Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu_t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu_o is a positive uniformly continuous function verifying –R¦u_o¦^m+u₀^q 0 in ^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 – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t)₊^N+1.Partially supported by the DGICYT No. 86/0405 project.