Abstract
In this paper we study initial value problems likeu t−R¦▽u¦m+λuq=0 in ℝn× ℝ+, u(·,0+)=uo(·) in ℝN, whereR > 0, 0 <q < 1,m ≥ 1, andu o is a positive uniformly continuous function verifying −R¦▽u o¦m+λu q0 ⩾ 0 in ℝN. We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t∞(·) defined byu(x, t) > 0 if 0<t<t ∞(x) andu(x, t)=0 ift ≥t ∞(x). Regularity, extinction rate, and asymptotic behavior of t∞(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u o(x − ξt))1−q −λ(1−q)t]+)1/(1−q): ¦ξ¦≤R}, (x, t)εℝ N+1+ .
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Communicated by A. Bensoussan
Partially supported by the DGICYT No. 86/0405 project.
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Diaz, G., Rey, J.M. Finite extinction time for some perturbed Hamilton-Jacobi equations. Appl Math Optim 27, 1–33 (1993). https://doi.org/10.1007/BF01182596
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DOI: https://doi.org/10.1007/BF01182596