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Extinction and decay estimates for viscous Hamilton-Jacobi equations in N . (English) Zbl 1001.35007

The Cauchy problem

u t -Δu+|u| p =0in(0,+)× N ,u(0)=u 0 in N ,

where p(0,+) and u 0 is a nonnegative function in C( n )L 1 ( N ), is considered. Here, C( N ) denotes the space of bounded and continuous functions in N . For such initial data, the existence and the uniqueness of nonnegative classical solutions to (1)-(2) have been obtained by Gilding, Guedda and Kersner. Within the framework of nonnegative solutions the term |u| p in (1) acts as an absorption term, and the smaller the exponent p is, the stronger is the absorption. The aim of this work is to investigate some qualitative properties of nonnegative solutions to (1)-(2) according to the values of p. More precisely, the authors proved that any nonnegative solution to (1)-(2) with initial data in C( N )L 1 ( N ) vanishes identically after a finite time when p(0,N/N(N+1)), this property called extinction in finite time.


MSC:
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE
35K15Second order parabolic equations, initial value problems