Gradient-Like Parabolic Semiflows on BUC(lR^{N})

Daniel Daners

School of Mathematics and Statistics University of Sydney, NSW 2006 Australia

daners@maths.usyd.edu.au

Sandro Merino

Mathematisches Institut Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich Switzerland

merino@math.unizh.ch

June 6, 1997

Abstract

We prove that a class of weighted semilinear reaction diffusion equations on R^{N} generates gradient-like semiflows on the Banach space of bounded uniformly continuous functions on R^{N}. If N = 1 we show convergence to a single equilibrium. The key for getting the result is to show the exponential decay of the stationary solutions, which is obtained by means of a decay estimate of the kernel of the underlying linear semigroup.

1 Introduction

We consider a class of semilinear reaction diffusion equations on the whole of R^{N}. A typical exponent being given by

\{ \begin{matrix}∂_{t}u − Δu & = & m(x)u − u^{3} & in R^{N} × (0, ∞) , \\ u(x, 0) & = & u_{0}(x) & in R^{N} . \end{matrix}  (1.1)

The positive and negative parts m^{+} and m^{−} of the bounded weight function m are assumed to have the following properties:

• m^{+} vanishes at infinity.

• There exist constants r, η > 0 such that for all x ∈ R^{N} the estimate ∫_{B(x,r)} m^{−}(x) dx ≥ η holds.

The nonnegative function m^{−} is called a strongly absorbing potential. We are interested in continuous initial data u_{0}(x) that do not have any specified decay at infinity but are required to be bounded. More precisely, we choose the phase space for (1.1) as BUC(R^{N}), the Banach space of bounded uniformly continuous real-valued functions on R^{N}. It is shown in [13] that (1.1) generates a global continuous semiflow (φ, BUC(R^{N})) of class AK on BUC(R^{N}) that consists of classical solutions of (1.1) if m is Hölder continuous.

We remark that in general the canonical Liapunov function associated with (1.1)

L(u(x)) := \frac{1}{2} ∫_{R^{N}} |∇u(x)|^{2}dx − ∫_{R^{N}} (\frac{1}{2}m(x)u(x)^{2} − \frac{1}{4}u(x)^{4}) dx (1.2)

is of little use for orbits starting in BUC(R^{N}). E.g. if the initial condition u_{0}(x) is a nonzero constant and if m^{+} has compact support, then

L(φ(t, u_{0})) = ∞

for t ≥ 0. To overcome this difficulty we will make use of the results in [13], where it is shown that the semiflow (φ, BUC(R^{N})) possesses a compact global B-attractor M of finite Hausdorff dimension. Furthermore M is contained in an order interval V := [\underline{w}, \overline{w}] of the ordered Banach space BUC(R^{N}). Here \underline{w} is the smallest and \overline{w} is the greatest stationary solution of (1.1). The principal part of this paper is devoted to proving that the stationary solutions of (1.1) have exponential decay at infinity. The underlying idea is to write (1.1) as

∂_{t}u − Δu + m^{−}(x)u = m^{+}(x)u − u^{3},

and to study the kernel k(t, x, y) of the semigroup associated with the generator Δ − m^{−}. In Section 2 we will prove the estimate

0 ≤ k(t, x, y) ≤ ct^{− \frac{N}{2}} e^{−ηt}e^{− \frac{|x−y|^{2}}{ct}}

for some constants c, η > 0. This estimate will then imply the exponential decay at infinity of the stationary solutions of (1.1). As a consequence the elements of the order interval V and, in particular, of the B-attractor M have exponential decay. This will imply that the Liapunov function is uniformly bounded below on the B-attractor M. As a consequence of the LaSalle invariance principle (e.g. [12, Theorem 2.3]) the ω-limit set ω(u_{0}) of each element u_{0} ∈ BUC(R^{N}) is contained in the set of stationary points of the semiflow, i.e., the semiflow induced by (1.1) is gradient-like. In one dimension it turns out that ω(u_{0}) consists of a single equilibrium.

Acknowledgement. We thank E.N. Dancer for valuable discussions, especially on the last part of this paper. This paper was initiated while the second author visited Australia. This visit was made possible by Yihong Du from the Department of Mathematics at the University of New England, Armidale, and we like to thank him for the generous support and the warm hospitality.

2 Kernel Estimates

Suppose that we are given a self adjoint strongly uniformly elliptic operator of second order on R^{N} of the form

A = − ∑_{i,j=1}^{N} \frac{∂}{∂x_{i}} (a_{ij}(x)\frac{∂}{∂x_{j}} u)  (2.1)

with a_{ij} = a_{ji} ∈ L_{∞}(R^{N}). By strongly uniformly elliptic we mean as usual that there exists α_{0} > 0 such that

∑_{i,j=1}^{N} a_{ij}(x)ξ_{i}ξ_{j} ≥ α_{0}|ξ|^{2}  (2.2)

for all x, ξ ∈ R^{N}. We consider perturbations of the above operator with a nonnegative potential satisfying a condition introduced in [1].

Definition 2.1 We say that a potential V ∈ L_{∞}(R^{N}) is strongly absorbing if V ≥ 0 and there exist constants r, η > 0 such that

∫_{B(x,r)} V (ξ) dξ ≥ η  (2.3)

for all x ∈ R^{N}. Here, B(x, r) = {y ∈ R^{N}: |x − y| < r}.

Using form methods one can show that an appropriate restriction of −A to L_{p}(R^{N}) generates a strongly continuous analytic semigroup on L_{p}(R^{N}) for all p ∈ [1, ∞). A similar statement is true for the operator −(A+V). One of the main results in [1] is that if V ≥ 0 then, the semigroup T_{V} (t) associated with −(A + V) is exponentially stable on L_{p}(R^{N}) for some (or equivalently all) p ∈ [1, ∞) if and only if V is strongly absorbing. By exponentially stable we mean that there exist γ_{0}, c_{0} > 0 such that

∥T_{V} (t)∥_{L(L_{p},L_{p})} ≤ c_{0}e^{−γ_{0}t}  (2.4)

for all t ≥ 0. Alternative proofs of the result can be found in [2, 5]. The key in the proof is that the kernel k_{0}(t; x, y) of T_{0}(t) satisfies upper and lower Gaussian estimate, that is, for some constant c > 0

c^{−1}t^{−N/2} exp(−\frac{c|x − y|^{2}}{t}) ≤ k_{0}(t; x, y) ≤ ct^{−N/2} exp(−\frac{|x − y|^{2}}{ct}) . (2.5)

By a simple comparison argument it is clear that for any nonnegative potential the upper estimate for the corresponding semigroup kernel remains valid. For strongly absorbing potentials we shall prove the following better upper estimate.

Theorem 2.2 Suppose that the above assumptions on A are satisfied and that V is a strongly absorbing potential. Denote the kernel of the semigroup T_{V} (t) by k_{V} (t; x, y). Then there exist constants c, η > 0 such that

0 ≤ k_{V} (t; x, y) ≤ ct^{−N/2} exp(−ηt) exp(−\frac{|x − y|^{2}}{ct})  (2.6)

for all t > 0 and x, y ∈ R^{N}.

Proof To prove (2.6) we make use of (2.4) and the results in [6, Section 3.2]. First of all note that

∥T_{V} (t)∥_{L(L_{p},L_{q})} ≤ ct^{− \frac{N}{2} ( \frac{1}{q} − \frac{1}{p})}  (2.7)

whenever 1 ≤ p ≤ q ≤ ∞. This follows for instance from the upper bound in (2.5), which, as we remarked earlier, is also valid for k_{V} (t; x, y). By using the semigroup property as well as (2.4) and (2.7) we obtain

\begin{aligned}∥T_{V} (t)∥_{L(L_{2},L_{∞})} & ≤ ∥T_{V} (t/2)∥_{L(L_{2},L_{∞})}∥T_{V} (t/2)∥_{L(L_{2},L_{2})} \\ & ≤ c_{0}(t/2)^{−N/4}e^{− \frac{η_{0}}{2} t} = ct^{−N/4}e^{−ηt} \end{aligned}  (2.8)

for some c, η > 0 and all t > 0. To get the “non-diagonal” estimate we modify the proof of the heat kernel estimate associated with an operator of the form (2.1) given in [6]. We shall use the notation from [6]. First of all it follows from [6, Theorem 2.2.4] and (2.8) that for all u ∈ W_{2}^{1}(R^{N}) the logarithmic Sobolev inequality

∫_{R^{N}} u^{2} log_{+} u dx ≤ εa(u, u) + β(ε)∥u∥_{2}^{2} + ∥u∥_{2}^{2} log ∥u∥_{2}  (2.9)

holds with

β(ε) = c − \frac{N}{4} log ε − ηε  (2.10)

for some constant c ∈ R. Here, a(· , ·) is the Dirichlet form associated with −A + V. Next note that the calculations in [6, Lemma 3.2.1] are not affected by the presence of a nonnegative potential, and so the result and its consequences, not using the explicit form of β(ε) remain valid. In particular, Theorem 3.2.4 remains true. Let us now modify the proof of Theorem 3.2.5. We make the same choice of parameters as there, that is, we set

μ = \frac{1}{2} and ε(p) = λ2^{λ}tp^{−λ},

where λ > 1. Then

t = ∫_{2}^{∞} p^{−1}ε(p) dp.

In order to estimate M(t) we only need to consider one additional term, namely

− ∫_{2}^{∞} p^{−2}ηλ2^{λ}tp^{−λ} dp = −\frac{λ}{2(1 + λ)}ηt.

Since λ > 1 this yields

M(t) ≤ −\frac{N}{4} log t + \frac{λ}{λ − 1}α^{2}t − \frac{η}{4}t + c_{λ}.

Choosing λ such that 1 + δ = λ(λ − 1)^{−1} we get an estimate similar to the one in Theorem 3.2.5, but with the additional exponentially decaying factor exp(−ηt/4). This factor does not affect the arguments given in Corollary 3.2.6, Theorem 3.2.7 and Corollary 3.2.8, so we get a bound of the form (2.6).

Corollary 2.3 If the potential V is strongly absorbing then the weak Green’s function g_{V} (x, y) associated with −(A + V) satisfies an estimate of the form

0 ≤ g_{V} (x, y) ≤ cη^{\frac{N}{2} −1}(\frac{|x − y|}{λ})^{1− \frac{N}{2}} K_{\frac{N}{2} −1}(λ|x − y|)  (2.11)

for some c > 0 and x ≠ y, where K_{\frac{N}{2} −1} is the Modified Bessel Function,

and λ := 2\sqrt{η/c}

Proof First note that g_{V} (x, y) is given by the Laplace Transform of the kernel k_{V} (t; x, y). More precisely,

g_{V} (x, y) = ∫_{0}^{∞} k_{V} (t; x, y) dt  (2.12)

for all x, y ∈ R^{N} with x ≠ y. Hence, by Theorem 2.2 we have

0 ≤ g_{V} (x, y) ≤ c ∫_{0}^{∞} t^{−N/2} exp(−ηt) exp(−\frac{|x − y|^{2}}{ct}) dt.  (2.13)

Recall that the modified Bessel Functions can be represented by

K_{ν}(ρ) = \frac{1}{2}(\frac{ρ}{2})^{ν} ∫_{0}^{∞} s^{−ν−1} exp(−\frac{ρ^{2}}{4s} − s) ds  (2.14)

for all ν ∈ R and ρ ∈ R \ {0} (e.g. [8, page 82, formula (23)]). Hence, by the substitution s = ηt, and setting λ := 2\sqrt{η/c} we get from (2.13) that (2.11) holds for x ≠ y.

Remark 2.4 Using asymptotic expansions of the Modified Bessel Functions (see [8, Section 7.2]) we get that

0 ≤ g_{V} (x, y) ≤ \{ \begin{matrix}c exp(−|x − y|/c) & if N = 1 \\ c log(|x − y|/c) exp(−|x − y|/c) & if N = 2 \\ c|x − y|^{−N+2} exp(−|x − y|/c) & if N ≥ 3 \end{matrix} (2.15)

for some constant c > 0. In particular, we see that the Green’s Function of −A + V decreases exponentially in space if V is strongly absorbing.

Remark 2.5 To prove (2.11) we only used an estimate of the form (2.6). Therefore, if we have a potential V = V_{1} − V_{2} with V_{1} strongly absorbing and V_{2} ∈ L_{∞}(R^{N}) and ∥V_{2}∥_{∞} small enough we still get an estimate of the form (2.6) and (2.11).

3 Nonlinear Elliptic Equations

Consider the nonlinear elliptic equation

Au = f(x, u), x ∈ R^{N}.  (3.1)

For simplicity we will only discuss properties of nonnegative solutions of (3.1). But under further restrictions on the nonlinearity f the arguments remain valid for arbitrary solutions (compare [13, Example 3.3b)]). Following [13, Section 3] we impose the following restrictions on the nonlinearity f(x, ξ).

f(·, ξ) ∈ BUC^{μ}(R^{N}), for some μ ∈ (1, 2), uniformly for ξ in bounded subsets of R.

(3.2)

The derivative ∂_{ξ}f exists and is uniformly Lipschitz continuous on sets of the form R^{N} × B, where B is a bounded subset of R.

(3.3)

There exists a constant M ≥ 0 such that f(x, c) ≤ 0 but

f(·, c) ≢ 0 for c ≥ M and x ∈ R^{N}. (3.4)

f is of the form f(x, ξ) = h(x, ξ)ξ. Moreover, for each u ∈ X^{+} the positive part h^{+}(· , u(·)) of h(· , u(·)) is in C_{0}(R^{N}), whereas the negative part h^{−}(· , u(·)) is strongly absorbing.

(3.5)

Under the assumptions considered in the introduction the nonlinearity in (1.1) satisfies all the above properties. We shall prove the following theorem.

Theorem 3.1 There exists an exponentially decaying function ψ ∈ C_{0}(R^{N}) such that any solution u ∈ L_{∞}(R^{N}) satisfies |u(x)| ≤ ψ(x) for all x ∈ R^{N}.

Proof Suppose that u ∈ L_{∞}(R^{N}) is a nonnegative solution of (3.1) and that ε > 0 is given. Setting

m_{1}(x) := h^{−}(x, u(x)) − ε, m_{2}(x) := h^{+}(x, u(x)) − ε,

we may rewrite (3.1) as

Au + m_{1}u = m_{2}u.  (3.6)

Furthermore note that by (3.5) the potential m_{1} + ε is strongly absorbing. Also note that

m_{2}(x) ≤ (h(x, u(x)) − ε)^{+}.

Therefore, by the results of the previous section, we see that

|u(x)| ≤ ∫_{R^{N}} g_{m_{1}}(x, y)(h(x, u(x)) − ε)^{+}(y) dy∥u∥_{∞}

for all x ∈ R^{N}. If we choose ε > 0 small enough and take into account Corollary 2.3 and Remark 2.5 we see that g_{m_{1}}(x, y) decays exponentially as |x − y| goes to infinity. Due to (3.5) the function (h^{+}(x, u(x)) − ε)^{+} has compact support and thus

ψ̃(x) := ∫_{R^{N}} g_{m_{1}}(x, y)(h(x, u(x)) − ε)^{+}(y) dy  (3.7)

decays exponentially as |x| goes to infinity. Next observe that due to (3.4) all nonnegative solutions of (3.1) are bounded by the same constant M. Hence, ψ := Mψ̃ has the required properties. This completes the proof of the theorem.

4 Convergence to the Set of Equilibria

In this section we study the semilinear parabolic evolution problem in the Banach space X := BUC(R^{N}). For simplicity we restrict our attention to nonnegative initial conditions u_{0} in the positive cone X^{+} of X.

\{ \begin{matrix}∂_{t}u − Δu & = & f(x, u) & in R^{N} × (0, ∞), \\ u(·, 0) & = & u_{0} & in R^{N}. \end{matrix}  (4.1)

As in the previous section we require that f has the properties (3.2)–(3.5). Under these assumptions the existence of a global semiflow (φ, X^{+}) that consists of classical solutions of (4.1) as well as the existence of the compact global B-attractor M ⊂ X^{+} is shown in [13, Theorem 3.1]. The proof of [13, Theorem 3.1] also shows that the attractor M is contained in an order interval V that is invariant under the flow. V is of the form

V := [0, \overline{w}] ⊂ C_{0}(R^{N}),  (4.2)

where \overline{w} is a steady state of (4.1). Making use of the results of the previous sections we will verify that the functional

L(u) := \frac{1}{2} ∫_{R^{N}} |∇u|^{2} dx − ∫_{R^{N}} F (x, u) dx  (4.3)

is a Liapunov function, where for (x, ξ) ∈ R^{N} × R we have set

F (x, ξ) := ∫_{0}^{ξ} f(x, η) dη.

We will see that the functional L is well-defined on the B-attractor M. This will be sufficient to prove that the ω-limit set of each point in X^{+} is contained in the set of equilibria of the semiflow (φ, X^{+}). To prove that L is a Liapunov function we show some regularity properties of solutions of (4.1).

Lemma 4.1 Let u_{0} ∈ L_{2}(R^{N}) ∩ BUC(R^{N}). Then, for some η > 0, the solution u of (4.1) lies in

C^{1}([ε, T], W_{2}^{1}(R^{N})) ∩ BUC^{3+η,(3+η)/2}(R^{N} × [ε, T])

for all ε > 0 and T > ε.

Proof Fix u_{0} ∈ L_{2}(R^{N}) ∩ BUC(R^{N}), ε > 0 and T > ε. We first show Hölder regularity. As a consequence of the classical regularity theorems for parabolic equations with Hölder continuous coefficients and the abstract theory of parabolic evolution equations we conclude that, for some η > 0, the solution u of (4.1) lies in BUC^{2+η,1+η/2}(R^{N} × [ε, T]) (see [4, Theorem 25.2]). Hence, by assumption (3.2) this implies that

m(x, t) := h(·, u(t, ·)) ∈ BUC^{1+η}(R^{N} × [ε, T]),  (4.4)

where we possibly need to decrease η. Applying [11, Theorem IV.5.1] we get the claimed Hölder regularity. For the Sobolev regularity we study the equation

\{ \begin{matrix}∂_{t}u − Δu − m(x, t)u & = & 0 & in R^{N} × (0, T), \\ u(·, 0) & = & u_{0} & in R^{N}. \end{matrix}  (4.5)

whose solution lies in C((0, T], W_{2}^{1}(R^{N})). To see this define the multiplication operator M(t)u(x) := m(x, t)u(x). Then,

M(·) ∈ C^{η}([0, T], L(L_{2}) ∩ L(W_{2}^{1})).  (4.6)

Thus, taking into account that Δ generates an analytic semigroup on L_{2} we find that the above claim is true (This follows e.g. from [4, Theorem 5.10 and Corollary 5.4] noting that D(Δ) = W_{2}^{2}(R^{N}) and [L_{2}, W_{2}^{2}]_{1/2} = W_{2}^{1}). This implies that u(ε) ∈ W_{2}^{1}(R^{N}). As Δ also generates a strongly continuous analytic semigroup on W_{2}^{1} it follows from (4.6) by the same argument that u ∈ C^{1}([ε, T], W_{2}^{1}(R^{N})). This concludes the proof of the lemma.

Lemma 4.2 For u_{0} ∈ V the function L(φ(t, u_{0})) is well defined for t > 0. Moreover, it is bounded from below on V, and

\frac{d}{dt}L(φ(t, u_{0})) < 0,

unless u_{0} is an equilibrium of (φ, X^{+}).

Proof Let u_{0} be an arbitrary element in V. We first show that L(φ(t, u_{0})) is finite for t > 0. By Theorem 3.1 the function \overline{w} in (4.2) is exponentially decreasing, and since V is invariant

0 ≤ u(t, x) := φ(t, u_{0})(x) ≤ Ce^{−α|x|}, x ∈ R^{N},  (4.7)

for some constants C, α > 0 independent of t and u_{0}. In particular, this implies that V ⊂ L_{2}(R^{N}). It follows now from Lemma 4.1 that the gradient term in (4.3) is finite for all t > 0. We next show the finiteness of the other term. Using (4.7) we conclude that

|F (x, u(x, t))| ≤ ∫_{0}^{u(x,t)} |f(x, ξ)| dξ ≤ ∫_{0}^{Ce^{−α|x|}} |f(x, ξ)| dξ.

The assumption (3.5) implies that |f(x, ξ)| ≤ c_{1} < ∞ for (x, ξ) ∈ R^{N} × [0, C]. We thus find that

|F (x, u(x, t))| ≤ c_{1}e^{−α|x|}.  (4.8)

This shows that

|∫_{R^{N}} F (x, u(x, t))dx| ≤ c_{0} < ∞

with c_{0} > 0 independent of u_{0} ∈ V, whence (4.3) is defined and bounded from below on V. This proves the first assertion. Next we show differentiability of L(u(t)) with respect to t. By Lemma 4.1 and the fact that the inner product on W_{2}^{1}(R^{N}) is differentiable we get that

\frac{1}{2} \frac{d}{dt} ∫_{R^{N}} |∇u(t)|^{2} dx = ∫_{R^{N}} (∇u|∇u_{t}) dx  (4.9)

Regarding the other term we already know the uniform bound (4.8). On the other hand,

\frac{d}{dt}F (x, u(x, t)) = f(x, u(x, t))u_{t}(x, t) = m(x, t)u(x, t)u_{t}(x, t)

is bounded by ∥m(t)u_{t}(t)∥_{∞}|u(x, t)|, and by (4.7) and Lemma 4.1 the derivative of F (x, u(x, t)) is bounded uniformly with respect to t in compact subsets of (0, ∞) by an integrable function. Hence, we can interchange the order of integration and differentiation to get

\frac{d}{dt} ∫_{R^{N}} F (x, u(x, t)) dx = ∫_{R^{N}} f(x, u(x, t))u_{t} dx.

Together with (4.9) we arrive at

\frac{d}{dt}L(u(t)) = ∫_{R^{N}} (∇u|∇u_{t}) dx − ∫_{R^{N}} f(x, u)u_{t} dx.  (4.10)

Using that div[u_{t}∇u] = u_{t}Δu+(∇u|∇u_{t}) and applying the Divergence Theorem in the form

f ∈ W_{1}^{1}(R^{N}, R^{N}) ⟹ ∫_{R^{N}} div f(x) dx = 0

we obtain

∫_{R^{N}} (∇u|∇u_{t}) dx = − ∫_{R^{N}} u_{t}Δu dx.

Finally, using that u is a solution of (4.1) we conclude from (4.10) that

\frac{d}{dt}L(φ(t, u_{0})) = − ∫_{R^{N}} u_{t}^{2} dx.

The last term is clearly negative if u is not stationary, which finishes the proof.

As a consequence of [12, Theorem 2.3] and the previous lemma we obtain the following result.

Theorem 4.3 The ω-limit set ω(u_{0}) of each point u_{0} ∈ X^{+} is contained in the set of stationary points of the semiflow (φ, X^{+}).

Proof The global B-Attractor M is contained in the order interval V. Hence the function L is well-defined on M. The result is now obtained similarly as in [12, Theorems 2.3 and 3.2]. Instead of using the continuity of the Liapunov function we make use of the fact that L is bounded below uniformly on V. Hence by the previous Lemma and the invariance of V

lim_{t→∞} L(φ(t, u_{0}))

exists for each u_{0} in V. This is the only nontrivial modification needed in [12, Theorem 2.3] to match the present situation. We also note that the semiflow (φ, X^{+}) is of class AK and that as a consequence the ω-limit set ω(u_{0}) is not empty and compact.

Remark 4.4 A class of problems with polynomial nonlinearities satisfying the conditions (3.2)–(3.5) is given in [13, Example 3.3.a)].

We note that the theorem above does not imply that ω(u_{0}) consists of a single stationary point of the semiflow. However, we will see in the next section that this is true in the scalar case N = 1. In [3] and [14], it is shown that the corresponding result is true for compact intervals

5 Convergence to a Single Equilibrium

In this section we shall employ a result due to Hale and Raugel [10] to show that in one dimension any trajectory converges to a single equilibrium. Suppose that u_{0} is any equilibrium of (4.1). We need show that zero is an eigenvalue of

Δφ + f_{u}(x, u_{0}(x))φ = λφ in R^{N}  (5.1)

of algebraic multiplicity at most one, and that there exist no purely imaginary eigenvectors.

Lemma 5.1 For all λ ∈ C with Re λ ≥ 0 the operator

Δ + f_{u}(· , u_{0}(·)) − λ  (5.2)

is Fredholm of index zero in BUC(R^{N}), C_{0}(R^{N}) and L_{2}(R^{N}). Moreover, all eigenvalues λ with Re λ ≥ 0 of (5.2) are real. They are algebraically simple in one dimension.

Proof Throughout the proof X denotes one of the spaces L_{2}(R^{N}), C_{0}(R^{N}) or BUC(R^{N}). By assumption (3.5) we have

m(x) := f_{u}(x, u_{0}) = h(x, u_{0}) + h_{u}(x, u_{0})u_{0}.

As u_{0} is exponentially decaying, and again using (3.5) it follows that m^{−} is strongly absorbing and m^{+} ∈ C_{0}(R^{N}). Due to Corollary 2.3 and Remark 2.5, we can write m = −V + g where g having compact support, and [Re z ≥ −ω_{0}] ⊂ ϱ(Δ − V) for some ω_{0} > 0 for all spaces under consideration. By interior regularity (e.g. [9, Theorem 8.22]) it follows that for all u ∈ L_{∞}(R^{N}) the function v := (Δ − V)^{−1}u is Hölder continuous with a Hölder constant independent on ∥u∥_{∞} ≤ M. Also, ∥∇v∥_{2} is bounded uniformly uniformly in ∥u∥_{2} ≤ ∞. Hence, embedding theorems for bounded domains imply that

v → g(Δ − V)^{−1}v

is compact on X. Hence, due to [7, Theorem IX.2.1] the first assertion of the lemma follows.

Note that the operator under consideration is selfadjoint in L_{2}(R^{N}), and therefore has real spectrum. From the above we in particular conclude that any point in the spectrum with positive real part is an eigenvalue. If we show that each eigenvalue in BUC is in L_{2} it follows that this part of the spectrum is real. Observe that if λ ∈ C then the heat kernel associated with −Δ + V + λ is k_{V} (t; x, y)e^{λt} and thus satisfies the estimate

|k_{(V +λ)}(t; x, y)| ≤ |k_{(V +Re λ)}(t; x, y)|

If now Re λ ≥ 0 we conclude as in the proof of Theorem 3.1 that any eigenvector to λ decays exponentially and thus belongs to L_{2}(R^{N}). This proves the second part of the lemma.

Suppose now that the dimension is one. We first show that all eigenvalues with Re λ ≥ 0 must be geometrically simple. Suppose to the contrary that (5.1) has two decaying solutions φ_{1}, φ_{2}. Then the Wronskian for the associated first order system of ordinary differential equations is

W (x) = φ_{1}(x)φ′_{2}(x) − φ_{2}(x)φ′_{1}(x) = constant.

for all x ∈ R. Note that by Lemma 4.1 the derivatives φ′_{1} and φ′_{2} are bounded. If φ_{1}, φ_{2} are both decaying also W (x) has to decay, which is impossible. Hence λ is geometrically simple. As algebraic and geometric multiplicity coincide for selfadjoint operators any eigenvalue λ ≥ 0 is algebraically simple in L_{2}. The same is true in BUC and C_{0}. Indeed, suppose an eigenvalue λ ≥ 0 has higher algebraic multiplicity. Then there exists u ∈ BUC such that (Δ − m − λ)^{2}u = 0 and v = (Δ − m − λ)u ≠ 0. Hence v is an eigenvalue of Δ + m, and, according to what we proved in the previous paragraph, v ∈ L_{2}. This means that λ has a higher multiplicity in L_{2} which is a contradiction.

Theorem 5.2 Suppose that N = 1. Then, any solution of (4.1) with nonnegative u_{0} ∈ BUC(R) converges to an equilibrium in BUC(R).

Proof From Theorem 4.3 we know that all solutions converge to the set of equilibria. Using Lemma 5.1 it is easy to see that all assumptions of [10, Theorem 2.4] satisfied. Hence, the assertion of the Theorem follows.

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