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

@article{Filippucci2019ALT,
  title={A Liouville-type theorem in a half-space and its applications to the gradient blow-up behavior for superquadratic diffusive Hamilton–Jacobi equations},
  author={Roberta Filippucci and Patrizia Pucci and Philippe Souplet},
  journal={Communications in Partial Differential Equations},
  year={2019},
  volume={45},
  pages={321 - 349}
}
Abstract We consider the elliptic and parabolic superquadratic diffusive Hamilton–Jacobi equations: and with p > 2 and 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) solutions of the parabolic problem in general bounded domains of with smooth boundaries… 
A note on one-dimensional symmetry for Hamilton–Jacobi equations with extremal Pucci operators and application to Bernstein type estimate
  • R. Fuentes, A. Quaas
  • Mathematics
    Nonlinear Differential Equations and Applications NoDEA
  • 2022
We prove a Liouville-type theorem that is one-dimensional symmetry and classification results for non-negative L-viscosity solutions of the equation −M±λ,Λ(D 2 u)± |Du| = 0, x ∈ Rn+, with boundary
A P ] 2 D ec 2 01 9 GRADIENT BLOW-UP RATES AND SHARP GRADIENT ESTIMATES FOR DIFFUSIVE HAMILTON-JACOBI EQUATIONS
Consider the diffusive Hamilton-Jacobi equation ut −∆u = |∇u| p + h(x) in Ω× (0, T ) with Dirichlet conditions, which arises in stochastic control problems as well as in KPZ type models. We study the
Gradient blow-up rates and sharp gradient estimates for diffusive Hamilton–Jacobi equations
Consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p+h(x)\ \ \text{ in } \Omega\times(0,T)$$ with Dirichlet conditions, which arises in stochastic control problems as well as
A nonlinear elliptic problem involving the gradient on a half space
We consider perturbations of the diffusive Hamilton-Jacobi equation { −∆u = (1 + g(x))|∇u| in RN+ , u = 0 on ∂RN+ , for p > 1. We prove the existence of a classical solution provided p ∈ ( 4 3 , 2)
Monotonicity of solutions for fractional p-equations with a gradient term
Abstract In this paper, we consider the following fractional p p -equation with a gradient term: ( − Δ ) p s u ( x ) = f ( x , u ( x ) , ∇ u ( x ) ) . {\left(-\Delta
Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem
We prove gradient boundary blow up rates for ergodic functions in bounded domains related to fully nonlinear degenerate/singular elliptic operators. As a consequence, we deduce the uniqueness, up to
Global Existence and Blow-Up of Solutions to a Parabolic Nonlocal Equation Arising in a Theory of Thermal Explosion
<jats:p>Focusing on the physical context of the thermal explosion model, this paper investigates a semilinear parabolic equation <jats:inline-formula> <math

References

SHOWING 1-10 OF 53 REFERENCES
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equation in domains with non-constant curvature
  • Carlos Esteve
  • Mathematics
    Journal de Mathématiques Pures et Appliquées
  • 2020
The profile of boundary gradient blow-up for the diffusive Hamilton-Jacobi equation
We consider the diffusive Hamilton-Jacobi equation $$u_t-\Delta u=|\nabla u|^p,$$ with Dirichlet boundary conditions in two space dimensions, which arises in the KPZ model of growing interfaces. For
Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena
We consider a one-dimensional semilinear parabolic equation with a gradient nonlinearity. We provide a complete classification of large time behavior of the classical solutions u: either the space
Asymptotic Behaviour of the Gradient of Large Solutions to Some Nonlinear Elliptic Equations
Abstract If h is a nondecreasing real valued function and 0 ≤ q ≤ 2, we analyse the boundary behaviour of the gradient of any solution u of −Δu + h(u) + |∇u|q = f in a smooth N-dimensional domain Ω
Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains
Consider the diffusive Hamilton-Jacobi equation ut = Δu + |∇u|p, p > 2, on a bounded domain Ω with zero-Dirichlet boundary conditions, which arises in the KPZ model of growing interfaces. It is known
Global solutions of inhomogeneous Hamilton-Jacobi equations
We consider the viscous Hamilton-Jacobi (VHJ) equationut-Δu=|∇u|p+h(x). For the Dirichlet problem withp>2, it is known thatgradient blow-up may occur in finite time (on the boundary). Whereas
Blow-up and regularization rates, loss and recovery of boundary conditions for the superquadratic viscous Hamilton-Jacobi equation
The problem of dirichlet for quasilinear elliptic differential equations with many independent variables
  • J. Serrin
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1969
This paper is concerned with the existence of solutions of the Dirichlet problem for quasilinear elliptic partial differential equations of second order, the conclusions being in the form of
...
...