Uniqueness without star-shapedness for the Crandall–Lions–Souganidis model problem $u_t-\Delta u+|\nabla u|^p=0$ with infinite initial data on a subdomain ($1<p<2$)

The model problem introduced by Crandall, Lions and Souganidis in 1989 asks for solutions of $u_t-\Delta u+|\nabla u|^p=0$ in $\Omega\times(0,\infty)$ with $u=0$ on $\partial\Omega$ and the singular initial datum $u(\cdot,0)=+\infty$ on a smooth subdomain $D\subset\subset\Omega$, $u(\cdot,0)=0$ on $\bar\Omega\setminus\bar D$, the initial conditions being uniform limits on compact subsets as $t\downarrow 0$. They proved existence for every $p>1$, the interface trichotomy $\lim_{t\downarrow 0}u=0$, $\ln 2$, $+\infty$ on $\partial D$ according as $p>2$, $p=2$, $1<p<2$, and uniqueness for $p\ge 2$; for $1<p<2$ uniqueness was proved only when $\Omega$ and $D$ are star-shaped with respect to a common point, and they conjectured that the geometric assumption can be dropped. Bidaut-Véron and Dao (2012-2015) recast the question in the initial-trace framework, constructing a minimal solution $Y_D^\Omega$ and a maximal solution $u_D^\Omega$ with this singular trace, proving uniqueness when $D$ alone is star-shaped, and showing that the weak and classical solution classes coincide for $1<p<(N+2)/(N+1)$; whether $Y_D^\Omega=u_D^\Omega$ for a general smooth $D$ was left open and is explicitly described as delicate. The problem: does uniqueness hold in the natural class for arbitrary smooth $D\subset\subset\Omega$ and all $1<p<2$?

Background

The study of evolution equations with infinite initial data was systematized in the late 1980s, motivated in particular by large-deviation asymptotics for diffusion processes: if $P_\varepsilon(x,t)$ is the probability that a small-noise diffusion remains in a bounded domain up to a prescribed time, the logarithmic transform $u_\varepsilon=-\varepsilon\log P_\varepsilon$ leads, as $\varepsilon\to 0$, to Hamilton–Jacobi equations whose initial datum is identically $+\infty$ on part of the domain. Crandall, Lions and Souganidis developed an abstract theory for such data: given an order-preserving semigroup $S(t)$ on a function lattice (the solution flow of an evolution equation, mapping smaller data to smaller solutions and satisfying $S(t)\circ S(s)=S(t+s)$), the supremum $M(t)=\sup\{S(t)\varphi\}$ over a suitable family of admissible data defines a maximal trajectory, a universal upper bound for all solutions with data below the given one. They proved universal boundary bounds of the form $u(x,t)\le B(t)\,d(x)$, where $d(x)$ is the distance from $x$ to $\partial\Omega$, for solutions of $u_t-\varepsilon Lu+H(x,\nabla u)=0$ with Hamiltonians $H$ growing superlinearly in the gradient, and used them to construct and characterize maximal solutions with data $u(\cdot,0)\equiv+\infty$. Here a viscosity solution is the standard weak notion for Hamilton–Jacobi and degenerate elliptic equations, defined by replacing derivatives at test points by those of smooth functions touching the graph from above (subsolution) and from below (supersolution), and lower semicontinuous means $u(x)\le\liminf_{y\to x}u(y)$. For the first-order equation $u_t+H(x,\nabla u)=0$ the maximal solution is in general only lower semicontinuous, and its continuity was among the issues the paper left open. The concrete model problem isolating the effect of the singular datum is $u_t-\Delta u+|\nabla u|^p=0 \quad\text{in } \Omega\times(0,\infty),$ with $u=0$ on $\partial\Omega\times(0,\infty)$ and initial conditions $u(\cdot,t)\to+\infty$ uniformly on compact subsets of $D$ and $u(\cdot,t)\to 0$ uniformly on compact subsets of $\bar\Omega\setminus\bar D$ as $t\downarrow 0$, where $\Omega$ is bounded and smooth and $D\subset\subset\Omega$ is smooth. The equation — a heat equation with nonlinear gradient absorption, also called the viscous Hamilton–Jacobi equation, arising in stochastic control and in KPZ-type (Kardar–Parisi–Zhang) models of surface growth (as recalled for instance in the recent classification results of Damagui Chabi and Souplet) — is exactly solvable for the borderline exponent $p=2$ by the Hopf–Cole transform $v=e^{-u}$, under which $v$ solves the heat equation with datum $0$ on $D$ and $1$ on $\Omega\setminus\bar D$, so that at the interface $v\to 1/2$ and $u\to\ln 2$. Crandall–Lions–Souganidis proved: existence of a solution for every $p>1$; for $x\in\partial D$ the trichotomy $\lim_{t\downarrow 0}u(x,t)=0$, $\ln 2$, $+\infty$ according as $p>2$, $p=2$, $1<p<2$; and uniqueness for $p\ge 2$ with no restriction on the geometry, while for $1<p<2$ uniqueness was established under the assumption that $\Omega$ and $D$ are star-shaped with respect to the same point (a set is star-shaped if it contains a point $x_0$ such that the segment from $x_0$ to every point of the set lies in the set). That proof is a scaling-comparison argument: the equation is invariant under $u(x,t)\mapsto\eta^{-\gamma}u(x/\eta,t/\eta^2)$ with $\gamma=(2-p)/(p-1)$, and star-shapedness makes the dilates of the pair $(\Omega,D)$ nested, so any solution is squeezed between the canonical solutions on the dilates $\eta D$ and forced to coincide with their limit. The authors conjectured that star-shapedness can be dropped. The question was taken up by Bidaut-Véron and Dao within a broader initial-trace program: every nonnegative weak solution (distributional, with $|\nabla u|^q$ locally integrable) of $u_t-\Delta u+|\nabla u|^q=0$ has an initial trace, a Borel measure which is $+\infty$ on a closed singular set $S$ and a locally finite measure off it; the model datum above is the trace $(\bar D,0)$. They proved universal decay estimates of order $t^{-1/(q-1)}$ valid for all weak solutions without any uniqueness assumption, and new uniqueness results for finite measure or $L^r$ data, but only in subcritical ranges ($q<(N+2)/(N+1)$ for measures, $q<(N+2r)/(N+r)$ for $L^r$ data), which do not cover the infinite datum on $D$. For the trace $(\bar D,0)$ they constructed the minimal solution $Y_D^\Omega$ (supremum of solutions with data supported in $D$) and the maximal solution $u_D^\Omega$ (infimum of solutions with data supported in small outer neighborhoods of $D$), proved uniqueness in the classical class whenever $D$ alone is star-shaped — dropping the condition on $\Omega$ — and proved that the weak and classical classes coincide for $1<q<(N+2)/(N+1)$. They also constructed a nonradial self-similar solution with trace $(\mathbb{R}^N_+,0)$ growing like $t^{-1/(q-1)}$ in the singular half-space and $t^{-(2-q)/(q-1)}$ along its boundary, showing the solution class is rich. Uniqueness for a general smooth $D$ — an annulus, or a disjoint union of balls — with $1<p<2$ remains open and is the surviving core of the conjecture.

Problem Statement

Let $N\ge 1$, let $\Omega\subset\mathbb{R}^N$ be a bounded domain with smooth boundary, let $D\subset\subset\Omega$ be a nonempty open set with smooth boundary, and let $1<p<2$. Consider the Crandall–Lions–Souganidis model problem $u_t-\Delta u+|\nabla u|^p=0 \quad\text{in } \Omega\times(0,\infty),$ $u=0 \quad\text{on } \partial\Omega\times(0,\infty),$ with the singular initial condition $\lim_{t\downarrow 0}u(x,t)=+\infty \text{ uniformly on compact subsets of } D, \qquad \lim_{t\downarrow 0}u(x,t)=0 \text{ uniformly on compact subsets of } \bar\Omega\setminus\bar D.$ Let $\mathcal{C}$ denote the class of nonnegative functions $u\in C^{2,1}(\Omega\times(0,\infty))\cap C(\bar\Omega\times(0,\infty))$ satisfying the equation, the Dirichlet condition, and both initial limits above (this is the solution class of the model problem as defined by Crandall–Lions–Souganidis, namely a classical solution for $t>0$ attaining the initial conditions as uniform limits on compact subsets; it is the class $\mathcal{C}$ of Bidaut-Véron and Dao). Question: does $\mathcal{C}$ contain exactly one solution for every such triple $(\Omega,D,p)$ — that is, does uniqueness hold without any geometric assumption such as star-shapedness of $D$ (or of $\Omega$ and $D$ with respect to a common point)? Equivalently, do the minimal solution $Y_D^\Omega$ and the maximal solution $u_D^\Omega$ of Bidaut-Véron and Dao coincide for every such triple? A single configuration $(\Omega,D,p)$ with $1<p<2$ and $D$ not star-shaped carrying two distinct elements of $\mathcal{C}$ would refute the conjecture. The same question for the Cauchy problem on $\mathbb{R}^N$ with bounded smooth singular set is the natural companion; the present statement concerns the bounded-domain Dirichlet case above.

References

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