Singularity formation and regularization at multiple times in the viscous Hamilton-Jacobi equation

TL;DR

This work analyzes the viscous Hamilton-Jacobi equation $u_t-\Delta u=|\nabla u|^p$ with $p>2$ on smooth bounded domains, revealing rich post-blow-up dynamics including multiple gradient blow-up events and losses/recoveries of boundary conditions. The authors introduce a multibump initial-data method and zero-number techniques to construct solutions exhibiting GBU and LBC at arbitrarily many times, classify one-dimensional behavior, and realize any prescribed sequence of GBU/LBC/bouncing events (even in radial higher dimensions). A key novelty is the bouncing phenomenon, where solutions transition from LBC to LBC via a recovery time, and a detailed transition-set analysis that links boundary behavior to intersection properties with a singular steady state $U^*$. The approach avoids relying on explicit special solutions and hinges on a continuum of critical parameters and compactness/regularity arguments, with implications for stochastic control interpretations of time-varying rewards. Overall, the paper broadens the understanding of post-blow-up dynamics for viscous HJ equations beyond Fujita-type models and provides tools potentially applicable to a wider class of nonlinear parabolic PDEs.

Abstract

The Cauchy-Dirichlet pbm for the superquadratic viscous Hamilton-Jacobi eqn (VHJ), which has important applications in stochastic control theory, admits a unique, global viscosity solution. Sol. thus exist in the weak sense after appearance of singularity in finite time, which occurs through gradient blow-up (GBU) on the boundary. Whereas theory of visc. sol. has been extensively studied and applied to many PDEs, there are less results on refined behavior of sol. In particular, detailed behavior of visc. sol. of VHJ after GBU has remained mostly open. Here, in general dim., for each $m\ge 1$ we construct sol. which undergo GBU and LBC at least at $m$ times and then recover regularity, as well as sol. that exhibit GBU without LBC at 1st blowup time. In 1d, we obtain the complete classification of visc. sol. at each time, which extends to radial cases in higher d. Furthermore for each $m\ge 2$ and arbitrarily given combination of GBU types with/without LBC at $m$ times in arbitrarily given order, we show exist. of a sol. with this exact combination of GBU. Some sol. display a new type of behavior called "bouncing". Global weak sol. of VHJ with multiple time singularity turn out to display larger variety of behaviors than for the Fujita eqn. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires a delicate argument. Since we do not rely on any known special sol. unlike in Fujita eqn, our method is expected to apply to other eqns. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. In this framework our results imply that for certain spatial distributions of rewards, if a controled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.

Figures (11)

• Figure 1.1: A solution with exactly $4$ losses and recoveries of boundary conditions
• Figure 1.2: A solution with exactly $1$ bouncing
• Figure 1.3: A solution with mixed behaviors (2 LBC, 2 GBU without LBC and 1 double bouncing)
• Figure 9.1: The times $\bar{t}_i$ and indices $\bar{\sigma}_i$ (here $d=3$)
• Figure 9.2: The functions in Proposition \ref{['lemconv1']} (the function $h$ is the dotted curve).

Theorems & Definitions (68)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Proposition 2.1
• Proposition 2.2
• proof