Abstract
In this paper, we study the asymptotic behavior of global solutions of the equation ut = �u + e jr uj in the annulus Br,R, u(x, t) = 0 on @Br and u(x, t) = M � 0 on @BR. It is proved that there exists a constant Mc > 0 such that the problem admits a unique steady state if and only
Highlights
Introduction and main resultsIn this paper we consider the problem ut = ∆u + e|∇u|, u(x, t) = 0, u(x, t) = M, u(x, 0) = u0(x), x ∈ Br,R, t > 0, x ∈ ∂Br, t > 0, x ∈ ∂BR, t > 0, x ∈ Br,R. (1.1)Here r > 0, Br,R = {x ∈ RN ; r < |x| < R}, ∂Br = {x ∈ RN ; |x| = r}, M ≥ 0, and u0(x) ∈ X, where X = {v ∈ C1(Br,R); v|∂Br = 0, v|∂BR = M }, endowed with the C1 norm
U(x, t), whose existence time will be denoted by T = T (u0) > 0, such that u ∈ C2,1(Br,R × (0, T )) ∩ C1,0(Br,R × [0, T ))
The differential equation in (1.1) possesses both mathematical and physical interest. This equation arises in the viscosity approximation of Hamilton-Jacobi type equations from stochastic control theory [2] and in some physical models of surface growth [4]
Summary
For M = 0 and u0 C1 sufficiently small, it is known that (I) occurs and u converges to the unique steady state S0 ≡ 0. If Problem (1.1) admits a steady state SM (x), it is unique and radial, and if M1 > M2, SM1 > SM2 in We can deduce that there exists Mc > 0 such that if M > Mc, Problem (1.1) does not admit a steady state, if 0 < M < Mc, Problem (1.1) admits a unique regular steady state SM ∈ C2([r, R]), and if M = Mc, Problem (1.1) still admits a steady state SMc ∈ C([r, R]) ∩ C2((r, R]), which is singular in the sense that it has infinite derivative on the boundary ∂Br. Theorem 2.1 Assume that M ≥ 0. If u is a global solution of Problem (1.1), (1) Problem (1.1) admits a steady state SM satisfying (2.1); (2) u(·, t) → SM (·) in C(Br,R) as t → ∞.
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