DECAY AT INFINITY FOR PARABOLIC EQUATIONS

OLIVER C. SCHNÜRER AND HARTMUT R. SCHWETLICK

Abstract. We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to infinity. We characterize velocity fields, so that positive solutions decay or lift-off at spatial infinity as time tends to infinity. This addresses the question of stability of the zero solution for decaying perturbations.

1. Introduction

Consider solutions u : ℝ^{n} × ℝ_{+} → ℝ of the parabolic equation

(1.1)  u̇ = Δu + ⟨b, ∇u⟩ in ℝ^{n} × ℝ_{+},

where b : ℝ^{n} → ℝ^{n} is of the form b(x) = \frac{x}{|x|}ψ(|x|), such that the flow equation preserves rotational symmetry of solutions. We are interested in the long-time behavior of solutions u that decay initially, u(x, 0) → 0 for |x| → ∞. For bounded vector fields b, a solution lifts off, i. e.

lim_{|x|→∞} lim_{t→∞} u(x, t) > 0,

or converges to zero uniformly as t → ∞. Convergence to zero corresponds to dynamical stability of the zero solution. In [2], stability of symmetric gradient Kähler-Ricci solitons was proved for perturbations that decay at infinity. The equation arising there behaves similarly to (1.1), when ⟨b, x⟩ ≥ 0. We prove in Theorem 3.1 that in this case bounded solutions u of (1.1) which decay initially, must tend to zero as t → ∞.

The situation is different when ⟨b, x⟩ ≤ 0. For b = −x, no positive solution tends to zero as t → ∞. However, it will be shown that linear growth at infinity is not necessary for the lift-off phenomenon. We characterize precisely the critical growth rate for large x for the vector fields b(x) = −\frac{x}{|x|}ψ(|x|) to be ψ(r) ≈ 1/r. In Section 2 we will show that vector fields of faster growth lead to the lift-off of positive solutions. On the contrary, we show in Section 3 that slower growth forces solutions to converge uniformly to zero.

The dependence of the behavior of solutions on the sign of ⟨b, x⟩ can be understood as follows. Rotationally symmetric solutions that decay monotonically in |x| may serve as barriers. For these functions, ⟨x, ∇u⟩ ≤ 0, so ⟨b, ∇u⟩ ≤ 0 for ⟨b, x⟩ ≥ 0 and u tends faster to zero than for the heat equation, where u is known to tend to zero. If ⟨b, x⟩ ≤ 0 comparison with the heat equation is not applicable anymore, and we may expect that large values of ⟨b, ∇u⟩ prevent the solution u to decay to zero.

Date: August 2005.

2000 Mathematics Subject Classification. 35B40.

Key words and phrases. Asymptotic behavior, long-time behavior, entire solutions.

We wish to thank Albert Chau, Klaus Ecker, Jürgen Jost, Stefan Müller, the Max Planck Institute for Mathematics in the Sciences, and Free University Berlin for discussions and support.

2. Lift-Off

2.1. Convergence to a Constant. Throughout that paper we will use the following lemma. Here, the vector field b(x) = \frac{x}{|x|}ψ(|x|) is not assumed to point in a specific direction. We remark that we use the space C^{α} (ℝ^{n}) for uniformly bounded functions having bounded Hölder semi-norm with exponent α.

Lemma 2.1. Let ψ : ℝ_{+} → ℝ, 0 < α < 1 be such that b(x) = \frac{x}{|x|}ψ(|x|) ∈ C^{α} (ℝ^{n}). Let u_{0} ∈ C^{2+α} (ℝ^{n}). Then there exists a unique positive solution u ∈ C^{2+α, 1+α/2} (ℝ^{n} × ℝ_{+}) of

(2.1)  \begin{cases}u̇ = Δu + ⟨b, ∇u⟩ & in ℝ^{n} × ℝ_{+}, \\ u(·, 0) = u_{0} & in ℝ^{n}. \end{cases}

If u_{0} is rotationally symmetric and monotonically decreasing (or increasing) in radial direction (i. e. [0, ∞) ∋ λ ↦ u_{0}(λx) is a monotonically decreasing (or increasing) function for all x ∈ ℝ^{n}), then u(·, t) shares these properties for any t > 0. Moreover, the assumptions above guarantee that u(·, t) tends locally uniformly to a constant as t → ∞. If u_{0} ≥ 0 but u_{0} is not identically zero, then u(x, t) > 0 for any x ∈ ℝ^{n}, t > 0.

Proof. First note that [4, Theorem 9.2.3] provides the claimed regularity for all time and [4, Theorem 8.11.1] gives uniform bounds in C^{2+α, 1+α/2}(ℝ^{n} × ℝ_{+}).

It is only here that we use the boundedness of ψ. It seems possible to weaken this hypothesis. However, already for bounded functions ψ, we observe both lift-off of solutions as well as convergence to zero. Therefore we will not pursue this issue any further.

Let u_{0} be rotationally symmetric and R be any orthogonal transformation on ℝ^{n}. Then u(Rx, t) is another solution to our initial value problem . As u(x, t)−u(Rx, t) vanishes at t = 0, this is preserved during the evolution [3]. Thus u(·, t) stays rotationally symmetric during the evolution.

Let u_{0} be monotonically decreasing in radial direction. As in [2, Appendix A], we obtain that u(·, t) is radially decreasing for any fixed t > 0.

Assume now that u_{0} is rotationally symmetric and monotonically decreasing in radial direction. If u_{0} is increasing, it suffices to consider −u_{0} as (2.1) is a linear equation. We wish to show that u tends to a constant as t → ∞. The following argument is similar to [1, 2, 5]. Observe that u(·, t) attains its maximum at x = 0. According to the strong maximum principle, u(0, t) is strictly decreasing in time or u is a constant. The maximum principle [3] implies that inf u(·, t) is non-decreasing in time.

Let h := lim_{t→∞} u(0, t). If lim_{t→∞} u(x, t) = h for every x ∈ ℝ^{n}, our uniform a priori estimates guarantee that u(·, t) converges locally uniformly to h as t → ∞. Otherwise, we find x_{0} ∈ ℝ^{n} and a sequence t_{k} → ∞ such that u(x_{0}, t_{k}) ≤ h − ε for some positive ε. Define u_{k}(x, t) := u(x, t + t_{k}). As u is uniformly bounded in C^{2+α, 1+α/2} (ℝ^{n} × ℝ_{+}), we can extract a subsequence of u_{k} that converges locally uniformly in C^{2, 1} (ℝ^{n} × ℝ) to a solution w of (2.1) in ℝ^{n} × ℝ. We obtain

w(x_{0}, 0) ≤ h − ε < h = w(0, 0),

so w is not constant. The function w attains its maximum at w(0, t) for all t ∈ ℝ. According to the strong maximum principle, this is impossible. We deduce that u(x, t) → h as t → ∞, locally uniformly in x.

Applying the strong maximum principle once again yields that a non-negative solution becomes immediately positive. □

2.2. Example for Lift-Off. Before we state our theorem concerning solutions lifting off at infinity for t → ∞, we wish to investigate the following model case.

Consider the evolution equation

(2.2)  \begin{cases}u̇ = Δu − ⟨x, ∇u⟩ & in ℝ^{n} × ℝ_{+}, \\ u(·, 0) = u_{0} & in ℝ^{n} \end{cases}

for some u_{0} ∈ C^{2+α}. Let w be the C^{2+α, 1+α/2}-solution to

\begin{cases}ẇ = Δw & in ℝ^{n} × [0, 1], \\ w(·, 0) = u_{0} & in ℝ^{n} \end{cases}

as in Lemma 2.1. It is easy to check that

u(x, t) := w (e^{−t}x, 1 − \frac{1}{2}e^{−2t})

solves (2.2). If u_{0} is positive, we see that u converges exponentially fast to a positive constant as t → ∞. In particular, this shows that solutions to (2.2), which decay at spatial infinity initially, do not necessarily decay at spatial infinity in the limit t → ∞. More precisely,

lim_{|x|→∞} lim_{t→∞} u(x, t)

may be different from zero. Note that the order of the limits is important. We remark that a formal calculation, that can be made precise for u with good decay at spatial infinity, suggests that |∫_{ℝ^{n}} u| increases exponentially in time,

\frac{d}{dt} ∫_{ℝ^{n}} u = ∫_{ℝ^{n}} Δu − ⟨x, ∇u⟩ = ∫_{ℝ^{n}} div(∇u − xu) + nu = n ∫_{ℝ^{n}} u.

2.3. Lift-Off Theorem. The following result shows that unbounded vector fields b are not necessary to let solutions lift off at spatial infinity as t → ∞.

Theorem 2.2. Let ψ : ℝ_{+} → ℝ be such that b(x) = −\frac{x}{|x|}ψ(|x|) ∈ C^{α} (ℝ^{n}) for some 0 < α < 1. We assume that

(2.3)  lim inf_{r→∞} \frac{1}{log r} ∫_{0}^{r} ψ(ρ)dρ > n.

Let 0 ≤ u_{0} ∈ C^{2+α}(ℝ^{n}) with u_{0} ≢ 0. In addition, we assume that u_{0} is rotationally symmetric and monotonically decreasing in radial direction. Then the unique positive solution u ∈ C^{2+α, 1+α/2}(ℝ^{n} × ℝ_{+}) of

(2.4)  \begin{cases}u̇ = Δu − ⟨\frac{x}{|x|}ψ, ∇u⟩ & in ℝ^{n} × ℝ_{+}, \\ u(·, 0) = u_{0} & in ℝ^{n} \end{cases}

is rotationally symmetric and satisfies

lim_{t→∞} sup_{x∈ℝ^{n}} u(x, t) = inf_{x∈ℝ^{n}} lim_{t→∞} u(x, t) > 0.

Proof. Recalling Lemma 2.1 it only remains to prove the last convergence claim. Define φ : ℝ_{+} → ℝ_{+} by

φ(r) := exp (− ∫_{0}^{r} ψ(ρ) dρ) .

Observe that φ is bounded and solves φ′ + φψ = 0. Define also

I_{R}(t) := ∫_{B_{R}(0)} φ(|x|) · u(x, t) dx

and

I(t) := ∫_{ℝ^{n}} φu.

Our assumptions ensure that there exists ε > 0, r_{0} > 0 such that φ(r) ≤ r^{−n−ε} for all r ≥ r_{0}. Thus, we deduce

∫_{ℝ^{n}} φ(|x|)dx < ∞,

I_{R}(0) ≤ I(0) < ∞, and 0 < I(0). We compute for φu = φ(|x|)u(x, t)

\frac{d}{dt}(φu) =φu̇
=φΔu − φ ⟨\frac{x}{|x|}ψ, ∇u⟩
= div(φ∇u) − ⟨φ′ \frac{x}{|x|}, ∇u⟩ − φ ⟨\frac{x}{|x|}ψ, ∇u⟩
= div(φ∇u) − (φ′ + φψ) ⟨\frac{x}{|x|}, ∇u⟩
= div(φ∇u).

So we obtain that

\frac{d}{dt}I_{R}(t) = ∫_{∂B_{R}} φ ⟨\frac{x}{|x|}, ∇u⟩ → 0 for R → ∞,

as |∇u| is bounded and φ decays faster than r^{−n} at infinity. Thus I(t) is time independent.

The solution u(·, t) stays non-negative during the evolution and tends to a constant as t → ∞, uniformly on compact subsets of ℝ^{n}. Since I(t) is time independent and ∫_{ℝ^{n}} φ(|x|)dx < ∞, this constant has to be positive. □

Remark 2.3. The conditions on ψ in Theorem 2.2 are fulfilled, if ψ is C^{α}, vanishes in a neighborhood of the origin, and, for r ≥ r_{0} > 0, it is of the form

ψ(r) = Ar^{β} with \begin{cases}A > 0, & β > −1; \\ A > n, & β = −1. \end{cases}

Remark 2.4. If u_{0} in Theorem 2.2 is not rotationally symmetric, u also lifts off. At any positive time ε, u is strictly positive. So there exists ũ_{0} fulfilling the assumptions on u_{0} in Theorem 2.2 and ũ_{0} ≤ u(·, ε). Let ũ be the solution to (2.4) with ũ(·, 0) = ũ_{0}. According to the maximum principle, ũ(x, t) ≤ u(x, t+ε). As ũ(·, t) converges locally uniformly to a positive constant, we obtain that

inf_{x∈ℝ^{n}} lim inf_{t→∞} u(x, t) > 0.

3. Convergence to Zero

In the following, we investigate the behavior of solutions in the sub-critical case, that is, the vector field b fails to obey the growth condition (2.3).

Theorem 3.1. Let ψ : ℝ_{+} → ℝ, 0 < α < 1 be such that b(x) = −\frac{x}{|x|}ψ(|x|) ∈ C^{α} (ℝ^{n}) and

lim sup_{r→∞} \frac{1}{log r} ∫_{0}^{r} max \{ψ(ρ), 0\} dρ < n.

Let 0 ≤ u_{0} ∈ C^{2+α}(ℝ^{n}) ∩ L^{1}(ℝ^{n}) with u_{0} ≢ 0. In addition, we assume that u_{0} is rotationally symmetric and monotonically decreasing in radial direction. Then the unique positive solution u ∈ C^{2+α, 1+α/2}(ℝ^{n} × ℝ_{+}) of

\begin{cases}u̇ = Δu − ⟨\frac{x}{|x|}ψ, ∇u⟩ & in ℝ^{n} × ℝ_{+}, \\ u(·, 0) = u_{0} & in ℝ^{n} \end{cases}

is rotationally symmetric and satisfies

lim_{t→∞} sup_{x∈ℝ^{n}} u(x, t) = 0.

Proof. According to Lemma 2.1, we only have to show that

lim_{t→∞} sup_{x∈ℝ^{n}} u(x, t) = 0.

Rotational symmetry and monotonicity in radial direction imply that

(3.1)  ⟨x, ∇u⟩ ≤ 0.

For φ : ℝ_{+} → ℝ_{+} given by

φ(r) = exp (− ∫_{0}^{r} ψ_{+}(ρ) dρ) ,

we observe that φ is bounded and solves φ′ + φψ_{+} = 0. Here we used the decomposition of ψ in its positive and negative part, ψ = ψ_{+} − ψ_{−}.

Define I_{R}(t) and I(t) as above in the proof of Theorem 2.2. Our assumptions on ψ and u_{0} ensure that

I_{R}(0) ≤ I(0) < ∞.

We compute for φu = φ(|x|)u(x, t) as above

\frac{d}{dt}(φu) = div(φ∇u) − (φ′ + φψ_{+}) ⟨\frac{x}{|x|}, ∇u⟩ + φψ_{−} ⟨\frac{x}{|x|}, ∇u⟩ .

Recalling φ′ + φψ_{+} = 0 and (3.1), we deduce that

\frac{d}{dt}(φu) ≤ div(φ∇u).

Using (3.1) again, we get

\frac{d}{dt}I_{R}(t) ≤ ∫_{∂B_{R}} φ ⟨\frac{x}{|x|}, ∇u⟩ ≤ 0.

So we obtain for 0 ≤ t_{1} ≤ t_{2} the inequality

I(t_{1}) ≥ I(t_{2}).

By assumption, there exists r_{0} > 0 such that φ(r) ≥ r^{−n} for all r ≥ r_{0}. Thus, we have

∫_{ℝ^{n}} φ(|x|)dx = ∞.

The solution u stays non-negative during the evolution. According to Lemma 2.1, the function u(·, t) tends to a constant as t → ∞, uniformly on compact subsets of ℝ^{n}. As I(t) is non-increasing in time, this constant has to be zero. □

Remark 3.2. The conditions on ψ in Theorem 3.1 are fulfilled, if ψ is smooth, vanishes in a neighborhood of the origin, and, for r ≥ r_{0} > 0, it is of the form

ψ(r) = Ar^{β} with \begin{cases}A ∈ ℝ, & β < −1; \\ A < n, & β = −1. \end{cases}

It is easy to check directly that the proof of Theorem 3.1 remains valid for a vector field with ψ(r) = nr^{−1} outside a compact set.

Remark 3.3. We want to note that our theorems provide a sharp characterization of the leading order of the growth rate. The functions

ψ(r) = \frac{1}{r} (n + \frac{α}{log r} )

have all critical growth as

lim inf_{r→∞} \frac{1}{log r} ∫_{2}^{r} ψ(ρ)dρ = n.

However, we can show that they lead to lift-off only if α > 1, whereas α ≤ 1 yields decay to zero. This follows from the respective proofs of the above theorems and a more detailed investigation of the integrability of φ(|x|).

Remark 3.4. Similar to Remark 2.4, there is also a version of Theorem 3.1 for u_{0} not being rotationally symmetric. Here, we may allow u_{0} to change sign too. We can find a barrier β, such that β ≥ u_{0} ≥ −β and β fulfills the conditions on u_{0} in Theorem 3.1. As the solution starting with initial datum β tends to zero, the maximum principle implies that u(·, t) converges uniformly to zero as t → ∞.

References

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2. Albert Chau and Oliver C. Schnürer, Stability of gradient Kähler-Ricci solitons, 2003, MPI-MIS Preprint 39/2003, http://www.mis.mpg.de/, Comm. Anal. Geom., to appear.

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Oliver Schnürer, Freie Universität Berlin, Arnimallee 2–6, 14195 Berlin, Germany

E-mail address: Oliver.Schnuerer@math.fu-berlin.de

Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

E-mail address: H.Schwetlick@bath.ac.uk