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

We determine the large-time behavior of unbounded solutions for the so-called viscous Hamilton–Jacobi equation, u_t − Δu + ∣Du∣^m = f(x), in the quadratic and subquadratic cases (i.e., for 1 < m ≤ 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.