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arXiv · 2111.03710

Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi equation: quadratic and subquadratic cases

Alexander Quaas·Andrei Rodríguez-Paredes
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Abstract

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 allowing arbitrary growth at infinity for $f$ and the prescribed initial data. The lack of a comparison principle for the associated ergodic problem is overcome by proving that a generalized simplicity holds for sub- and supersolutions of the ergodic problem. Moreover, as the uniqueness of solutions of the parabolic problem remains open in the current setting, our result on large-time holds for any solution, even if multiple solutions exist.

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Alexander Quaas, Andrei Rodríguez-Paredes. 2021-11-05. Large-time behavior of unbounded solutions of the viscous Hamilton-Jacobi equation: quadratic and subquadratic cases. https://arxiv.org/abs/2111.03710

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