Approximate Solutions to the Abstract Cauchy Problem

BORIS BÄUMER Department of Mathematics/MS 084, University of Nevada, Reno, NV 89557.

Abstract. We introduce the notion of approximate solutions to deal with Cauchy problems with generalized solutions. We show the equivalence of this notion of solvability with the regularized solutions introduced by I. Cioranescu and G. Lumer. As an example, we show that the backwards heat equation in Ω ⊂ IR^{d} is well posed in the approximate (or regularized) sense if and only if d = 1.

1. INTRODUCTION

The aim of this article is to give an alternative point of view on regularized solutions^{(1)} to the abstract Cauchy problem

(ACP)  u′(t) = Au(t);  u(0) = x;  0 ≤ t < T ≤ ∞

as introduced by I. Cioranescu and G. Lumer [5],[6],[7]. A regularized solution is a continuous function v that satisfies

v = A(1 ⋆ v) + (1 ⋆ k)x

on [0, T) for some regularizer k ∈ L_{loc}^{1}[0, T) with 0 ∈ supp(k), where

f ⋆ g := t ↦ ∫_{0}^{t}f(t − s)g(s) ds

(1)We investigate only scalar valued kernels in this article. Regularized solutions were defined with even more general operator valued kernels. Conditions on these kernels that still allow approximate solutions can be found in [2].

denotes the convolution of f and g on IR^{+}. If v is a regularized solution, it is the k-convoluted image of the “true” solution; i.e., if (ACP) admits a classical solution u, then v = k ⋆ u.

Our approach uses the fact that we can always invert the process of convolution (see [2]): For any function v ∈ C_{0}([0, T); X) and any regularizer k we can construct a sequence u_{n} ∈ C([0, T); X) such that k ⋆ u_{n} → v uniformly on compact intervals. This suggests the following definition of an approximate solution, converging to the “true” solution:

DEFINITION: A k-approximate solution to (ACP) for some closed^{(2)} linear operator A on a Banach space X is a sequence of continuous functions u_{n} such that

(i) k ⋆ u_{n} → v for some v ∈ C_{0}([0, T); X),

(ii) 1 ⋆ u_{n} ∈ C([0, T); [D(A)]),

(iii) k ⋆ (u_{n} − A(1 ⋆ u_{n}) − x) → 0.

Clearly, any k-approximate solution [u_{n}] gives rise to a k-regularized solution v := lim k ⋆ u_{n}. One can think of the sequences [u_{n}] as elements in the generalized function space^{(3)}

C^{[k]}([0, T); X) := \overline{C([0, T); X)}^{∥·∥_{k}},

the completion of the space of continuous functions with respect to the seminorms ∥f∥_{k,T_{n}} := ∥k ⋆ f∥_{T_{n}} := sup_{0≤t≤T_{n}} ∥∫_{0}^{t}k(t − s)f(s) ds∥. In other words, we consider existence, uniqueness and continuous dependence on initial data of the problem

[u_{n}] = A[1 ⋆ u_{n}] + x

for some [u_{n}] ∈ C^{[k]}([0, T); X). Furthermore, for any regularized solution we will actually construct a sequence of exponential functions that form an approximate solution.

We say that a Cauchy problem is well posed if there exists k ∈ L_{loc}^{1}[0, T) with 0 ∈ supp(k) such that (ACP) has a unique k-approximate solution [u_{n}] ∈ C^{[k]}([0, T); X) for all x ∈ X. With this definition of wellposedness we show that the backwards heat equation on a regular set Ω ⊂ IR^{d} with Dirichlet or Neumann boundary conditions is well posed if and only if d = 1.

2. PRELIMINARIES

We briefly recall some theorems about asymptotic Laplace transforms and its inversion [1]. The asymptotic Laplace transform is an operator extension of the usual Laplace transform to all of L_{loc}^{1}, including functions that are not exponentially bounded. It was first introduced by J.C. Vignaux in 1939 [12] and further developed by G. Lumer and F. Neubrander [11]. The asymptotic Laplace transform L_{A}(f) of a function f ∈ L_{loc}^{1}([0, T); X) is defined to be an equivalence class of analytic functions defined in a post sectorial region Ω of the complex plane;i.e.,

(2)Techniques for investigating the abstract Cauchy problem with nonclosed operators are found in [3] and [2].

(3)For a more detailed introduction to these generalized function spaces see [4] or [2].

L_{A}(f) = {r ∈ A(Ω, X) : r ≈_{t} λ ↦ ∫_{0}^{t} e^{−λs}f(s) ds for all 0 < t < T }

where r ≈_{t} q if r − q is of exponential decay^{(4)} t. This set is always nonempty, meaning that L_{A}(f) exists and is well defined for all f, and the properties of the Laplace transform extend fully (see [11] or [2]).

The asymptotic Laplace transform can be extended to transform generalized functions^{(5)} f ∈ C^{[k]}([0, T); X) for k ∈ L_{loc}^{1}[0, T) with 0 ∈ supp(k). These spaces are the completions of the space of continuous functions with respect to the Frechét topology that is obtained by measuring the convolution image (∥f∥_{k} := ∥k⋆f∥). Thus we can define k ⋆f := k ⋆[f_{n}] := [k ⋆f_{n}] = g for any element f := [f_{n}] ∈ C^{[k]}([0, T); X), where g ∈ C_{0}([0, T); X). The asymptotic Laplace transform is defined via

L_{A}(f) := \frac{L_{A}(k ⋆ f)}{L_{A}(k)}.

Thus we obtain co-classes of (in general) meromorphic functions. They have the property that for each element f̂ ∈ L_{A}(f) we can construct a Müntz sequence^{(6)} (β_{n})_{n∈IN} such that

lim_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ = −S ≤ 0,

where S := min{x; x ∈ supp(k ⋆ f)}. For a proof see [1].

The following theorem is the backbone of this article. The proof is a straightforward adjustment of [1], Thm. 9.

THEOREM 1 (The Phragmén-Mikusiński Inversion of the Laplace Transform). Let f ∈ C_{0}([0, T); X), or let f be a generalized function in one of the Frechét spaces C^{[k]}([0, T); X) for some regularizer k ∈ L_{loc}^{1}[0, T) with 0 ∈ supp(k). Let f̂(λ) ∈ L_{A}(f). Let (β_{n})_{n∈IN} be a Müntz sequence such that lim_{n→∞} \frac{1}{β_{n}} ln ∥f̂(β_{n})∥ ≤ 0.

Let ϵ_{n} = max\{\frac{2}{\sqrt{n}}, \frac{2}{β_{n}} ln ∥β_{n}f̂(β_{n})∥\}. Let N_{n} be such that ∑_{i=1}^{N_{n}} \frac{1}{β_{ni}} → T, as n → ∞. Define

α_{n,i} := β_{ni}e^{−β_{ni} ∑_{j=1}^{Nn} \frac{1}{β_{nj}}} ∏_{\begin{matrix}j=1 \\ j≠i \end{matrix}}^{N_{n}} \frac{β_{nj}}{β_{nj} − β_{ni}} .

Then

f = lim_{n→∞} ∑_{i=1}^{N_{n}} α_{n,i}f̂(β_{ni})e^{β_{ni}(·−ϵ_{n})},

where the limit is uniform on compact sets, or in C^{[k]}([0, T); X) respectively. ♢ 3. APPROXIMATE SOLUTIONS

(4)A function r ∈ A(Ω, X) is of exponential decay t if lim sup_{λ→∞} \frac{1}{λ} ln ∥r(λ)∥ ≤ −t (5)We use C^{[k]}([0, T); X) for simplicity of notation, of course the same construction also works for f ∈ L_{loc}^{p,[k]}([0, T); X).

(6)(β_{n})_{n∈IN} ⊂ IR^{+} is a Müntz sequence if β_{n+1} − β_{n} ≥ 1 and ∑_{n=1}^{∞} \frac{1}{β_{n}} = ∞

In this section we apply the Laplace transform results to the abstract Cauchy problem. We show the equivalence of the notions of approximate solutions and the notion of regularized solutions for scalar valued kernels. Furthermore, for any problem that is well posed in the regularized sense, we construct a sequence of exponential functions forming an approximate solution that converges in the best possible topology.

THEOREM 2. Let A be a closed linear operator on a Banach space X. Let k ∈ L_{loc}^{1}[0, T) with 0 ∈ supp(k). The following are equivalent.

(i) u_{n} is a k-approximate solution of (ACP).

(ii) v := lim k ⋆ u_{n} is a k-regularized solution of (ACP).

PROOF. (i) => (ii). Obvious by the closedness of A.

(ii) => (i). Since 1 ⋆ v ∈ C_{0}([0, t); [D(A)]) there exists v̂ ∈ L_{A}(v) with v̂(λ) ∈ D(A) for all λ > 0. Let

u_{n}(t) := ∑_{i=1}^{N_{n}} α_{n,i} \frac{v̂(β_{ni})}{k̂(β_{ni})}e^{β_{ni}(t−ϵ_{n})},

where β_{n} is such that lim_{n→∞} \frac{1}{β_{n}} ln |k̂(β_{n})| = 0 and the variables N_{n} and α_{n,i} are chosen as in Theorem 1. Then u_{n} ∈ C^{∞}([0, t); [D(A)]). By Theorem 1, k ⋆ u_{n} → v uniformly in X on compact sets, and k ⋆ 1 ⋆ u_{n} → 1 ⋆ v uniformly in [D(A)] on compact sets. Hence k ⋆ (u_{n} − A(1 ⋆ u_{n}) − x) → v − A(1 ⋆ v) − k ⋆ x = 0. ♢

Thus, the existence and uniqueness results for regularized solutions carry over to approximate solutions.

COROLLARY 3. Let ∅ ≠ ρ(A). Let k ∈ L_{loc}^{1}[0, T) with 0 ∈ supp(k). Let f ∈ C^{[k]}([0, T); X). Suppose λ ↦ k̂(λ)R(λ, A)λf̂(λ) ∈ L_{A}(g) for some g ∈ C_{0}([0, T); X). Then

[∑_{i=1}^{N_{n}} α_{n,i}β_{ni}R(β_{ni}, A)f̂(β_{ni})e^{β_{ni}(t−ϵ_{n})}] ,

is a k-approximate solution to u = A(1 ⋆ u) + f.

PROOF. By assumption there exists ĝ ∈ L_{A}(g) with

(λ − A)ĝ(λ) = k̂(λ)λf̂(λ).

Thus g is a regularized solution, and with the same argument as above, the assertion follows. ♢

Sufficient conditions on when an analytic function in some region in the complex plane is in the asymptotic Laplace transform of a continuous function can be found in [11] or [2].

4. WELLPOSEDNESS

In this section we explore some of the consequences and conditions for wellposedness in the following non-traditional sense. We say that (ACP) is well posed if there exists a regularizer k such that (ACP) admits a unique k-approximate solution for all x ∈ X that depend continuously on the initial data (Uniqueness and continuity in the sense of [u_{n}] ∈ C^{[k]}([0, T); X)). We will show that existence and uniqueness already imply continuous dependence on the initial data and a nonempty resolvent set. Furthermore we obtain a sharper version of Lyubich’s uniqueness theorem on the growth of the resolvent. These theorems are extensions of results obtained by G. Lumer and F. Neubrander in [11].

THEOREM 4 (Existence+Uniqueness => Cont. Dependence). Suppose the abstract Cauchy problem has a unique approximate solution u(·, x) ∈ C^{[k]}([0, T); X) for all x ∈ X. Then (ACP) is well posed; i.e., for all 0 < t < T there exists M_{t} > 0 such that ∥u(·, x)∥_{k} ≤ M_{t}∥x∥ on [0, t].

PROOF. Let S(·)x := k ⋆ u(·, x). Thus, by Theorem 2,

Sx = A(1 ⋆ Sx) + (1 ⋆ k)x.

By the existence and uniqueness property, we know that S : X → C([0, T); X) is a linear operator. Furthermore, S is a closed operator, since x_{n} → x and Sx_{n} → v implies that 1 ⋆ Sx_{n} → 1 ⋆ v and, by the closedness of A, that

A(1 ⋆ v) = v − (1 ⋆ k)x.

By the uniqueness property, Sx = v, and hence S is a closed linear operator. By the closed graph theorem, S is bounded and hence for all 0 ≤ t < T there exists M_{t} > 0 such that

∥u(·, x)∥_{k,t} = ∥Sx∥_{t} = sup_{s∈[0,t]} ∥S(s)x∥ ≤ M_{t}∥x∥.

♢

Recall the fact shown by G. Lumer and F. Neubrander [11] that any function that is analytic in a postsectorial region and satisfies a certain growth condition is in the asymptotic Laplace transform of a continuous function. Suppose a meromorphic function on such a region multiplied by the asymptotic Laplace transform of a regularizer yields an analytic function satisfying the growth conditions. Then, this meromorphic function is an element of the asymptotic Laplace transform of a generalized function. Thus, the existence of a meromorphic resolvent on a postsectorial region with some growth condition implies wellposedness.

The next theorem shows that wellposedness has implications on the resolvent set and on the growth behavior of the resolvent of the operator A.

THEOREM 5 (Existence of Resolvent). If (ACP) is well posed, then every Müntz sequence has a subsequence (β_{n_{j}})_{j∈IN} such that β_{n_{j}} is in the resolvent set of A and lim sup_{j→∞} \frac{1}{β_{n_{j}}} ln ∥R(β_{n_{j}}, A)∥ ≤ 0.

PROOF. If (ACP) is well posed, then there exists a strongly continuous operator family (S(t))_{t∈[0,T)} such that

Sx = A(1 ⋆ Sx) + (1 ⋆ k)x.

Furthermore, let 0 < t < T, and

S̃(λ)x := ∫_{0}^{t} e^{−λs}S(s)x ds.

By the previous Theorem, (1 ⋆ S)(t) ∈ L(X, [D(A)]) for all 0 ≤ t < T, and thus S̃(λ) ∈ L(X, [D(A)]) and (S̃(λ))_{λ∈C} is a strongly analytic operator family. Let

k̂ ∈ L_{A}(k). Then

(λ − A)S̃(λ)x = k̂(λ)x + r_{x}(λ),

where r_{x} ≈_{t} 0 for all x ∈ X. Furthermore, r_{(·)}(λ) ∈ L(X). Let 0 < ϵ < t/2. Since r_{x} ≈_{t} 0, there exists M_{x} such that

e^{λ(t−ϵ)}∥r_{x}(λ)∥ ≤ M_{x}

for all λ > 0. By the uniform boundedness principle there exists a constant M such that r(λ) : x ↦ r_{x}(λ) satisfies

∥r(λ)∥ ≤ Me^{−λ(t−ϵ)}.

Let (β_{n})_{n∈IN} be a Müntz sequence. Then there exists a subsequence (β_{n_{j}})_{j∈IN} such

that lim_{j→∞} \frac{1}{β_{n_{j}}} ln |k̂(β_{n_{j}})| = 0 and thus there exists a constant J such that

|k̂(β_{n_{j}})| ≥ e^{−β_{n_{j}} ϵ} > 2∥r(β_{n_{j}})∥

for all j > J. Then ∥\frac{r(β_{n_{j}})}{k̂(β_{n_{j}})}∥ < 1/2. Thus Id + \frac{r(β_{n_{j}})}{k̂(β_{n_{j}})} is an invertible operator that

maps onto X and

∥ (Id + \frac{r(β_{n_{j}})}{k̂(β_{n_{j}})})^{−1} ∥ = ∥ ∑_{i=0}^{∞}(−1)^{i} (\frac{r(β_{n_{j}})}{k̂(β_{n_{j}})})^{i} ∥ ≤ 2.

Since k̂(β_{n_{j}}) ≠ 0 is scalar-valued, the operator

(β_{n_{j}} − A)S̃(β_{n_{j}}) = k̂(β_{n_{j}})(Id + \frac{r(β_{n_{j}})}{k̂(β_{n_{j}})})

is a continuously invertible operator. Let

Q(β_{n_{j}}) := (k̂(β_{n_{j}})(Id + \frac{r(β_{n_{j}})}{k̂(β_{n_{j}})}))^{−1}

and define

R(β_{n_{j}}) := S̃(β_{n_{j}})Q(β_{n_{j}}).

Then, ∥Q(β_{n_{j}})∥ ≤ _{\frac{2}{k̂(β_{n_{j}})}} and (β_{n_{j}} − A)R(β_{n_{j}}) = Id.

We show next that (β_{n_{j}} − A) is one-to-one. Suppose β_{n_{j}}x = Ax for some x ∈ D(A). Then

(e^{β_{n_{j}} t} − 1)x = A\frac{1}{β_{n_{j}}} (e^{β_{n_{j}} t} − 1)x = A∫_{0}^{t}e^{β_{n_{j}} s}x ds.

Since the solutions to ACP are unique, we deduce that S(t)x = (k ⋆ e^{β_{n_{j}} (·)})(t)x. Thus S̃(λ)x is a scalar multiple of x for all λ ∈ C. Hence,

0 = (β_{n_{j}} − A)S̃(β_{n_{j}})x = k̂(β_{n_{j}})(x + \frac{r(β_{n_{j}})x}{k̂(β_{n_{j}})}).

Therefore, ∥ − x∥ = ∥\frac{r(β_{n_{j}})x}{k̂(β_{n_{j}})}∥ ≤ ∥x∥/2, and thus x = 0. Hence (β_{n_{j}} − A) is

one-to-one and thus R(β_{n_{j}}) = R(β_{n_{j}}, A) is the resolvent of A at β_{n_{j}}.

Since ∥Q(β_{n_{j}})∥ ≤ 2/|k̂(β_{n_{j}})|, we know that

lim sup_{j→∞} \frac{1}{β_{n_{j}}} ln ∥Q(β_{n_{j}})∥ = 0

and hence the same holds for R(β_{n_{j}}) = S̃(β_{n_{j}})Q(β_{n_{j}}). ♢

As a Corollary to the Theorem 1 we obtain that existence and growth of the resolvent yield the following extension of Lyubich’s uniqueness condition, see [10] or [3].

COROLLARY 6 (Lyubich’s Uniqueness Theorem). Suppose there exists a Müntz sequence (β_{n})_{n∈IN} such that β_{n} ∈ ρ(A) for all n and

lim sup_{n→∞} \frac{1}{β_{n}} ln ∥R(β_{n}, A)∥ ≤ 0.

Then every approximate solution is unique.

PROOF. Suppose u is a k-approximate solution of u′ = Au; u(0) = 0. Let û ∈ L_{A}(u). Then there exists r̂ ∈ L_{A}(0) such that û(β_{n}) = R(β_{n}, A)r(β_{n}) and thus

∥û(β_{n})∥ ≤ ∥R(β_{n}, A)∥∥r(β_{n})∥ ≈_{T} 0.

Thus u = 0. ♢ 5. EXAMPLE: THE BACKWARDS HEAT EQUATION

Consider the Laplace operator −Δ on a bounded open set Ω ⊂ IR^{d} with smooth boundary and Dirichlet or Neumann boundary conditions. Then it is well known (see for example [8]) that the eigenvalues λ_{n} grow like n^{2/d}. Thus, if d ≥ 2, the point spectrum is forming a Müntz sequence. This implies that any regularizer k̂ that regularizes the resolvent would have to be zero on a Müntz sequence; otherwise λ ↦ k̂(λ)R(λ, A) can not be analytic on some region containing (ω, ∞). But that in turn implies that k = 0.

In one dimension, consider Ω = (0, π). We use the eigenfunctions of the Laplacian; i.e., let e_{n}(x) := sin(nx) for the Dirichlet problem, and e_{n}(x) := cos(nx) for the Neumann problem. Every function f ∈ L^{2}[0, π] can be expressed in terms of its eigenfunction expansion f = ∑_{n=0}^{∞} a_{n}e_{n} and thus R(λ, A)f = ∑_{n=0}^{∞} \frac{a_{n}}{λ−n^{2}} e_{n}. Let

k̂(λ) := \frac{1}{λ^{2}} ∏_{n=0}^{∞} \frac{n^{2} − λ}{n^{2} + λ}.

Clearly, λ^{2}k̂ is bounded and analytic in a right halfplane and therefore k̂ is the Laplace transform of a continuous function (see, for example, [3]). In order to show that k satisfies the conditions for a regularizer we have to show that 0 ∈ supp(k). We use the basic fact of Laplace transform theory that min{x : x ∈ supp(k)} = − lim sup_{λ→∞} \frac{1}{λ} ln |k̂(λ)|. See, for example, [9] Satz 14.3.1 or [1]. Since

0 ≤ − lim sup_{λ→∞} \frac{1}{λ} ln |∏_{n=0}^{∞} \frac{n^{2} − λ}{n^{2} + λ}| = lim inf_{λ→∞} \frac{1}{λ} ∑_{n=0}^{∞} ln |\frac{n^{2} + λ}{n^{2} − λ}|

≤ lim inf_{λ→∞} \frac{1}{λ} ∑_{n=0}^{∞} ln(1 + \frac{2λ}{|n^{2} − λ|})

≤ lim inf_{λ→∞} ∑_{n=0}^{∞} \frac{2}{|n^{2} − λ|} = 0,

we obtain that 0 ∈ supp(k). Since

∥k̂(λ)R(λ, A)∥ ≤ \frac{1}{|λ^{2}|} ∑_{n=0}^{∞} |\frac{1}{n^{2} + λ} ∏_{j≠n} \frac{j^{2} − λ}{j^{2} + λ}| ≤ \frac{c + \frac{1}{|λ|}}{|λ^{2}|} ,

for λ in the open right halfplane, k̂(λ)R(λ, A) is the Laplace transform of a strongly continuous operator family.

Thus we can apply Corollary 3. Let (β_{n})_{n∈IN} be the sequence of positive integers excluding the square numbers. Then

u_{n}(t) := ∑_{i=1}^{N_{n}} α_{n,i}β_{ni}R(β_{ni}, A)fe^{β_{ni}(t−2/\sqrt{n})} = ∑_{i=1}^{N_{n}} α_{n,i}β_{ni} ∑_{j=0}^{∞} \frac{a_{j}}{β_{ni} − j^{2}} e_{j}e^{β_{ni}(t−2/\sqrt{n})}

is a sequence of exponential functions that converges in the best possible topology towards the solution of the backwards heat equation

u′(t, x) = −Δu(t, x); u(0, x) = f(x).

Notice that the sequence is independent of our choice of k, which might not have been the optimal choice.

ACKNOWLEDGMENT. I would like to thank Professor F. Neubrander for many valuable suggestions, especially concerning the above example.

REFERENCES

[1] B. Bäumer: On the inversion of the Laplace and convolution transforms. Submitted. See http:/math.lsu.edu/∼tiger/evolv/evolv.html.

[2] B. Bäumer: A Vector-Valued Operational Calculus and Abstract Cauchy Problems. Dissertation, Louisiana State University, 1997. See http:/math.lsu.edu/∼tiger/evolv/evolv.html.

[3] B. Bäumer and F. Neubrander: Laplace transform methods for evolution equations, Conference del Seminario di Matematica dell’ Universitá di Bari 259 (1994), 27–60.

[4] B.Bäumer, G. Lumer, and F. Neubrander: Convolution kernels and generalized functions, in Generalized Functions, Operator Theory, and Dynamical Systems. C.R.C. Research Notes in Math., Chapman & Hall, 1998.

[5] I. Cioranescu: Local convoluted semigroups, in Evolution Equations (Baton Rouge, LA, 1992), 107–122. Lecture Notes in Pure and Appl. Math., 168, Dekker, New York, 1995.

[6] I. Cioranescu and G. Lumer: Problèmes d’évolution régularisés par un noyau général K(t). Formule de Duhamel, prolongements, théorèmes de génération. C.R. Acad. Sci. Paris 319 (1994), séries I, 1273-1278.

[7] I. Cioranescu and G. Lumer: On K(t)-convoluted semigroups, in Recent Developments in Evolution Equations, Pitman Research Notes in Math. 324, 1995, 86–93.

[8] R. Courant and D. Hilbert: Methods of Mathematical Physics, Volume I, Interscience publishers (a division of John Wiley and Sons, New York, 1953.

[9] G. Doetsch: Handbuch der Laplace Transformation. Birkhäuser Verlag, Basel-Stuttgart, 1950.

[10] Yu. L. Lyubich: The classical and local Laplace transformation in an abstract Cauchy problem, Usepi Math. Nauk 21 (1966), 3–51.

[11] G. Lumer and F. Neubrander: Asymptotic Laplace transforms and evolution equations, in Evolution Equations, Feshbach Resonances, Singular Hodge Theory, Advances in Partial Differential Equations,16, Wiley-VCH, 1999.

[12] J. C. Vignaux: Sugli integrali di Laplace asintotici, Atti Accad. naz. Lincei, Rend. Cl. Sci. fis. mat. (6) 29 (1939), 345–400.