301 research outputs found
Decay estimates for a viscous Hamilton-Jacobi equation with homogenious Dirichlet boundary conditions
20 pagesInternational audienceGlobal classical solutions to the viscous Hamilton-Jacobi equation with homogenious Dirichlet boundary conditions are shown to converge to zero at the same speed as the linear heat semigroup when p > 1. For p = 1, an exponential decay to zero is also obtained in one space dimension but the rate depends on a and differs from that of the linear heat equation. Finally, if 0 < p < 1 and a < 0, finite time extinction occurs for non-negative solutions
Large time behavior of solutions of viscous Hamilton-Jacobi Equations with superquadratic Hamiltonian
We study the long-time behavior of the unique viscosity solution of the
viscous Hamilton-Jacobi Equation with inhomogeneous Dirichlet boundary conditions,
where is a bounded domain of . We mainly focus on the
superquadratic case () and consider the Dirichlet conditions in the
generalized viscosity sense. Under rather natural assumptions on the
initial and boundary data, we connect the problem studied to its associated
stationary generalized Dirichlet problem on one hand and to a stationary
problem with a state constraint boundary condition on the other hand
Convergence to separate variables solutions for a degenerate parabolic equation with gradient source
The large time behaviour of nonnegative solutions to a quasilinear degenerate
diffusion equation with a source term depending solely on the gradient is
investigated. After a suitable rescaling of time, convergence to a unique
profile is shown for global solutions. The proof relies on the half-relaxed
limits technique within the theory of viscosity solutions and on the
construction of suitable supersolutions and barrier functions to obtain optimal
temporal decay rates and boundary estimates. Blowup of weak solutions is also
studied
Well-posedness and gradient blow-up estimate near the boundary for a Hamilton-Jacobi equation with degenerate diffusion
This paper is concerned with weak solutions of the degenerate viscous
Hamilton-Jacobi equation with
Dirichlet boundary conditions in a bounded domain ,
where and . With the goal of studying the gradient blow-up
phenomenon for this problem, we first establish local well-posedness with
blow-up alternative in norm. We then obtain a precise gradient
estimate involving the distance to the boundary. It shows in particular that
the gradient blow-up can take place only on the boundary. A regularizing effect
for is also obtained.Comment: 20 pages 1 figur
Large time behavior for a viscous Hamilton-Jacobi equation with Neumann boudary condition
We prove the existence and the uniqueness of strong solutions for the viscous
Hamilton-Jacobi Equation with Neumann boundary condition and initial data a
continious function. Then, we study the large time behavior of the solutions.Comment: 32 page
Global Regularity vs. Finite-Time Singularities: Some Paradigms on the Effect of Boundary Conditions and Certain Perturbations
In light of the question of finite-time blow-up vs. global well-posedness of
solutions to problems involving nonlinear partial differential equations, we
provide several cautionary examples which indicate that modifications to the
boundary conditions or to the nonlinearity of the equations can effect whether
the equations develop finite-time singularities. In particular, we aim to
underscore the idea that in analytical and computational investigations of the
blow-up of three-dimensional Euler and Navier-Stokes equations, the boundary
conditions may need to be taken into greater account. We also examine a
perturbation of the nonlinearity by dropping the advection term in the
evolution of the derivative of the solutions to the viscous Burgers equation,
which leads to the development of singularities not present in the original
equation, and indicates that there is a regularizing mechanism in part of the
nonlinearity. This simple analytical example corroborates recent computational
observations in the singularity formation of fluid equations
Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton-Jacobi equation
Convergence to a single steady state is shown for non-negative and radially
symmetric solutions to a diffusive Hamilton-Jacobi equation with homogeneous
Dirichlet boundary conditions, the diffusion being the -Laplacian operator,
, and the source term a power of the norm of the gradient of . As a
first step, the radially symmetric and non-increasing stationary solutions are
characterized
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