ISSN:
1432-0606
Keywords:
Hamilton-Jacobi equations
;
Viscosity solutions
;
Extinctiontime property
;
Representation formulae
;
Primary 49C10
;
35L60
;
Secondary 35D05
;
35C99
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
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 0 q ⩾ 0 in ℝ N . 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 .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01182596
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