On normal stability for nonlinear parabolic equations

We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable.


Introduction
In this short note we consider quasilinear as well as fully nonlinear parabolic equations and we study convergence of solutions towards equilibria in situations where the set of equilibria forms a C 1 -manifold. Our main result can be summarized as follows: suppose that for a nonlinear evolution equation we have a C 1 -manifold of equilibria E such that at a point u * ∈ E, the kernel N (A) of the linearization A is isomorphic to the tangent space of E at u * , the eigenvalue 0 of A is semi-simple, and the remaining spectral part of the linearization A is stable. Then solutions starting nearby u * exist globally and converge to some point on E. This situation occurs frequently in applications. We call it the generalized principle of linearized stability, and the equilibrium u * is then termed normally stable.
A typical example for this situation to occur is the case where the equations under consideration involve symmetries, i.e. are invariant under the action of a Lie-group.
The situation where the set of equlibria forms a C 1 -manifold occurs for instance in phase transitions [13,25], geometric evolution equations [12,14], free boundary problems in fluid dynamics [15,16], stability of traveling waves [26], and models of tumor growth, to mention just a few.
A standard method to handle situations as described above is to refer to center manifold theory. In fact, in that situation the center manifold of the problem in question will be unique, and it coincides with E near u * . Thus the so-called shadowing lemma in center manifold theory implies the result. Center manifolds are well-studied objects in the theory of nonlinear evolution equations. For the parabolic case we refer to the monographs [17,20], and to the publications [5,6,10,19,21,27,28].
However, the theory of center manifolds is a technically difficult matter. Therefore it seems desirable to have a simpler, direct approach to the generalized principle of linearized stability which avoids the technicalities of center manifold theory.
Such an approach has been introduced in [26] in the framework of L p -maximal regularity. It turns out that within this approach the effort to prove convergence towards equilibria in the normally stable case is only slightly larger than that for the proof of the standard linearized stability result -which is simple.
The purpose of this paper is to extend the approach given in [26] to cover a broader setting and a broader class of nonlinear parabolic equations, including fully nonlinear equations. This approach is flexible and general enough to reproduce the results contained in [7,12,13,14,15,16,25,26], and it will have applications to many other problems.
Our approach makes use of the concept of maximal regularity in an essential way. As general references for this theory we refer to the monographs [1,11,20].

Abstract nonlinear problems in a general setting
Let X 0 and X 1 be Banach spaces, and suppose that X 1 is densely embedded in X 0 . Suppose that F : where U 1 is an open subset of X 1 . Then we consider the autonomous (fully) nonlinear problemu (t) + F (u(t)) = 0, t > 0, u(0) = u 0 , for u 0 ∈ U 1 . In the sequel we use the notation | · | j to denote the norm in the respective spaces X j for j = 0, 1. Moreover, for any normed space X, B X (u, r) denotes the open ball in X with radius r > 0 around u ∈ X.
Let E ⊂ U 1 denote the set of equilibrium solutions of (2), which means that Given an element u * ∈ E, we assume that u * is contained in an m-dimensional manifold of equilibria. This means that there is an open subset U ⊂ R m , 0 ∈ U , and a C 1 -function Ψ : U → X 1 such that • the rank of Ψ ′ (0) equals m, and We assume further that near u * there are no other equilibria than those given by Ψ(U ), i.e. E ∩ B X1 (u * , r 1 ) = Ψ(U ), for some r 1 > 0.
Let u * ∈ E be given and set A := F ′ (u * ). Then we assume that A ∈ H(X 1 , X 0 ), by which we mean that −A, considered as a linear operator in X 0 with domain X 1 , generates a strongly continuous analytic semigroup {e −At ; t ≥ 0} on X 0 . In particular we may take the graph norm of A as the norm in X 1 .
In addition, we assume that there is a constant c 0 > 0 independent of We refer to [1, Section III.1.4] for further information on trace spaces. Moreover, we assume that (A2)w ∈ E 1 (J) and |w(t)| 0 ≤ |w(t)| 1 , t ∈ J, imply ||w|| E0(J) ≤ ||w|| E1(J) ; for ω > 0 fixed, there exists a constant c 1 > 0 not depending on J and such that Our key assumption is that (E 0 (J), E 1 (J)) is a pair of maximal regularity for A.
To be more precise we assume that (A3) the linear Cauchy problemẇ + Aw = g, w(0) = w 0 has for each (g, We impose the following assumption for the sake of convenience. For all examples that we have in mind the condition can be derived from (A3). Suppose that σ(A), the spectrum of A, admits a decomposition σ(A) = σ s ∪ σ ′ , where σ s ⊂ {z ∈ C : Re z > ω} for some ω > 0 and σ ′ ⊂ {z ∈ C : Re z ≤ 0}. Let P s denote the spectral projection corresponding to the spectral set σ s . Then we assume that (A4) there exists a constant M 0 > 0 such that for any J = [0, a), a ∈ (0, ∞], any σ ∈ [0, ω], and any function g with e σt P s g ∈ E 0 (J) there is a unique solution w ofẇ + A s w = P s g, t ∈ J, w(0) = 0, satisfying there exists a constant M 1 > 0 such that for any J = [0, a), a ∈ (0, ∞], and for any z ∈ X γ there holds We again refer to [1, Chapter III] for more background information on the notion of maximal regularity. In order to cover the case X γ = X 1 we assume the following structure condition on the nonlinearity G: (A5) there exists a uniform constant C 1 such that for any η > 0 there is r > 0 such that Observe that condition (A5) trivially holds in the case where U γ ⊂ X γ is an open set.
Lastly, concerning solvability of the nonlinear problem (4) we will assume that (A6) given b > 0 there exists . Note that since v = 0 is an equilibrium of (4), condition (A6) is satisfied whenever one has existence and uniqueness of local solutions in the described class as well as continuous dependence of the maximal time of existence on the initial data.
We conclude this section by describing three important examples of admissible pairs (E 0 (J), E 1 (J)).
Example 1: (L p -maximal regularity.) In our first example, the spaces (E 0 (J), E 1 (J)) are given by The trace space is a real interpolation space given by γE 1 = X γ = (X 0 , X 1 ) 1−1/p,p and we have E 1 (J) ֒→ BUC(J; X γ ), see for instance [1,Theorem III.4.10.2]. For a proof of (6) we refer to [23,Proposition 6.2]. This yields Assumption (A1). For Assumption (A2) we note that for t ≥ 0 and w ∈ E 1 (R + ). We refer to [11,18,24], [1, Section III.4.10] and the references therein for conditions guaranteeing that the crucial Assumption (A3) on maximal regularity is satisfied. It is clear that the property of maximal regularity is passed on from A to A s in the spaces E s 0 (J) := L p (J; X s 0 ), E s 1 (J) := H 1 p (J; X s 0 )∩L p (J; X s 1 ), and this implies Assumption (A4), see for instance [1, Remark III.4.10.9(a)]. Assumption (A5) is satisfied in case that the nonlinear mapping F has a quasilinear structure, see [26]. Assumption (A6) follows in case that F has a quasilinear structure from (A3) and [22,Theorem 3.1], see also [2, Theorem 2.1, Corollary 3.3]. We remark that the case of L p -maximal regularity has been considered in detail in [26].

The main result
In this section we state and prove our main theorem about convergence of solutions for the nonlinear equation (2) towards equilibria.
Theorem 3.1. Let u * ∈ X 1 be an equilibrium of (2), and assume that the above conditions (A1)-(A6) are satisfied. Suppose that u * is normally stable, i.e. assume that (i) near u * the set of equilibria E is a C 1 -manifold in X 1 of dimension m ∈ N, (ii) the tangent space for E at u * is given by N (A), Re z > ω} for some ω > 0. Then u * is stable in X γ , and there exists δ > 0 such that the unique solution u(t) of (2) with initial value u 0 ∈ X γ satisfying |u 0 − u * | γ < δ exists on R + and converges at an exponential rate to some u ∞ ∈ E in X γ as t → ∞.
Proof. The proof to Theorem 2.1 will be carried out in several steps, as follows.
(a) We denote by P l , l ∈ {c, s}, the spectral projections corresponding to the spectral sets σ s and σ c := {0}, respectively, and let A l = P l AP l be the part of A in X l 0 = P l (X 0 ) for l ∈ {c, s}. Note that A c = 0. We set X l j := P l (X j ) for l ∈ {c, s} and j ∈ {0, γ, 1}. It follows from our assumptions that X c 0 = X c 1 . In the following we set X c := X c 0 and equip X c with the norm of X 0 . Moreover, we take as a norm on X j |v| j := |P c v| 0 + |P s v| j for j = 0, γ, 1.
(b) Next we show that the manifold E can be represented as the (translated) graph of a function φ : B X c (0, ρ 0 ) → X s 1 in a neighborhood of u * . In order to see this we consider the mapping It follows from our assumptions that g ′ (0) = P c ψ ′ (0) : R m → X c is an isomorphism. By the inverse function theorem, g is a C 1 -diffeomorphism of a neighborhood of 0 in R m onto a neighborhood, say B X c (0, ρ 0 ), of 0 in X c . Let g −1 : B X c (0, ρ 0 ) → U be the inverse mapping. Then g −1 : B X c (0, ρ 0 ) → U is C 1 and g −1 (0) = 0. Next we set Φ(x) := ψ(g −1 (x)) for x ∈ B X c (0, ρ 0 ) and we note that and that {u * + x + φ(x) : x ∈ B X c (0, ρ 0 )} = E ∩ W, where W is a neighborhood of u * in X 1 . This shows that the manifold E can be represented as the (translated) graph of the function φ in a neighborhood of u * . Moreover, the tangent space of E at u * coincides with N (A) = X c . By applying the projections P l , l ∈ {c, s}, to equation (5) and using that x + φ(x) = ψ(g −1 (x)) for x ∈ B X c (0, ρ 0 ), and that A c ≡ 0, we obtain the following equivalent system of equations for the equilibria of (4) Finally, let us also agree that ρ 0 has already been chosen small enough so that This can always be achieved, thanks to (13).
(c) Introducing the new variables we then obtain the following system of evolution equations in X c × X s 0 ẋ = T (x, y), x(0) = x 0 , y + A s y = R(x, y), y(0) = y 0 , with x 0 = P c v 0 and y 0 = P s v 0 − φ(P c v 0 ), where the functions T and R are given by Using the equilibrium equations (14), the expressions for R and T can be rewritten as Equation (17) immediately yields showing that the equilibrium set E of (2) near u * has been reduced to the set Observe also that there is a unique correspondence between the solutions of (2) close to u * in X γ and those of (16) close to 0. We call system (16) the normal form of (2) near its normally stable equilibrium u * .
Clearly, v ∞ is an equilibrium for equation (4), and u ∞ := u * + v ∞ ∈ E is an equilibrium for (2). It follows from (A2), the estimate for T in (18), and from (20) that T (x(s), y(s)) ds ≤ β This shows that x(t) converges to x ∞ at an exponential rate. Due to (15), (21) and the exponential estimate for |x(t) − x ∞ | we now get for the solution u(t) = u * + v(t) of (2) thereby completing the proof of the second part of Theorem 3.1. Concerning stability, note that given r > 0 small enough we may choose 0 < δ ≤ r such that the solution starting in B Xγ (u * , δ) exists on R + and stays within B Xγ (u * , r).
Remarks: (a) Theorem 3.1 has been proved in [26] in the setting of L p -maximal regularity, and applications to quasilinear parabolic problems with nonlinear boundary conditions, to the Mullins-Sekerka problem, and to the stability of travelling waves for a quasilinear parabolic equation have been given. (b) It has been shown in [26] by means of examples that conditions (i)-(iii) in Theorem 3.1 are also necessary in order to get convergence of solutions towards equilibria u ∞ ∈ E.