Related papers: Uniqueness for unbounded solutions to stationary v…

Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellman equations whose Hamiltonians are not always defined, especially when the diffusion term is unbounded with respect to the control. We obtain existence and uniqueness…

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi equation [u_{t}-\Delta u+|\nabla u|^{q}=0] in $Q_{\Omega,T}=\Omega\times(0,T),$ where $q>1,T\in(0,\infty] ,$ and $\Omega$ is a smooth bounded domain of $\mathbb{R}%…

In quantitative genetics, viscosity solutions of Hamilton-Jacobi equations appear naturally in the asymptotic limit of selection-mutation models when the population variance vanishes. They have to be solved together with an unknown function…

Uniqueness of positive solutions to viscous Hamilton-Jacobi-Bellman (HJB) equations of the form $-\Delta u(x) + \frac{1}{\gamma} |D{u}(x)|^\gamma = f(x) - \lambda$, with $f$ a coercive function and $\lambda$ a constant, in the subquadratic…

We consider the evolutionary Hamilton-Jacobi equation \begin{align*} w_t(x,t)+H(x,Dw(x,t),w(x,t))=0, \quad(x,t)\in M\times [0,+\infty), \end{align*} where $M$ is a compact manifold, $H:T^*M\times R\to R$, $H=H(x,p,u)$ satisfies Tonelli…

Here we study the nonnegative solutions of the viscous Hamilton-Jacobi problem \[ \left\{\begin{array} [c]{c}% u_{t}-\nu\Delta u+|\nabla u|^{q}=0, u(0)=u_{0}, \end{array} \right. \] in $Q_{\Omega,T}=\Omega\times\left(0,T\right) ,$ where…

We discuss a class of time-dependent Hamilton-Jacobi equations, where an unknown function of time is intended to keep the maximum of the solution to the constant value 0. Our main result is that the full problem has a unique viscosity…

We consider a path-dependent Hamilton--Jacobi equation with coinvariant derivatives over the space of continuous functions. We prove two uniqueness results for viscosity (generalized) solutions defined in terms of coinvariantly smooth test…

We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally H\"older continuous with H\"older exponent depending only on the…

In this note, we discuss a class of time-dependent Hamilton-Jacobi equations depending on a function of time, this function being chosen in order to keep the maximum of the solution to the constant value 0. The main result of the note is…

We prove non-uniqueness and study the behaviour of viscosity solutions of a class of uniformly elliptic fully nonlinear equations of Hamilton-Jacobi-Bellman-Isaacs type, with quadratic growth in the gradient. The crucial a priori bound for…

If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we…

Motivated by the vanishing contact problem, we study in the present paper the convergence of solutions of Hamilton-Jacobi equations depending nonlinearly on the unknown function. Let $H(x,p,u)$ be a continuous Hamiltonian which is strictly…

We show that if a Hamilton-Jacobi equation admits a differentiable solution whose gradient is Lipschitz, then this solution is the unique semi-concave weak solution. Our result does not rely on any convexity (nor concavity) assumptions on…

We determine the large-time behavior of unbounded solutions for the so-called viscous Hamilton Jacobi equation, $u_t - \Delta u + |Du|^m = f(x)$, in the quadratic and subquadratic cases (i.e., for $1<m\leq 2$), with a particular focus on…

The goal of this paper is to study uniqueness of a one-dimensional Hamilton-Jacobi equation \begin{equation*} \begin{cases} u_t=|u_x|^2+R(x,I(t)) &\text{in }\mathbb{R} \times (0,\infty), \max_{\mathbb{R}} u(\cdot,t)=0 &\text{on }[0,\infty),…

Viscosity solutions of fully nonlinear, local or non local, Hamilton-Jacobi equations with a super-quadratic growth in the gradient variable are proved to be H\"older continuous, with a modulus depending only on the growth of the…

We study existence, uniqueness and regularity properties of classical solutions to viscous Hamilton-Jacobi equations with Caputo time-fractional derivative. Our study relies on a combination of a gradient bound for the time-fractional…

We study an ergodic problem associated to a non-local Hamilton-Jacobi equation defined on the whole space $\lambda-\mathcal{L}[u](x)+|Du(x)|^m=f(x)$ and determine whether (unbounded) solutions exist or not. We prove that there is a…

We prove the uniqueness property for a class of entire solutions to the equation \begin{equation*} \left\{ \begin{array}{ll} -{\rm div}\, \mathcal{A}(x,\nabla u) = \sigma, \quad u\geq 0 \quad \text{in } \mathbb{R}^n, \\…