. e.g. [36, Proposition 3.3]) that $\bar{v}\geq\lambda u$ in $\Omega\times(t_{0},T(u_{0})]$. Let $a\in\partial\Omega$ be a GBU point of $u$. By Theorem 1.4, we have

[TABLE]

Hence

[TABLE]

But this contradicts (6.6). Consequently, (6.5) cannot hold, i.e. $\bar{v}$ loses boundary conditions before $T(u_{0})$. Hence in particular $T(v_{0})<T(u_{0})$. This proves assertion (ii) and completes the proof. ∎

7. Appendix: proof of Proposition 5.1

It is based on three lemmas (that we state together and will prove afterwards). The first one, based on estimate (3.1) from Proposition 3.1, shows that sufficiently weak singularity will propagate from small finite distance up to the boundary.

Lemma 7.1**.**

Let $p>2$, let $\omega\subset\mathbb{R}^{n}$ be an open set. Assume that the open line segment $(a,a+Re)\subset\omega$, where $a,e\in\mathbb{R}^{n}$, $|e|=1$, $R>0$. Let $t_{0}<t_{1}$ and let $v\in C^{2,1}(\omega\times[t_{0},t_{1}))$ be a solution of

[TABLE]

satisfying

[TABLE]

and with the property that for any $\varepsilon>0$ there exists a constant $C_{\varepsilon}>0$ such that for any unit vector $\tau\perp e$

[TABLE]

For each $k\in(0,d_{p})$, there exists $r_{0}>0$, depending only on $p,k,M$ and on the constants $C_{\varepsilon}$, such that, if

[TABLE]

then

[TABLE]

The second lemma provides regularization in time of the boundary derivative, provided that sufficiently weak singularity occurs. It is in the spirit of [30, Lemma 2.2], and [34, Lemma 4.1] and, like these results, its proof relies on a barrier argument. However, the statements there are not sufficient. In particular we need a uniform version for the purposes of the present paper.

Lemma 7.2**.**

Let $p>2$, let $\omega$ be a bounded domain of class $C^{2}$, $a\in\partial\omega$, $\rho>0$. Set $D=B_{\rho}(a)\cap\omega$, assume that the distance function $\delta(x)={\rm dist}(x,\partial\omega)$ is of class $C^{2}$ in $\overline{D}$, and set

[TABLE]

Let $t_{0}<t_{1}$ and let $v\in C^{2,1}(\omega\times[t_{0},t_{1}))$ be a solution of

[TABLE]

For any $k\in(0,d_{p})$, there exists $\rho_{0}=\rho_{0}(n,p,k,L)>0$ such that, if $0<\rho\leq\rho_{0}$ and

[TABLE]

then

[TABLE]

The last lemma provides a bound of the gradient near a boundary point assuming a control of the normal derivative on the boundary. The result and the proof (based on a local Bernstein-type argument) are similar to [30, Lemma 2.1], but again we require a more quantitative form.

Lemma 7.3**.**

Let $p>2$, let $\omega$ be any domain of $\mathbb{R}^{n}$ and let $x_{0}\in\partial\omega$. Let $R>0$, $t_{0}<t_{1}$ and v\in C^{2,1}\bigl{(}(\overline{B}_{R}(x_{0})\cap\overline{\omega})\times[t_{0},t_{1})\bigr{)} be a solution of

[TABLE]

such that

[TABLE]

and

[TABLE]

for some $M,N>0$. Then there exists $C(n,p)>0$ such that

[TABLE]

in \big{(}B_{R/2}(x_{0})\cap\omega\big{)}\times(t_{0},t_{1}).

Proof of Lemma 7.1. Set

[TABLE]

Then

[TABLE]

Take any $\eta\in(0,1)$. By (7.3), there exists $\delta_{\eta}>0$, depending only on $\eta,p$ and on the constants $C_{\varepsilon}$, such that, for any unit vector $\tau\perp e$,

[TABLE]

for all $s\in(0,\min\{\delta_{\eta},R\}]$ and $t\in(t_{0},t_{1})$. Using the elementary inequality

[TABLE]

with $c_{0}=c_{0}(p)>0$, it follows that

[TABLE]

for all $s\in(0,\min\{\delta_{\eta},R\}]$ and $t\in(t_{0},t_{1})$. Using (7.1), (7.2), (7.11) and (7.12) and taking $\delta_{\eta}$ even smaller if necessary (depending only on $\eta,p,M$ and on the constants $C_{\varepsilon}$), we obtain

[TABLE]

where $c_{1}=c_{1}(n,p)>0$. Let now $\psi(s)=ks^{-\beta}$. Since $k<d_{p}$, hence $-\beta k+k^{p}<0$, we may choose $\eta>0$ sufficiently small (depending only on $p,k,n$) so that $-\beta k+(1+\eta)k^{p}+c_{1}\eta<0$. A simple computation gives

[TABLE]

Now choose $r_{0}=\delta_{\eta}$. Since $\phi(r)\leq\psi(r)$ by (7.4), it follows from ODE comparison that $\phi(s)\leq\psi(s)$ for all $s\in(0,r]$, i.e. (7.5). ∎

Proof of Lemma 7.2. Fix a function $\Theta\in C^{\infty}_{0}(\mathbb{R}^{n})$ with $\Theta(x)=1$ for $|x|\leq 1/2$, $\Theta(x)=0$ for $|x|\geq 1$, and $0\leq\Theta\leq 1$. Next set $\psi=\Theta^{2}$, $\tau=t_{1}-t_{0}$ and $\kappa=(1-\beta)^{-1}k$. We define

[TABLE]

with

[TABLE]

where $\eta\in(0,1)$ will be chosen suitably small below. Note that $\varphi\in C^{2,1}(\mathbb{R}^{n}\times[t_{0},t_{1}])$, $\varphi\geq 0$ and $0\leq h\leq 1$. In this proof we denote by $C_{1},C_{2},\dots$ various positive constants depending only on $p$ and $n$ (through $\psi$). We have

[TABLE]

Morevover,

[TABLE]

and $\psi^{2\beta-1\over 1-\beta}|\nabla\psi|^{2}=4\Theta^{{2(2\beta-1)\over 1-\beta}+2}|\nabla\Theta|^{2}=4\Theta^{2\beta\over 1-\beta}|\nabla\Theta|^{2}$ where $\psi\neq 0$. Hence,

[TABLE]

Setting Z=\bigl{(}\delta+\varphi\bigr{)}^{1-\beta}, we compute

[TABLE]

It follows that

[TABLE]

Also, for $\varepsilon\in(0,1)$ to be chosen below, by the elementary inequality

[TABLE]

we have

[TABLE]

Therefore,

[TABLE]

Using $\beta p=\beta+1$, we obtain in the set $D\times(t_{0},t_{1})$

[TABLE]

On the other hand, owing to $|\nabla\delta|=1$ and (7.19), we have

[TABLE]

Also, since $\beta>k^{p-1}$, we may choose $\varepsilon=\varepsilon(p,n,k)\in(0,1)$ sufficiently small, so that

[TABLE]

Assuming $\eta\in(0,\varepsilon]$, using $(1-\beta)\kappa=k$, (7.14) (7.16)-(7.18), (7.20), $\delta\leq 2\rho$ in $D$, and recalling (7.6), we derive

[TABLE]

Now taking $\varepsilon=\varepsilon(p,n,k)\in(0,1)$ possibly smaller and $\rho_{0}=\rho_{0}(p,n,k,L)\in(0,1)$ sufficiently small, we may assume that

[TABLE]

Then, choosing

[TABLE]

with $c_{1}=c_{1}(p,n,k)\in(0,\varepsilon)$ sufficiently small, we get

[TABLE]

Hence,

[TABLE]

On the other hand, we have $v=0=V$ on $(B_{\rho}(a)\cap\partial\omega)\times(t_{0},t_{1})$ and, by assumption (7.7),

[TABLE]

It then follows from the comparison principle that $v\leq V$ in $D\times(t_{0},t_{1})$. In particular, for all $(x,t)\in(B_{\rho/2}(a)\cap\partial\omega)\times(t_{0},t_{1})$, we obtain

[TABLE]

which is the desired conclusion. ∎

Proof of Lemma 7.3. Put $w=|\nabla v|^{2}$, then

[TABLE]

where $|D^{2}v|^{2}=\Sigma_{i,j}(v_{x_{i}x_{j}})^{2}$ and

[TABLE]

Let $m\in(0,1)$. We select a cut-off function $\eta\in C^{2}(\overline{B}_{R}(x_{0}))$, with

[TABLE]

and such that

[TABLE]

where $C=C(m)>0$. Such a function $\eta$ is given for instance in the proof of [42, Theorem 3.2]. Put

[TABLE]

Then

[TABLE]

Since $2|\nabla\eta\cdot\nabla w|\leq 4\eta^{-1}|\nabla\eta|^{2}w+\eta|D^{2}u|^{2}$, it follows that

[TABLE]

Using $|w^{p/2}-u_{t}|=|\Delta u|\leq\sqrt{n}|D^{2}u|^{2}$, hence $w^{p}/(2n)\leq|D^{2}u|^{2}+|u_{t}|^{2}$, we get

[TABLE]

Taking $m=(p+1)/(2p)$ and using Young’s inequality, we obtain

[TABLE]

Let $c=[4n/(p-1)]^{1/(p-1)}$ and set

[TABLE]

We have $\phi\geq B\geq z$ on $\partial((B_{R}(x_{0})\cap\omega)\times(t_{0},t_{1})$, and

[TABLE]

Since $z$ remains bounded as $t\to t_{0}$, whereas $\phi(t)\to\infty$, it follows from the comparison principle (cf. e.g. [42, Proposition 2.2]) that $\phi\geq z$ in $(B_{R}(x_{0})\cap\omega)\times(t_{0},t_{1})$. Hence in particular

[TABLE]

which implies the desired conclusion. ∎

Proof of Proposition 5.1. Assumption (5.1) guarantees that, for each $b\in B_{\sigma}(a)\cap\partial\omega$, the line segment $(b,b+R\nu_{b})$ lies in $D$ and that

[TABLE]

Let $r_{0}=r_{0}(p,k,M,C_{\varepsilon})$ be given by Lemma 7.1. We deduce from assumptions (5.3)-(5.5) and Lemma 7.1 that

[TABLE]

Since $v=0$ on $B_{\sigma}(a)\cap\partial\omega$, it follows by integration that

[TABLE]

Using (7.21) and $\sigma\leq r\leq R$, we have in particular

[TABLE]

We may then apply Lemma 7.2 (taking $r_{0}$ possibly smaller, which may also depend on $L$) to infer that

[TABLE]

Since $z:=-v$ satisfies $z_{t}-\Delta z=-|\nabla z|^{p}\leq|\nabla z|^{p}$, we may apply Lemma 7.2 to $z$ as well, so that actually

[TABLE]

From this estimate, (5.6) finally follows from Lemma 7.3. ∎

Acknowledgements. Part of this work was done during a visit of PhS at the Dipartimento di Matematica e Informatica of the Università degli Studi di Perugia within the auspices of the INdAM – GNAMPA Projects 2018. He wishes to thank this institution for the kind hospitality. PhS is partly supported by the Labex MME-DII (ANR11-LBX-0023-01).

RF and PP were partly supported by the Italian MIUR project Variational methods, with applications to problems in mathematical physics and geometry (2015KB9WPT_009) and are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Projects 2018 Problemi non lineari alle derivate parziali (Prot_U-UFMBAZ-2018-000384).

Declaration of interest statement. No conflict of interests.