diffusion-convection equation on annular domains

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

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].
On the other hand, it can serve as a typical model-case in the theory of parabolic PDEs.Indeed, it is the one of the simplest examples (along with Burger's equation) of a parabolic equation with a nonlinearity depending on the first-order spatial derivatives of u.
A basic fact about (1.1) is that the solutions satisfy a maximum principle: min Br,R u 0 ≤ u(x, t) ≤ max Br,R u 0 , x ∈ B r,R , 0 ≤ t < T. (1.2) Since Problem (1.1) is well-posed in C 1 locally in time, only three possibilities can occur: (I) u exists globally and is bounded in (II) u blows up in finite time in C 1 norm (finite time gradient blowup): (III) u exists globally but is unbounded in C 1 (infinite time gradient blowup): For M = 0 and u 0 C 1 sufficiently small, it is known that (I) occurs and u converges to the unique steady state S 0 ≡ 0. On the contrary, if u 0 suitably large, (II) occurs (see [5] and [8]).
For M > 0, the situation is slightly more complicated.There exists a critical value M c (see Section 2 below for its existence) such that (1.1) has a unique, regular and radial (S For one dimensional case (see [8]), it was proved among other things that, if M > M c , then all solutions of (1.1) satisfy (II), and if 0 < M < M c , then both EJQTDE, 2012 No. 39, p. 2 (I) and (II) can occur.Moreover, in [9], it was shown that if 0 ≤ M < M c , then all global solutions of (1.1) are bounded in C 1 , and they converge to S M in C 1 .If (II) occurs, with the assumption on the initial data so that the solution is monotonically increasing both in time and in space, Zhang and Hu in [8] studied the blowup estimate and obtained that the blowup rate is close to ln 1 T −t but not exactly equal to ln 1 T −t , which is very interesting because the blowup estimate can not be predicted by the usual self-similar transformations.For N (> 1) dimensional and zero-Dirichlet problem, in [10], Zhang and Li considered the gradient estimate near the boundary and the blowup rate of the radial case.
The purpose of this paper is to extend the results of [5,8,9,10] to Problem (1.1), i.e., if M = M c and u 0 ≤ S Mc , then (III) occurs and, u converges in C(B r,R ) exponentially to S Mc , as well as u ρ (r, t) grows up exponentially to infinity.Therefore, we provide a classification of large time behavior of the solutions of (1.1) for arbitrary spatial dimension.Our main results are as follows: then all global solutions of (1.1) converges in C(B r,R ) to S M .Moreover, if u 0 ≤ S M , then the solution of (1.1) is global in time and converges in C 1 (B r,R ) to S M , and we have the uniform exponential convergence where λ 1 is the first eigenvalue of (3.2) (see Section 3 below). ( Moreover, if u 0 ≤ S M , then the solution of (1.1) is global in time and converges in C 1 (B r,R ) to S M , and we have the uniform exponential convergence as well as the blowup estimate where λ 1 is the first eigenvalue of (4.1) (see Section 4 below).

Stationary states and global existence
From the maximum principle, if Problem (1.1) admits a steady state S M (x), then it is unique and radial, and if (2.1) EJQTDE, 2012 No. 39, p. 3 For M > 0, from the existence theory of ODEs, we know that S M,ρ > 0 in (r, R].Then S M,ρ satisfies e SM,ρ ≤ −S M,ρρ ≤ ce SM,ρ in (r, R], where c > 1 is some constant.We consider a special case where S M,ρ (r) = ∞, so we have Proof.
(1) Let χ(ρ) be the solution of and κ(ρ) be the solution of So by the maximum principle, we have u t ≥ 0 in B r,R for all t > 0. As a consequence, there exists a function S M ∈ B r,R such that for all x ∈ B r,R , u(x, t) → S M (x) as t → ∞.Similar to the proof of [7, Theorem 3.2] or [10, Theorem 3.1], we have where δ(x) = dist(x, ∂B r,R ).Parabolic estimates imply that for any small ε > 0, for some 0 < α < 1, there holds By the diagonal procedure, there exists a sequence (2) Define w(t) = u(t) − S M , φ(t) = w(t) ∞ .It follows from [7] that φ(t) is non-increasing for all t > 0. Set We know that Choose a sequence t n → ∞ and set u n (•, Then the functions u n then satisfy Moreover, (2.5) implies that {u(τ ); τ ≥ 0} is relatively compact in C(Q).For each fixed t ≥ 0, we may thus find a subsequence n k such that Setting w(t) := z(t) − S M , then w(t) satisfies w| ∇SM +s∇ e w |∇SM +s∇ e w| ds ∈ C(Q).Assume for contradiction that l > 0. Since w(•, 2) ∈ C 0 (B r,R ), there exists )), we may apply the strong maximum principle to deduce that | w| = l in B(x 0 , ρ) × [1,2].But by letting ρ → δ(x 0 ), this contradicts w(•, 2) ∈ C 0 (B r,R ).Therefore, l = 0. Since the sequence t n was arbitrary, we conclude that lim t→∞ u(t) − S M ∞ = 0, and the assertion (2) is proved.

Subcritical case M < M c
In this section, we assume that u 0 ≤ S M in B r,R .By the maximum principle, we have −χ − µκ ≤ u ≤ S M for t < T , where µ is a suitably large constant.Similar to the proof of [7, Theorem 3.2] or [10, Theorem 3.1], we can get that ∇u blows up only on the boundary.So u exists globally and ∇u is uniformly EJQTDE, 2012 No. 39, p. 5 bounded in B r,R × [0, ∞).So standard arguments imply that u(•, t) → S M (•) as t → ∞.
We consider the eigenvalue problem So Equation (3.1) can be written as It is equivalent to where a(ρ) satisfies Let ϕ(ρ) be the first eigenfunction and λ 1 be the corresponding eigenvalue.
Let u be the (global) solution of (1.1) with −χ − µκ as the initial data for some µ > 0 such that −χ − µκ ≤ u 0 .By the comparison principle, we get u ≤ u.Therefore S M − u ≤ v := S M − u.Since u is radially symmetric, then, by Taylor's expansion up to second order, we obtain where Let ϕ(ρ) be the first eigenfunction of (3.2) and choose a constant C > 0 such that u 0 + χ + µκ ≤ Cϕ.We observe that Ce −λ1t ϕ is a super-solution of (3. which implies Theorem 2.1 (1).

Critical case M = M c
In this section, we assume that u 0 ≤ S Mc in B r,R .We claimed that u exists globally.Assume for contradiction that T * < ∞.By the maximum principle, we have u ≥ −χ − µκ for some µ, so ∇u blows up only on the boundary ∂B r by the similar proof of contradicting to the blowup of ∇u at t = T * .
In the following, we use the idea of [6] to study the asymptotic behavior of the radial solution of Problem (1.1).
3).Then by the comparison principle, we get S M −u ≤ v ≤ Ce −λ1t ϕ.By the strong maximum principle, we get u(•, t 0 ) < S M (•) and −u ν (•, t 0 ) < −S M,ν (•) on the boundary of B r,R .Without loss of generality we assume that t 0 = 0.So there is a radially symmetric function ϑ(ρ) such that u 0 < ϑ < S M .Let u be the EJQTDE, 2012 No. 39, p. 6 solution of (1.1) with ϑ as the initial data.Then by comparison principle, we have u ≤ u ≤ S M .Let v = S M − u, by the Taylor's expansion up to the second order, we also get (3.3) with replaced v by v. Since |F
.2) So we can deduce that there exists M c > 0 such that if M > M c , then Problem (1.1) does not admit a steady state, if 0 < M < M c , then Problem (1.1) admits a unique regular steady state S M ∈ C 2 ([r, R]), and if M = M c , then Problem (1.1) still admits a steady state S Mc ∈ C([r, R]) ∩ C 2 ((r, R]), which is singular in the sense that it has infinite derivative on the boundary ∂B r .Theorem 2.1 Assume that M ≥ 0. If u is a global solution of Problem (1.1), then (1) Problem (1.1) admits a steady state S M satisfying (2.1);