Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi Equations in $\mathbb{R}^N$ - Archive ouverte HAL Access content directly
Journal Articles Communications in Partial Differential Equations Year : 2021

Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi Equations in $\mathbb{R}^N$

Abstract

We study the large-time behavior of bounded from below solutions of parabolic viscous Hamilton-Jacobi Equations in the whole space $\mathbb{R}^N$ in the case of superquadratic Hamiltonians. Existence and uniqueness of such solutions are shown in a very general framework, namely when the source term and the initial data are only bounded from below with an arbitrary growth at infinity. Our main result is that these solutions have an ergodic behavior when $t\to +\infty$, i.e., they behave like $\lambda^*t + \phi(x)$ where $\lambda^*$ is the maximal ergodic constant and $\phi$ is a solution of the associated ergodic problem. The main originality of this result comes from the generality of the data: in particular, the initial data may have a completely different growth at infinity from those of the solution of the ergodic problem.
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Dates and versions

hal-02531455 , version 1 (03-04-2020)

Identifiers

  • HAL Id : hal-02531455 , version 1

Cite

Guy Barles, Alexander Quaas, Andrei Rodríguez. Large-time behavior of unbounded solutions of viscous Hamilton-Jacobi Equations in $\mathbb{R}^N$. Communications in Partial Differential Equations, 2021, 46 (3), pp.547-572. ⟨hal-02531455⟩
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