STABILITY FOR A FAMILY OF DEGENERATE SEMILINEAR PARABOLIC PROBLEMS

. This paper deals with the initial value problem for a class of degenerate nonlinear parabolic equations on a bounded domain in R N for N ≥ 2 with the Dirichlet boundary condition. The assumptions ensure that u ≡ 0 is a stationary solution and its stability is analysed. Amongst other things the results show that, in the case of critical degeneracy, the principle of linearized stability fails for some simple smooth nonlinearities. It is also shown that for levels of degeneracy less than the critical one linearized stability is justiﬁed for a broad class of nonlinearities including those for which it fails in the critical case.

Since k(0) = 0, the function u ≡ 0 is a stationary solution and the objective here is to discuss its stability. For τ ∈ [0, 2) the principle of linearized stability holds.

CHARLES A. STUART
That is, the stability of u ≡ 0 in F τ for the equation (1.1) is the same as that for its linearization at u ≡ 0, However, this is not the case for τ = 2 which will therefore be referred to as critical degeneracy, the cases where τ ∈ [0, 2) being subcritical. To be a little more precise, we note that F τ is continuously and densely embedded in L 2 (B) and that a selfadjoint operator S τ : D(S τ ) ⊂ L 2 (B) → L 2 (B) is defined by D(S τ ) = {u ∈ F τ : ∇ · (|x| τ ∇u) ∈ L 2 (B)} and S τ u = −∇ · (|x| τ ∇u).

STABILITY FOR DEGENERATE PARABOLIC PROBLEMS 5299
Under the hypothesis (A) τ the conditions (1.5) and (1.7) are satisfied by considering solutions of (1.4) and (1.9) having the property that u(·, t) ∈ H A for all t ∈ [0, T ), where H A is defined as follows.
Definition. Setting u, v A = Ω A∇u · ∇v dx and u A = { Ω A|∇u| 2 dx} 1/2 , (1.11) the space H A is the completion of the space C ∞ 0 (Ω) with respect to the norm · A . It is proved in Section 2 that there exists a constant C such that and it is shown that when (A) τ is satisfied, H A = F τ except in the case N = τ = 2. In fact, for N = 2, F 2 is not complete for norms equivalent to · A when A satisfies (A) 2 . A second family of weighted spaces, E τ , is introduced in Section 2.2 using generalized derivatives on Ω\{0} rather than Ω. In Proposition 2.2 it is shown that for all N ≥ 2 and τ ∈ [0, 2], H A = E τ when (A) τ is satisfied. In Section 3 it is shown that the assumptions (A) τ and V ∈ L ∞ (Ω) ensure that a self-adjoint operator S : D(S) ⊂ L 2 (Ω) → L 2 (Ω) is defined by Su = −∇ · {A∇u} + V u on D(S) = {u ∈ H A : ∇ · {A∇u} ∈ L 2 (Ω)}. (1.13) The linearized problem (1.9) can be written as d dt u(t) + [S − λ]u(t) = 0 (1.14) and its stability depends on the position of λ relative to the spectrum of S. This operator is bounded below, the space H A is the domain of |S| 1/2 and · A is equivalent to the graph norm of |S| 1/2 on H A . Propositions 3.1 and 3.2 deal with the spectral theory of S. Denoting the spectrum and essential spectrum of S by σ(S) and σ e (S) respectively, it is shown in Proposition 3.1 that σ e (S) = ∅ if A satisfies (A) τ with τ ∈ [0, 2), whereas Proposition 3.2 concerns the case τ = 2 where σ e (S) = ∅. The assumptions concerning the nonlinear term g in (1.4) are set out in Section 4. For a Caratheodory function g : Ω×R → R and a measurable function u : Ω → R letg denote the Nemytskii operator defined bỹ g(u)(x) = g(x, u(x)) for x ∈ Ω.
(1. 15) Defining G : Ω × R → R by G(x, s) = s 0 g(x, t) dt, let ψ(u) = Ω G(x, u(x)) dx (1.16) when u : Ω → R is measurable and G(u) is integrable over Ω. For a coefficient A which satisfies the condition (A) τ two types of assumption, (G1) τ and (G2) τ , on the function g are introduced in order to ensure thatg maps H A continuously and boundedly into L 2 (Ω). Since the space H A becomes bigger as τ increases, these conditions become more restrictive with increasing τ . For all τ ∈ [0, 2], the hypothesis (G1) τ admits functions of the form g(x, s) = k(s) under suitable restrictions on k. In particular, if ±k satisfies the condition (K), then g satisfies (G1) τ for all τ ∈ [0, 2]. The conditions (G2) τ accommodate stronger growth of g(x, s) as |s| → ∞ provided that g(x, s) → 0 as x → 0 sufficiently fast. In all cases these assumptions ensure theg : H A → L 2 (Ω) is locally Lipschitz continuous and bounded. In fact, for τ ∈ [0, 2) we even haveg ∈ C 1 (H A , L 2 (Ω)), but as Lemma 4.2 shows, this is not ensured by (G1) 2 and, for the case where τ = 2 and g(x, s) = ±k(s) with k satisfying (K),g : H A → L 2 (Ω) is Fréchet differentiable at 0 only when k ≡ 0. Hence in the critical case it is important treat situations whereg : H A → L 2 (Ω) is not Fréchet differentiable at 0. It is in such cases that the principle of linearized stability is shown to fail in Section 7.2 due to the behaviour of the energy functional φ λ associated with (1.4). It is defined by where ψ is defined in (1.16) and in all cases φ λ ∈ C 1 (H A , R) and the stationary solution u ≡ 0 is a critical point of φ λ for all λ. In the subcritial case φ λ ∈ C 2 (H A , R) but this is not true for τ = 2 wheng is not Fréchet differentiable at 0. Nonetheless the second variation φ λ (0) of φ λ at u ≡ 0 is well defined in all cases and it is the quadratic form which is positive definite on H A if and only if λ < m where m = inf σ(S). However, when τ = 2 this does not imply that u ≡ 0 is a local minimum of φ λ for λ < m. This issue is investigated in [34] and the criteria established there form the basis for our analysis of stability in the critical case. The hypotheses of Sections 2 to 4 allow us to deal with strong solutions of the initial value problem (1.4) to (1.7) as a dynamical system on the space H A . More precisely, for every u 0 ∈ H A there exist T > 0 and a unique function u ∈ C([0, T ), H A ) ∩ C((0, T ), D(S)) ∩ C 1 ((0, T ), H A ) such that u(0) = u 0 and the equation (1.4) is satisfied in the sense that 18) where this equation expresses the equality of elements in L 2 (Ω). These conclusions are derived from the theory of analytic semigroups in Section 6. The underlying results are recalled in a simple but convenient abstract setting in Section 5.1 and then the relevant definitions of stability and instability of the solution u ≡ 0 are recalled in the same abstract setting in Section 5.2. In preparation for the treatment of the critical case in Section 7.2, Theorem 5.3 collects several conclusions about stability and instability obtained using the energy as a Lyapunov function. Although this is a classical approach, existing references do not seem to cover the precise setting required here so proofs are given. Criteria for the stability and instability of the solution u ≡ 0 of the problem (1.4) to (1.7) are established in Section 7.1 for the subcritical case and in Section 7.2 for the critical one. The claims made about the special case presented at the beginning of this introduction are justified by Theorem 7.2 for τ ∈ [0, 2) and by Corollary 7.4 for τ = 2, since Proposition 6.2 shows that inf σ(S) = inf σ e (S) when Ω = B(0, 1), A(x) = |x| 2 and V ≡ 0.
Having summarized the main contents of this paper, it is time to comment on some limitations of this study. The stability analysis of stationary solutions is restricted to the solution u ≡ 0 and the case N = 1 is excluded. Concerning the first point, we observe that in the critical case which is our main focus, it is shown in Proposition 6.2 that in some circumstances the set of all stationary solutions is {(λ, u ≡ 0) : λ ∈ R}. For the second issue we refer to [33] where it is shown that, in the critical case, it is more natural to deal with the problem on the intervals (p, 0) and (0, q) separately rather than on Ω = (p, q) where p < 0 < q. It is in this form that bifurcation is studied for the stationary problem in [35] and we claim that the present approach to the parabolic problem could easily be adapted to that setting.
Let us end this introduction with a few remarks about the position of the problem treated here with respect to the general theory of degenerate second order elliptic and parabolic problems. The classical treatment of degenerate linear elliptic operators of second order in general divergence form is mainly due to Kruskov [26], Murthy and Stampacchia [29] and Trudinger [36]. When specialized to the operator −∇ · {A∇} + V − λ with A satisfying (A) τ and V ∈ L ∞ (Ω), their main hypotheses are satisfied if and only if 0 ≤ τ < 2 and their results deal with weak solutions in Concerning the eigenvalue problem (1.19) it is known from [26], [29] and [36] that for τ < 2, eigenfunctions of −∇·{A∇}+V are Hölder continuous and bounded on Ω and the spectrum is discrete. The case τ = 2 is not covered by these results and many of the conclusions obtained there fail in this case. As is shown in Section 3.2 below and in greater generality in Section 4 of [33], eigenfunctions may become unbounded as x → 0 and the spectrum is no longer discrete, N 2 a 4 + V 0 being the infimum of the essential spectrum. For the case where A satisfies (A) τ with 0 ≤ τ < 2, existence theory for the linear parabolic equation (1.9) is treated by Ivanov in [25] in a setting analogous to that used for the stationary problem in [26], [29] and [36]. Again, the results in [25] do not cover the case τ = 2, which is the subject of the present paper. Properties, such as Hölder continuity, of solutions of degenerate parabolic equations are discussed in [11,12,22,23]. See also [1] and [14] for weak solutions of degenerate parabolic equations containing singular coefficients.
2. Properties of space H A . In this section the space H A is considered in some detail under the assumption that A satisfies the condition (A) τ . The main objectives here are to characterize the elements of this completion and to establish some embeddings which are crucial for the subsequent discussion of the nonlinear term in (1.4). It turns out that the case N = τ = 2 presents some particular features so it is dealt with in a separate subsection where a family of spaces E τ similar to those used in [17,18,34] is introduced.
Spaces like H A and F τ appear in almost all work concerning degenerate elliptic problems often in much greater generality than treated here. See [26,29,36] for linear problems and [8,9,17,18,28] for nonlinear problems having features similar to the stationary version of (1.4) and (1.5). The following exposition contains some information about the critical case not found in these references and it incorporates the Cafferelli-Kohn-Nirenberg estimates in a way which will be useful when dealing with the nonlinear term in (1.4).
2.1. The spaces F τ and H A . The first step is to establish some crucial properties of the space F τ defined by F τ is a Banach space when considered with the norm |∇u| L 1 + u τ . For u ∈ F τ and 1 ≤ p < 2, by Hölder's inequality where q = 2/(2 − p). Since r −τ p 2−p ∈ L 1 (Ω) for p < 2N/(N + τ ) and 2N/(N + τ ) > 1 except in the case N = τ = 2, it follows that for 1 ≤ p < 2N/(N + τ ) there exists a constant C such that |∇u| L p ≤ C u τ for all u ∈ F τ . In the case τ = 0, we have the better conclusion that |∇u| L 2 = u 0 for all u ∈ F 0 . Hence, except for the case N = τ = 2, the norm · τ on F τ is equivalent to |∇ · | L 1 + · τ and (F τ , · τ ) is a Hilbert space.
From (2.1) the following properties of F τ can be deduced, the case N = τ = 2 being excluded, of course. Using Lemmas 7.12 and 7.14 of [20], there exists a constant C such that u L q ≤ C |∇u| L p for all u ∈ W 1,p 0 (Ω) provided that p < N and 1 ≤ p ≤ q < N p/(N − p). Combining these observations it follows that · τ is a norm on F τ with (F τ , · τ ) being continuously embedded in W 1,p (Ω) for 1 ≤ p < 2N/(N + τ ) and in L q (Ω) for 1 ≤ q < 2N/(N − 2 + τ ) = 2 * τ , except in the case N = τ = 2. Furthermore, Since Ω is bounded, r α ∈ L 1 (Ω) for α > −N and so the embedding of F τ in L q (Ω) and Hölder's inequality show that there exists a constant C = C(Ω, τ, γ, p) such that N −2+τ −2γ . Some of these conclusions are sharpened in the following result which also covers the case N = τ = 2 which was excluded from the preceding discussion.
(1) Except for the case N = 2 with τ = 0, r and (4) There exists a constant C such that, τ in all cases except N = 2 with τ = 0 where the embedding holds for 1 ≤ p < ∞.
For any function A satisfying the condition (A) τ , F τ can be identified with a subset of H A and then the norms · τ and · A are equivalent on F τ . Except for the case N = τ = 2, F τ = H A and (F τ , · τ ) is a Hilbert space. (7) For N = 2, (F 2 , · 2 ) is not complete but its completion is isomorphic to the Hilbert space (H A , · A ) for any A satisfying (A) 2 . For such A, (2.4) holds for u ∈ H A and (2.5) extends to u ∈ H A for γ ∈ [0, 1) with 1 ≤ p ≤ 2 1−γ and for γ = 1 with 1 ≤ p < ∞.
Remark 2.1. The inequality (2.2) is proved directly for all u ∈ F τ and then used in the proof of (2). These proofs extend without change to cover the spaces E τ defined in Section 2.2. The inequality (2.5) is due to Caffarelli, Kohn and Nirenberg [7] for u ∈ C ∞ 0 (R N ). It can be extended to all of F τ by part (2) and (2.2) is then a special case of (2.5). See also [31] for very general results of this kind, including necessary conditions for such inequalities and many historical remarks. The C-K-N inequalities have inspired a large amount of work on degenerate and singular nonlinear elliptic equations, [10,2,6] being early examples of this.
Remark 2.2. For N = 2, the completion of F 2 mentioned in part (7) is identified concretely in Section 2.2 as the space E 2 .
(5) To establish the compactness of the embedding it is enough to show that if {u n } is a sequence converging weakly to zero in F τ as n → ∞, then u n L p → 0 as n → ∞ for 1 ≤ p < 2 * τ . Since F τ is continuously embedded in L 2 (Ω), L 2 (Ω) * ⊂ F * τ and hence {u n } converges weakly to zero in L 2 (Ω). Consider ε ∈ (0, ε 0 ]. If {u n } does not converge weakly to zero in W 1,2 (Ω ε ), there exist f ∈ W 1,2 (Ω ε ) * , a subsequence {u n k } and δ > 0 such that |f (u n k )| ≥ δ for all n k . Passing to a further subsequence if necessary, we can suppose that {u n k } converges weakly to an element v in W 1,2 (Ω ε ). Thus {u n k } converges weakly to v in L 2 (Ω ε ) and so v = 0 a.e. on Ω ε since {u n } converges weakly to zero in L 2 (Ω) and hence also on L 2 (Ω ε ). But then f (u n k ) → f (v) = f (0) = 0 as n k → ∞, contradicting the choice of δ. Hence {u n } converges weakly to zero in W 1,2 (Ω ε ) and therefore u n L p (Ωε) → 0 as by Hölder's inequality. The weak convergence of {u n } in F τ and (2.5) with γ = 0 imply that this sequence is bounded in L q (Ω) and so there exists a constant M such that Letting ε → 0+ shows that u n L p → 0 as n → ∞, completing the proof (6) Identifying C ∞ 0 (Ω) with a subspace of H A , the equivalence of the norms on C ∞ 0 (Ω) follows immediately from (1.10). Then part (2) implies that F τ ⊂ H A . Exclude now the case N = τ = 2 and consider a Cauchy sequence {u n } in (F τ , · τ ). It follows from (2.1) that it is also a Cauchy sequence in W 1,1 0 (Ω), showing that F τ is complete and hence F τ = H A .
(7) Fix ε ∈ (0, ε 0 ) with ε < 1 and consider the function u(x) = (ε − r)/(r ln r) for 0 < r ≤ ε and u(x) = 0 on Ω ε , and the sequence of truncations where n ∈ N with n > 1/ε. Then u n ⊂ H 1 0 (Ω) ⊂ F 2 and a short calculation shows that It follows that {u n } is a Cauchy sequence in (F 2 , · 2 ) and if it converges, its limit must be u. However, a similar calculation shows that |∇u| dx → ∞ as n → ∞, so u ∈ W 1,1 (Ω) and hence {u n } does not have a limit in F 2 . Therefore (F 2 , · 2 ) is not complete and so F 2 = H A for N = 2.

2.2.
The spaces E τ and H A . In Section 2.1 it is shown that for a coefficient A satisfying (A) τ , H A = F τ , except in the case N = τ = 2. In order to obtain a characterisation of the elements of H A using generalized derivatives when N = τ = 2, we begin by introducing a space E τ consisting of functions having generalized derivatives on Ω * = Ω\{0}. To put the case N = τ = 2 in proper perspective we consider N ≥ 2 and τ ∈ [0, 2]. Setting where ∇u ∈ [L 1 loc (Ω * )] N consists of generalized derivatives of u on Ω * , it is easy to check that E τ with the scalar product is a Hilbert space and (by a slight abuse of notation) E τ ⊂ W 1,2 (Ω ε ). Then E 2 coincides with the space H defined in [17,18] for the case N ≥ 3 and τ = 2. Now let where Γ : W 1,2 (Ω ε ) → L 2 (∂Ω) is the usual trace operator, see [3] A 5.7 for example. The continuity of Γ ensures that (E τ , (·, ·) τ ) is a Hilbert space. Since F τ is continuously embedded in L 2 (Ω) by Proposition 2.1, F τ ⊂ E τ . The exact relationships between the spaces E τ , F τ and H A is established in the following result. Note that part (ii) shows that if u ∈ E τ then u admits generalized derivatives on Ω, except in the case N = τ = 2. (i) The inequality (2.2) holds for all u ∈ E τ , except in the case N = 2 with τ = 0.
(iii) In all cases, · τ is a norm on E τ equivalent to (·, ·) (iv) If A satisfies the condition (A) τ , H A can be identified with E τ and then H A is continuously embedded in L 2 (Ω). Hence H A = E τ = F τ except in the case (v) With a slight abuse of notation, E τ is continuously embedded in W 1,2 (Ω ε ) and weak convergence in E τ implies both weak convergence in W 1,2 (Ω ε ) and strong convergence in L p (Ω ε ) for 1 ≤ p < 2 * .
(vi) E 2 is continuously but not compactly embedded in L 2 (Ω).
Proof. For part (i) it suffices to copy the proof of part (1) of Proposition 2.1.
(ii) The inequality (2.1) shows that |∇u| ∈ L 1 (Ω * ) for u ∈ E τ except in the case N = τ = 2. The proof of Lemma 6.1(i) in [17] now shows that u admits a generalized derivative on Ω. It follows that E τ is continuously embedded in W 1,1 (Ω) and hence that E τ = F τ .
(iii) To establish the equivalence of the norms it suffices to prove that there is a constant C = C(N, τ ) such that u L 2 ≤ C u τ for all u ∈ E τ . For the case N = τ = 2, this follows from part (i) and from (2.5) in all other cases since then E τ = F τ . The density of C ∞ 0 (Ω) in E τ is now implied by part (ii) of Proposition 2.1 except in the case N = τ = 2. However, since the inequality (2.2) holds for u ∈ E 2 , the proof of Proposition 2.1(ii) can be repeated simply replacing F τ by E 2 .
(iv) Combining part (ii) with parts (6) and (7) of Proposition 2.1, it is sufficient to show that in the case and so E τ is continuously embedded in W 1,2 (Ω ε ). The argument at the beginning of the proof of part (5) of Proposition 2.1 shows that weak convergence in E τ implies weak convergence in W 1,2 (Ω ε ). Strong convergence in L p (Ω ε ) for 1 ≤ p < 2 * then follows from the compactness of the Sobolev embedding since Ω ε has a Lipschitz boundary.
(vi) The continuity of the embedding is proved in part (iv). To show that it is not compact choose any u ∈ E 2 such that u ≡ 0 but u = 0 on Ω ε and then, for n ≥ 1, set u n = 0 on Ω ε/n and u n (x) = n N/2 u(nx) for r ≤ ε/n. It is easy to check that Since r 2 |∇v| 2 ∈ L 1 (Ω), showing that u n 0 weakly in E 2 as n → ∞. But u n L 2 = u L 2 = 0 for all n ≥ 1 and so the embedding of E 2 in L 2 (Ω) is not compact.
3. The operators S A and S. Under the assumptions that A satisfies the condition (A) τ and that V ∈ L ∞ (Ω), two self-adjoint operators are introduced to deal with the differential expressions −∇ · {A∇} and −∇ · {A∇} + V and the boundary conditions (1.5).

3.1.
The self-adjoint operator S A . In this section we introduce a self-adjoint operator acting in L 2 (Ω) associated with the expression −∇ · [A∇] and the Dirichlet boundary condition on Ω, in such a way that H A appears as its form space. To this end we use a version the Friedrich's extension procedure which we now recall, following [37].
It follows from parts (iii) and (iv) of Proposition 2.2 that the Hilbert spaces (L 2 (Ω), ·, · L 2 ) and (H A , ·, · A ) form an admissible pair for this construction provided that the condition (A) τ is satisfied. Henceforth, S A : D(S A ) ⊂ L 2 (Ω) → L 2 (Ω) will denote the unique self-adjoint operator associated with this pair through the Friedrich's extension procedure. Using the properties (1) to (4) we have To proceed further we need a more explicit characterisation of the space D(S A ). As noted in property (1), u ∈ D(S A ) if and only if u ∈ H A and there exists w ∈ L 2 (Ω) such that

if and only if
u ∈ H A and − ∇ · {A∇u} = w ∈ L 2 (Ω) in the sense of distributions on Ω, (3.8) even though in the case N = τ = 2 we may have |∇u| ∈ L 1 loc (Ω). In this sense we can write without making assumptions about the differentiability of A on Ω. This will not be required here.
3.2. The operator S = S A +V and its spectrum. For V ∈ L ∞ (Ω) multiplication by V defines a bounded self-adjoint operator from L 2 (Ω) into itself and hence by Theorem 9.1 of [37], provided that A satisfies the condition (A) τ . Furthermore, the graph norms of S A and S = S A + V are equivalent norms on D(S). The graph norm of S will be denoted by · S . Therefore, . Since the linearization of (1.4) about the solution u ≡ 0 is (1.9), more precisely (1.14), the location of the spectrum of S can be expected to play an crucial role in the stability theory of the solution u ≡ 0.
Let σ(S) and σ e (S) denote the spectrum and essential spectrum of S, respectively. For the simpler subcritical case the essential spectrum of S is empty for all V ∈ L ∞ (Ω).
It is bounded below and has no essential spectrum. In fact, setting m = inf σ(S), m is a simple eigenvalue of S with a positive eigenfunction and All eigenfunctions of S are Hölder continuous on Ω.
Proof. The self-adjointness of S has already been noted at the beginning of this subsection. Hence by (2.2), proving the lower bound for m Since H A = F τ is compactly embedded in L 2 (Ω) by Proposition 2.1, it is easily seen that S has a compact resolvent and consequently a discrete spectrum. The fact that ker(S − m) is spanned by a positive eigenfunction can be proved by the usual minimization argument, as in part (iv) of Theorem 4.1 in [33]. The regularity theory in [29,36] shows that the eigenfunctions of S are Hölder continuous on Ω. See Corollaries 5.5 and 6.1 in [36], for example.
In the case of critical degeneracy the properties of the operator S are quite different. Its essential spectrum is not empty and since inf σ e (S) plays a prominent role in the stability criteria in Section 7 it is important to have a sharp estimates for it. In fact, when the potential V has a limit as x tends to 0, inf σ e (S) can be calculated exactly.
The following properties of S are established as part of Theorem 4.1 in [33].
(iv) If m < m e , then m is a simple eigenvalue of S and there exists an element ϕ ∈ D(S) ∩ C(Ω\{0}) such that ϕ > 0 on Ω\{0} and ker (S − m) = span{ϕ}. Proposition 6.2 provides conditions under which m = m e and S has no eigenvalues. If S does have eigenvalues below the essential spectrum the corresponding eigenfunctions can be singular at the origin even when A is smooth and V is zero in a neighbourhood of the origin. In the following example, χ J denotes the characteristic function of an interval J.
The coefficients in the equation (3.11) are radially symmetric and radially symmetric solutions can be calculated explicitly since the resulting ordinary differential equation is of Euler type. A positive, radially symmetric eigenfunction of S α with eigenvalue λ = N 2 ( 1 4 − α 2 ) is given by Observe that u α is singular at x = 0 for α ∈ (0, 1/2). In connection with Proposition 3.2, note also that the conditions (A) 2 and (V) are satisfied in this example with , so it follows from part (iv) that m α is a simple eigenvalue of S α with a positive eigenfunction. From the orthogonality of eigenfunctions of S α associated with different eigenvalues we conclude that m α = N 2 ( 1 4 − α 2 ). A more general, not necessarily radially symmetric situation where eigenfunctions of S are singular at x = 0 is established in Theorem 4.2 of [33]. 4. The nonlinearity. The results of this section concern the properties of the term g(x, u) in (1.4), viewed as a mapping from H A into L 2 (Ω), under the assumption (A) τ .
The first step is to formulate hypotheses on the function g which ensure that it generates a continuously differentiable Nemytskii operator from H A into L 2 (Ω). Since the space H A gets bigger as τ is increased the assumptions required to ensure this become more restrictive as τ ∈ [0, 2] increases. A lemma then shows that when τ = 2 Fréchet differentiability is too restrictive to admit nonlinearities where g(x, s) is independent of x. For such cases only Gâteaux differentiability at 0 can be obtained and a weaker condition (G1) * 2 is formulated to ensure this property. (4.1) For τ = 0, this condition is the standard assumption used to ensure thatg ∈ C 1 (H 1 0 (Ω), L 2 (Ω)). For τ > 0 the range of admissible growth of g(x, s) with respect to s can be extended at the expense of imposing suitable decay to zero as x tends to 0.
For τ = 0 this condition is vacuous. If the estimate (4.2) holds for some α > (with a different value of K) since Ω is bounded. It should also be noted that, for a given value of τ ∈ (0, 2], it is easy to find functions which satisfy both (G1) τ and (G2) τ . For some purposes the following stronger version of (G2) 2 will be required.
Theng maps bounded subsets of F τ into bounded subsets of L 2 (Ω) and g ∈ C 1 (F τ , L 2 (Ω)) withg(0) = 0 andg (0) = 0. Also, ψ ∈ C 2 (F τ , R) andg maps bounded subsets of F τ into bounded subsets of B(F τ , L 2 (Ω)). If g satisfies (G2) * 2 , there exists a constant C such that g(u) (C) For a coefficient A satisfying the condition (A) τ , the conclusions of parts (A) and (B) also hold when the space F τ is replaced by H A . Proof. We begin by recalling a more or less standard result concerning the differentiability of Nemytskii operators acting between Lebesgue spaces. Let f : where K and σ are positive is constants and b ∈ L q(1+σ)/σ (Ω).
. These conclusions are justified in Theorem 2.6 of [15], for example. By integration and Young's inequality, showing thatf maps bounded subsets of L q(1+σ) (Ω) into bounded subsets of L q (Ω).
(A) Suppose that g satisfies the condition (G1) τ and consider the Caratheodory . But ψ(u) = Ω G(u) dx for all u ∈ F τ and so the conclusion follows in this case. Now suppose instead that g satisfies the condition (G2) τ . This means that τ > 0 and σ > 0 and we can take . Set γ = α/(2 + σ) and consider the Caratheodory function defined by Then integrating (4.2) and using Young's inequality, By the restrictions on σ and β, . But G(x, s) = F (x, r γ s) so G = F oT and the conclusion follows as in the previous case.
Suppose now that g satisfies the condition (G2) τ . Recalling that this means that τ and σ are positive and that we can take α = σ(N −2+τ )−(2−τ ) Then the restrictions on σ ensure that 0 < 2γ ≤ τ for N ≥ 3 and 0 < 2γ < τ for it follows from (2.5) that T u = r γ u defines a bounded linear operator from F τ into L 2(1+σ) (Ω). Now define a Caratheodory function f : Ω × R → R by setting f (x, s) = g(x, r −γ s) for almost all x ∈ Ω and s ∈ R.
Thus u ρ 2 = 0 for all ρ ∈ (0, R/2) and u ρ 2 → 0 as ρ → 0. But, Therefore the Fréchet differentiability ofg at 0 implies that The necessary condition follows from this since Consider a function A satisfying (A) τ . To see that the same conclusions are valid forg : H A → L 2 (Ω) it is sufficient to recall the following properties established in Proposition 2.1. For τ ∈ [0, 2], H A is continuously embedded in L 2 (Ω) and F τ ⊂ H A , with equality except in the case N = τ = 2. Also the norms · τ and · A are equivalent on F τ for τ ∈ [0, 2]. This lemma shows that for a function g(x, s) satisfying (G1) 2 which does not decay to zero as x tends to 0, Fréchet differentiability ofg : H A → L 2 (Ω) cannot be obtained for a function A satisfying (A) 2 . If the goal of ensuring Fréchet differentiability is abandoned and replaced by simply requiring Gâteaux differentiability at 0, it is possible to proceed using the following assumption (G1) * 2 which is weaker than (G1) 2 . For τ < 2, the corresponding modification of (G1) τ still ensures Fréchet differentiability at 0.
Hence for all s ∈ R and x ∈ B(0, R), From the continuity of Γ at 0 it follows easily that the condition (C) is satisfied with γ = Γ(0).
Proof. (I) Using (4.8) the proof is the same as in part (A) of Proposition 4.1 for (G1) τ .
The first step is to prove that J =g − γI : H A → L 2 (Ω) is compact. Let {u n } be a sequence in H A which converges weakly to u in H A . It is enough to prove that J(u n ) − J(u) L 2 → 0 as n → ∞. Since {u n } is bounded in H A it is also bounded in L 2 (Ω) by Proposition 2.2. Suppose that u L 2 and u n L 2 ≤ M for all n.

From Proposition 2.2 (v)
, Since ε > 0 is arbitrary, this proves the compactness of J =g − γI : H A → L 2 (Ω). By Proposition 2.2(vi), I : H A → L 2 (Ω) is not compact and sog −λI : H A → L 2 (Ω) cannot be compact for λ = γ. Propositions 4.1 and 4.3 provide hypotheses which are sufficient to ensure that the initial value problem (1.4) to (1.7) is well-posed as is shown in Section 6. The following results will be used in the analysis of the asymptotic stability of the stationary solution u ≡ 0.
(A) The mapping g 2 : H A → L 2 (Ω) is compact. (B) For λ < m e +γ where γ is given by the condition (C), the mapping S +g −λ : D(S) → L 2 (Ω) is proper when restricted to closed bounded subsets of (D(S), · S ).
Proof. (A) Let {u n } be a sequence in H A which converges weakly to u in H A . It is enough to prove that g 2 (u n ) − g 2 (u) L 2 → 0 as n → ∞. Since {u n } is bounded in H A there is a constant M such that u A and u n A ≤ M for all n.
(B) It is sufficient to show that if {u n } is a bounded sequence in (D(S), · S ) such that the sequence {Su n +g(u n ) − λu n } converges in L 2 (Ω), then {u n } has a subsequence converging in (D(S), · S ). Recalling from (3.1) and the properties of S that (D(S), · S ) is continuously embedded in (H A , · A ),g − γ =g 1 − γ +g 2 is a compact mapping of (D(S), · S ) into L 2 (Ω) by Proposition 4.3 and part (A). Hence by passing to a suitable subsequence we can suppose that {Su n − (λ − γ)u n } converges in L 2 (Ω). But, since λ − γ < m e , S − (λ − γ) : D(S) → L 2 (Ω) is a Fredholm operator of index zero and so this implies that {u n } has a subsequence converging in (D(S), · S ) by Yood's criterion. See Proposition 9.3 in Chapter 2 of [16] for example.
(C) Fix λ < min{m, m e − 1 }. Suppose that {u n } ⊂ D(S)\{0} is a sequence such that Su n +g(u n ) = λu n for all n ∈ N and u n A → 0 as n → ∞. Then λ is a bifurcation point of weak solutions in the sense of Section 6.1 in [33]. By Remark 6.1(I) in [33] the condition (6.3) in Theorem 6.1 in [33] is satisfied since λ < m e − 1 and so, by part (ii) of that result, λ must be an eigenvalue of S. But this is not the case since λ < m. Hence there exists R(λ) > 0 such that Su +g(u) − λu = 0 for 0 < u A ≤ R(λ). Fix r ∈ (0, R(λ)] and suppose that there exists a sequence {u n } ⊂ D(S) such that r ≤ u n A ≤ R(λ) for all n and Su n +g(u n ) − λu n L 2 → 0 as n → ∞. Sinceg maps any bounded subset of H A into a bounded subset of L 2 (Ω) by Propositions 4.3 for g 1 and 4.1(C) for g 2 , it follows that the sequence {u n } is bounded in (D(S), · S ). From (G1) * 2 and (C) we have that |γ| ≤ 1 and hence λ < m e + γ. By part (B), there exist a subsequence {u n k } and w ∈ D(S) such that u n k − w S → 0 as n k → ∞. Then r ≤ w A ≤ R(λ) and Sw +g(w) − λw = 0, contradicting the choice of R(λ).
5. An initial value problem. In the first part of this section, we recall a result concerning existence, uniqueness and regularity of solutions of an abstract initial value problem in a form which is convenient to deal with the problem (1.4) to (1.7) in Sections 6 and 7. The basic notions related to stability are defined at the beginning of the second part. Then conclusions about the stability of the stationary solution u ≡ 0 of a gradient system are established in Theorem 5.3 by using the energy as a Lyapunov function. This result is used to treat the problem (1.4) to (1.7) in the case of critical degeneracy in Section 7.2. For the subcritical situation, stability is determined in Theorem 7.1 using standard results on linearization.

5.
1. An abstract IVP. Let (H, ·, · , · ) be a real Hilbert space and A : D(A) ⊂ H → H a self-adjoint operator which is bounded below. Let (E α , · α ) denote D(|A| α ) equipped with a norm · α which is equivalent to the graph norm of |A| α . Let A mapping F : E α → H is locally Lipschitz continuous if, for every u ∈ E α there exist a radius r = r u > 0 and a constant L = L u such that In this setting we consider the following abstract initial value problem d dt Given u 0 ∈ E α , a solution of (5.1) is defined to be a function u : [0, T ) → H, for some T > 0, having the following properties.
The following result concerning the existence, uniqueness and regularity of solutions of (5.1) can be derived from the theory of analytic semigroups.
Theorem 5.1. Suppose that A : D(A) ⊂ H → H is a self-adjoint operator which is bounded below and that F : E α → H is locally Lipschitz continuous for some α ∈ [0, 1).
(B) For u 0 ∈ H A , let u(·, u 0 ) : [0, T (u 0 )) → E α be the unique maximal solution of (5.1) for the initial condition u 0 ∈ H A . If F maps bounded subsets of E α into bounded subsets of H, then lim sup t→T (u0) u(t, The proof of Theorem 5.1 can be pieced together from the following results in [24]. See Theorem 3.3.3 for local existence and uniqueness, Theorem 3.3.4 for part (B) and Theorem 3.5.2 for additional regularity. Part (C) follows from Corollary 3.3.5.
Remark 5.1. Observe that in Theorem 5.1 there is no loss of generality in supposing that A is positive definite (i.e. there exists a constant c > 0 such that Au, u ≥ c u 2 for all u ∈ D(A)) since the hypotheses are still satisfied with A and F replaced by A + kI and F + kI for any constant k where I is the identity on H.
Remark 5.2. Sometimes local Lipschitz continuity is taken to mean that for every r > 0 there exists a constant L r such that F (u) − F (v) ≤ L r u − v α for all u, v ∈ B α (0, r). This stronger assumption implies that F maps bounded subsets of 5322 CHARLES A. STUART E α into bounded subsets of H and Theorem 5.1 is proved in Chapters 2 and 9 of [13] under this hypothesis. This definition of local Lipschitz continuity is also used in Theorem 3.1 of [5] where the conclusion in part (B) is sharpened to In fact, the proof in [5] yields this conclusion under the hypotheses of Theorem 5.1. The case where (5.1) is an infinite dimensional gradient system is particularly useful in dealing with (1.4) to (1.7) when there is critical degeneracy.

Stability and instability for (5.1).
If, in addition to the hypotheses of Theorem 5.1, F (0) = 0, then the function u(t) = 0 for all t ≥ 0 is a solution of (5.1) and its stability will be investigated in this part. Since most of the results are obtained using the energy J introduced in Corollary 5.2 as a Lyapunov function only the case α = 1/2 will be treated from now on and so the notation will be simplified as follows. X = E 1/2 and · X = · 1/2 , (5.7) and it should be recalled that (X, · X ) is continuously embedded in (H, · ). Under the hypotheses of Theorem 5.1 with α = 1/2 and F (0) = 0, (5.1) generates a dynamical system on X and in this context the usual definitions of stability for the stationary solution u ≡ 0 are now recalled for convenience. For u 0 ∈ X, u(·, u 0 ) ∈ C([0, T (u 0 )), X) ∩ C 1 ((0, T (u 0 )), X) ∩ C((0, T (u 0 )), D(A)) denotes the unique maximal solution with u(0) = u 0 .
The solution u ≡ 0 is said to be stable if there exists δ > 0 such that for u 0 X < δ, T (u 0 ) = ∞ and, for all ε > 0 there exists δ(ε) ∈ (0, δ) such that if u 0 X < δ(ε), then u(t, u 0 ) X < ε for all t ≥ 0. Otherwise, it is said to be unstable, in which case, there exist ε 0 > 0 and a sequence {u n } ⊂ X such that u n X → 0 as n → ∞ and sup 0≤t<T (un) u(t, u n ) X ≥ ε 0 for all n.
The solution u ≡ 0 is said to be asymptotically stable provided that it is stable and that δ > 0 can be chosen such that if u 0 X < δ, then T (u 0 ) = ∞ and u(t, u 0 ) X → 0 as t → ∞. In cases where the nonlinearity defines a Fréchet differentiable mapping from X into H, the stability and instability of the solution u ≡ 0 follow immediately from standard results on nonlinear perturbation of linear systems. See Section 5 of [24], for example. Broadly speaking, the conclusion is that, provided 0 = Σ ≡ inf σ(A − F (0)), the stability of u ≡ 0 is the same as that of the linearized equation u (t) + Au(t) = F (0)u(t), namely, asymptotic stability if Σ > 0 and instability if Σ < 0. This situation is referred to as the principle of linearized stabilty in the Introduction. As is pointed out by Lemma 4.2, Fréchet differentiability is not always an appropriate property when dealing with (1.4), even for smooth nonlinearities. An alternative way of studying stability is through the use of Lyapunov functionals and the rest of this section is devoted to developing this in a general setting which will cover the problem (1.4) to (1.7), even in non-Fréchet differentiable situations.
Under the hypotheses of Corollary 5.2, the energy J : X → R is continuously differentiable and if F (0) = 0, J (0) = 0. In this case the stability of the stationary solution u ≡ 0 of (5.1) depends on nature of this critical point of J. It lies in a potential well of J provided that there exists δ > 0 such that inf{J(u) : u X = r} > J(0) for all r ∈ (0, δ). Clearly, this implies that J has a strict local minimum at 0 and the relationship is explored more closely in Proposition A.1 of [34].
The Lipschitz continuity that is assumed in Corollary 5.2 limits how steep a potential well of J can be. Let r > 0 and L ≥ 0 be such that F (u)−F (0) ≤ L u X for all u ∈ B X (0, r) = {u ∈ X : u X < r}. Then for u ∈ B X (0, r), Thus under the hypotheses of Corollary 5.2 and assuming that F (0) = 0, it follows easily that there exist r > 0 and K > 0 such that If there exist r > 0 and ξ > 0 such that then 0 is said to lie in a quadratic potential well of J. In [34] sufficient conditions for 0 to lie in a quadratic potential well are derived without requiring the potential to be twice Fréchet differentiable at 0. These criteria will be used in Section 7.2 to exploit the following result.
Theorem 5.3. Under the hypotheses of Corollary 5.2 set E 1/2 = X and suppose in addition that F (0) = 0 and that F takes bounded subsets of X to bounded subsets of H. Then u ≡ 0 is a stationary solution of (5.1).
then u ≡ 0 is unstable. If (5.13) holds for all ε > 0, there exists a sequence {u n } ⊂ X such that u n X → 0 as n → ∞ and for all n, J(u n ) < 0, u(t, u n ) 2 ≥ u n 2 + 4|J(u n )|t for 0 ≤ t < T (u n ) and u(t, u n ) X → ∞ as t → T (u n ).
(iii) Seeking a contradiction, suppose that the solution u ≡ 0 is stable. For the ε given by (5.13), there exists δ > 0 such that for all initial conditions u 0 ∈ B X (0, δ), T (u 0 ) = ∞ and u(t, u 0 ) X < ε for all t ≥ 0. Since J : X → R does not have a local minimum at 0 there exists u 0 ∈ B X (0, δ) such that J(u 0 ) < J(0) = 0. For this u 0 simplify the notation by setting u(t) = u(t, u 0 ) for all t ≥ 0 and then h(t) = u(t) 2 .
Recalling that there exists a constant C > 0 such that · ≤ C · X on X, h(t) ≤ C 2 u(t) 2 X ≤ C 2 ε 2 for all t ≥ 0. As for (5.16), h ∈ C([0, ∞)) ∩ C 1 ((0, ∞)) and h (t) = −4J(u(t)) − 4Π(u(t)) + 2Π (u(t))u(t) ≥ −4J(u(t)) (5.17) by (5.13) for all t > 0 since u(t) X < ε. But J(u(t)) ≤ J(u 0 ) < 0 for all t ≥ 0 by (5.6) and so h (t) ≥ 4|J(u 0 )| > 0 for all t > 0 from which it follows that h(t) → ∞ as t → ∞. This contradiction shows that the solution u ≡ 0 is unstable. Suppose now that (5.13) holds for all ε > 0. Since 0 is not a local minimum of J there exists a sequence {u n } ⊂ X such that J(u n ) < J(0) = 0 for all n and u n X → 0 as n → ∞. Let h n (t) = u(t, u n ) 2 . As before (5.17) holds for 0 < t < T (u n ) and yields h n (t) ≥ h n (0) + 4|J(u n )|t for 0 ≤ t < T (u n ). (5.18) If there is a subsequence {u n k } such that T (u n k ) < ∞ for all n k it follows from (5.2) that lim t→T (un k ) u(t, u n k ) X = ∞, as required. On the other hand, if there exists n 0 such that T (u n ) = ∞ for all n ≥ n 0 it suffices to let t → ∞ in (5.18). The final result in this section gives another criterion for instability without requiring Fréchet differentiability at u ≡ 0. It will be useful even in the subcritical case where the nonlinearity is Fréchet differentiable at u ≡ 0 since it requires less regularity that similar results. For example, Theorem 5.1.3 in [24] requires what is sometimes called strict Fréchet differentiability at u ≡ 0.
Theorem 5.4. Let A : D(A) ⊂ H → H be a self-adjoint operator which is bounded below and let X be given by (5.7). Suppose that F : X → H is locally Lipschitz continuous on X and Gâteaux differentiable at 0 with F (0) = L| X where L ∈ B(H, H) is self-adjoint. Suppose also that F − F (0) = R 1 + R 2 for mappings R 1 and R 2 having the properties, Hence A 1/2 (·) and S 1/2 (·) define equivalent norms on D(A) = D(S). It follows easily from this that D(S 1/2 ) = D(A 1/2 ) = X and that S 1/2 (·) is equivalent to · X on X. Henceforth we consider the Hilbert space (X, ·, · 1 , · 1 ) where u, v 1 = S 1/2 u, S 1/2 v and u 1 = S 1/2 u for u, v ∈ X.
Setting m = inf σ(S) = inf σ(A − L) + λ and m e = inf σ e (S) = inf σ e (A − L) + λ, the hypotheses imply that λ > m and m < m e . Hence m is an isolated eigenvalue of S having finite multiplicity. Set E = ker{S − mI} and let P and Q denote the orthogonal projections of H onto E and F = E ⊥ , respectively. Noting that Qu = u − P u ∈ X if and only if u ∈ X since E ⊂ D(S) ⊂ X, it is easily seen that P and Q are orthogonal projections on the space (X, ·, · 1 ) onto E and F 1 = F ∩ X, respectively.
6. The problem (1.4) to (1.7). Throughout this section the initial value problem (1.4) to (1.7) is considered under the following assumptions. Either (SC) (subcritical case) A satisfies (A) τ for some τ ∈ [0, 2), V ∈ L ∞ (Ω) and g = g 1 + g 2 where g 1 satisfies (G1) * τ and g 2 satisfies (G2) τ , or (CC) (critical case) A satisfies the condition (A) 2 , V satisfies (V) and g = g 1 + g 2 where g 1 satisfies (G1) * 2 and g 2 satisfies (G2) 2 . The development is based on the results in Section 5 using the Hilbert space For all λ ∈ R, the mapping F λ defined by such that u(0) = u 0 and u (t) + S A u(t) = F λ (u(t)) for t ∈ (0, T ) as an equality in L 2 (Ω). This differential equation can be written as Setting By (3.6) this energy functional can also be written as and From the uniqueness of the solution of the intial value problem this means that φ λ (u(·)) is strictly decreasing on (0, T ) unless u is a stationary solution.
For future reference, the main conclusions, which have just been derived, concerning the degenerate parabolic problem (1.4) to (1.7) are collected the following result.
Furthermore, for the energy functional defined by (1.17), φ λ (u(·, u 0 )) ∈ C([0, T (u 0 )), R) ∩ C 1 ((0, T (u 0 )), R) and where S = −∇ · {A∇} + V . This is a degenerate elliptic nonlinear eigenvalue problem for which it is also natural to consider weak solutions, defined as elements However, the hypotheses of Theorem 6.1 ensure that V u 0 +g(u 0 ) − λu 0 ∈ L 2 (Ω) for all u 0 ∈ H A and so by (3.8), (6.12) implies that u 0 ∈ D(S A ) = D(S) and that (λ, u 0 ) satisfies (6.11). Clearly, (λ, u 0 ≡ 0) is a stationary solution for all λ ∈ R since g(x, 0) ≡ 0. The existence of non-trivial stationary solutions can be established using bifurcation theory. In the subcritical case this is straight forward sinceg ∈ C 1 (H A , L 2 (Ω)), if g 1 satisfies (G1) τ instead of (G1) * τ , and S has discrete spectrum so the simple eigenvalue m is a bifurcation point (Theorem 4.1 in Chapter 5 of [4] for example). In fact, by the well-known result concerning variational problems (Theorem 6.6 in [27] for example) there is bifurcation at every eigenvalue of S since φ λ ∈ C 2 (H A , R) if g 1 satisfies (G1)τ . The situation is much more complicated in the critical case since σ e (S) = ∅ andg may not be Fréchet differentiable at 0. However a variety of sufficient conditions for bifurcation can be found in [32] and [33], where the conditions imposed on the nonlinearity are considerably weaker than those required here for Theorem 6.1. But there the also situations where the hypothesis (CC) of Theorem 6.1 is satisfied and (λ, u 0 ≡ 0) are the only stationary solutions. The following result is a special case of Corollary 7.8 in [33]. Proposition 6.2. Consider the problem (6.11) under the following assumptions.
(i) The bounded open set Ω is star-shaped with respect to 0 and its boundary ∂Ω is of class The conclusions of Theorem 6.1 hold and the stationary solution (λ, u ≡ 0) is asymptotically stable for λ < m and unstable for λ > m.
Remark 7.1. Unlike the situation mentioned in Remark 7.3 where diffusion does not eliminate a singularity at the origin, the initial value problem does have a regularizing effect on all of Ω in the subcritical case. See [30,23].
Proof. Setting F λ (u) = λu − V u −g(u) for u ∈ H A , as in (6.1), Propositions 4.1 and 4.3 show that F λ : H A → L 2 (Ω) is locally Lipschitz continuous. Setting L λ u = (λ − V )u, L λ ∈ B(L 2 (Ω), L 2 (Ω)) is self-adjoint and these results also show that Theorem 7.2. Consider the initial value problem (1.4) to (1.7) under the hypothesis (CC) with g 1 ≡ 0. The conclusions of Theorem 6.1 hold and the stationary solution (λ, u ≡ 0) is asymptotically stable for λ < m. It is unstable for λ > m provided that g is a finite sum of terms satisfying the condition (G2) * 2 .
The rest of this section is devoted to the more interesting situation where (CC) is satisfied and g 1 ≡ 0. As pointed out by Lemma 4.2, the condition (G1) 2 does not imply that g 1 ; H A → L 2 (Ω) is Fréchet differentiable at 0 and the results used to prove Theorem 7.2 do not apply. In fact, they would lead to an incorrect conclusion in cases where the principle of linearized stability fails. Most of the following conclusions about stability and instability will be obtained from Theorem 5.3 with X = H A by using the energy (6.4) as a Lyapunov functional. In verifying the hypotheses of Theorem 5.3 extensive use will be made of the results in [34] concerning the existence of a potential well for a functional whose gradient is not necessarily Fréchet differentiable. where G 1 (x, s) = s 0 g 1 (x, y) dy. Note that by (G1) * 2 , where 1 is the best Lipschitz constant for g 1 given by (4.9).
The conclusions of Theorem 6.1 hold and (λ, u ≡ 0) is a stationary solution for all λ ∈ R.
In all three cases, 0 lies in a quadratic potential well of φ λ : H A → R and, if 2G(x, s) ≤ g(x, s)s for all x ∈ Ω and s ∈ R, then G(x, s) ≥ 0 and there exist δ > 0 and η > such that u(t, u 0 ) (B, asymptotic stability) (i) Suppose that the function g 1 satisfies the conditions (G1) * 2 and (C) and that the function g 2 satisfies the condition (G2) 2 for some exponents α > N σ/2 and 0 < σ < 2/(N − 2).
(ii) If g 1 (x, s)s ≥ 0 for all x ∈ Ω and s ∈ R, then G 1 (x, s) ≥ 0 for all x ∈ Ω and s ∈ R and (λ, u ≡ 0) is asymptotically stable for all λ < m.
(ii) If in addition to the assumption in part (i), then the sequence given in part (i) has the stronger properties that (iii) Suppose that m < m e and that, in addition to satisfying the condition (G1) * 2 , the function g 1 (x, ·) : R → R is non-decreasing for all x ∈ Ω. Then G 1 (x, s) ≥ 0 and (λ, u ≡ 0) is unstable for λ > m.
Remark 7.2. Observing that 2G(x, s) ≥ g(x, s)s for all s ∈ R is equivalent to the property that s∂ s { G(x,s) s 2 } ≤ 0 for all s = 0, it follows that G(x, s) ≤ 0 for all s ∈ R and consequently, g(x, s)s ≤ 0 for all s ∈ R. Similarly, if 2G(x, s) ≤ g(x, s)s for all s ∈ R then G(x, s) and g(x, s) ≥ 0 for all s ∈ R.
Remark 7.3. The conclusion lim t→T (un) u(t, u n ) L ∞ = ∞ in part C(ii) needs to be placed in the proper perspective since elements in H A , and even D(S A ), are not necessarily bounded in a neighbourhood of the origin. Hence it conceivable that there is a solution of (6.2)(6.3) for which u(t, u 0 ) L ∞ = ∞ for all t ∈ [0, T (u 0 )). This can be demonstrated explicitly for the linearised equation (1.14). In the setting of Example 3.1, for any λ ∈ R, u(t, u α ) = e (λ−mα)t u α for all t ≥ 0 is the unique solution of (6.2) and (1.14) with initial condition the eigenfunction u α associated with the eigenvalue m α . For α ∈ (0, 1/2), u(t, u α ) has a singularity at x = 0 for all t ≥ 0.
Remark 7.4. In view of Theorem 7.2 which concerns the case g 1 ≡ 0 it would seem reasonable to expect that the conclusion in part (C)(iii) also holds when m = m e , but so far I have not found a proof of this.
(B) Parts (A)(i) and (iii) show that 0 lies in a quadratic potential well of φ λ . By part (ii) of Theorem 5.3 it suffices to show that there exists R(λ) > 0 such that inf{ Su +g(u) − λu L 2 : u ∈ D(S) and r ≤ u A ≤ R(λ)} > 0 for all r ∈ (0, R(λ)]. This is established in part (C) of Lemma 4.5 for the case (i) and in Lemma 4.4 for case (ii).
(C) (i) As explained in the proof of part (A), the conclusions of Theorem 4.1 in [34] are valid under the hypothesis (CC). For λ > min{m, m e + γ 1 }, Theorem 4.1(iii) in [34] shows that there exists a sequence {u n } ⊂ H A such that u n A → 0 as n → ∞ and φ λ (u n ) < 0 for all n. From (7.3) it follows that the condition (5.13) is satisfied for all ε > 0 and Theorem 5.3(iii) yields the conclusion.
Thus for the sequence given by part (i), by (6.10) for 0 ≤ t < T (u 0 ) and it follows that lim t→T (un) u(t, u n ) L 2 = ∞ for all n. Recalling that Ω is bounded, this implies that lim t→T (un) u(t, u n ) L ∞ = ∞ for all n.
For a more restricted class of nonlinearities the criteria for stability and instability in Theorem 7.3 become sharp. To illustrate this consider a nonlinearity of the form g(x, s) = ±k(s) + g 2 (x, s) where the function k satisfies the condition (K) mentioned in the Introduction and g 2 satisfies the condition (G2) 2 . As mentioned in the Introduction some real analytic functions such as k(s) = s − tanh s and k(s) = s − arctan s satisfy the condition (K), as do functions of the form k(s) = C |s| σ s 1 + |s| σ for s ∈ R with C ≥ 0 and 0 < σ ≤ 1.
(I) In this case g 1 (x, s)s ≥ 0 for all x ∈ Ω and s ∈ R. The conclusions follow from parts (B)(ii) and (C)(iii) of Theorem 7.3 since k is non-decreasing.
(III) This follows from part (II) with k ≡ 0 and g 2 ≡ 0. In fact it also follows from Theorem 7.2.
Remark 7.6. In the case g(x, s) = −k(s), parts (II) and (III) justify the remarks made at the beginning of the Introduction concerning the failure of the principle of linearized stability. Furthermore, Theorem 7.3 provides some additional information about the nature of the instability in this case. Namely, T (u 0 ) = ∞ for all u 0 ∈ H A and if λ > min{m, m e − k (∞)}, there is a sequence {u n } ⊂ H A of initial conditions such that u n A → 0 as n → ∞ and lim inf t→∞ u(t,un) L 2 √ t ≥ |φ λ (u n )| > 0 for all n.
A final example illustrates many of the main features of the more general results established in this paper in a simple setting where they are particularly sharp. (ii) A(x) ≡ |x| 2 , (iii) V ∈ C 1 (Ω) with V (0) = 0 and x · ∇V (x) ≥ 0 for all x ∈ Ω, (iv) g(x, s) = −k(s) where k is a function satisfying the condition (K) with k(s)/s strictly increasing on (0, ∞) and hence k (∞) > 0.
The linearization of (7.5) at u ≡ 0 is ∂ t u − r 2 ∆u + V u = 2x · ∇u + λu (7.6) and for this equation the initial value problem is asymptotically stable for λ < N 2 4 and unstable for λ > N 2 4 .