2004-Fez conference on Differential Equations and Mechanics Electronic Journal of Differential Equations, Conference 11, 2004, pp. 23–32. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

ON THE WELL-POSEDNESS OF THE HEAT EQUATION ON UNBOUNDED DOMAINS

WOLFGANG ARENDT, SOUMIA LALAOUI RHALI

Abstract. This work concerns the well-posedness of the heat equation in an unbounded open domain, under basic regularity assumptions on this domain.

1. Introduction

Let Ω be an open set of ℝ^{n} with boundary Γ = ∂Ω and consider the problem

\begin{matrix}u′(t) = Δu(t), t ∈ [0, τ] \\ u(t)|_{Γ} = φ(t), t ∈ [0, τ] \\ u(0) = u_{0}, \end{matrix}  (1.1)

where u_{0} ∈ C(\overline{Ω}), φ ∈ C([0, τ]; C(Γ)), τ > 0.

The aim of this work is to study the well-posedness of (1.1) when Ω is unbounded. The case where Ω is bounded has been studied in [2, Chapter 6], and sufficient conditions on the initial data u_{0} and the boundary condition φ are given to show that the problem (1.1) is well-posed in C(([0, τ]; C(\overline{Ω})) whenever Ω is regular (See definition 2.1).

We point out here that the regularity assumption is equivalent when Ω is bounded to that the Dirichlet problem, (1.2),

\begin{matrix}u ∈ C(\overline{Ω}) \\ Δu = 0 in D(Ω)′ \\ u|_{Γ} = ϕ, \end{matrix}  (1.2)

has for all ϕ ∈ C(Γ) a classical solution u, that means that u is a solution of (1.2) and u ∈ C^{2}(Ω). (See [5], [9] for instance). The situation is more complicated when Ω is unbounded since one must take into account the condition at infinity that the solution of (1.2) satisfies (see Theorem 2.2), and the choice of the space X ⊂ C(\overline{Ω}) in which the solution u(t) of (1.1) belongs will be imposed by this condition at infinity and then by the choice of the unbounded regular open set Ω.

2000 Mathematics Subject Classification. 35K05, 35K20, 47D06.

Key words and phrases. Unbounded domains; heat equation; Dirichlet problem;

resolvent positive operators.

\begin{matrix}c \\ ◯ \end{matrix}2004 Texas State University - San Marcos.

Published October 15, 2004.

The second author gratefully acknowledges his support by the DAAD.

In this work, the unbounded set is taken in the case n = 1 as an interval of ℝ and as the exterior of a ball of ℝ^{n} for n ≥ 2. When n = 2, we deal with the heat equation with homogeneous boundary conditions and for n ≥ 3, the heat equation with inhomogeneous boundary conditions is studied for an exterior domain.

The organization of this work is as follows: In Section 2, we recall some preliminaries results lying between the regularity property for unbounded sets and the well-posedness of the Dirichlet problem (1.2). We also recall some existence results for Cauchy problems with resolvent positive operators. We present in Section 3 our main result, the method of proof consists on reformulating (1.1) as a Cauchy problem with the Poisson operator. Section 4 is devoted to the study of the well-posedness of this Cauchy problem, we first show that the Poisson operator has a positive resolvent in X × C(Γ). Using results of Section 2, we then show the well-posedness of (1.1).

2. Preliminaries

The Dirichlet Problem. Let Ω be an open set of ℝ^{n} with boundary Γ = ∂Ω.

Definition 2.1 ([5]). (a) Let z ∈ Γ. we say that z is a regular boundary point of Ω if there exists r > 0, and w ∈ C(\overline{Ω ∩ B(z, r)}) such that

Δw ≤ 0, in D(Ω ∩ B(z, r))′
w(x) > 0, x ∈ (Ω ∩ B(z, r))\{z}
w(z) = 0 .

Then the function w is called a barrier.

(b) We say that Ω is regular if all boundary points are regular.

This regularity property is related to the Dirichlet problem (1.2) as follows.

Theorem 2.2 ([5]). Let Ω be an unbounded set, not dense in ℝ^{n}(n ≥ 2) with boundary Γ. Then the following two assertions are equivalent:

(i) For every continuous ϕ with compact support in Γ, there exists a classical solution of (1.2) satisfying the following null condition at infinity (NC) There exists h harmonic on Ω such that h ∈ C(\overline{Ω}), with h(x) > 0 for |x|

large so that lim_{|x|→+∞} \frac{u(x)}{h(x)} = 0.

(ii) All boundary points of Ω are regular.

Example 2.3 ([5]). Let n ∈ ℕ^{∗}.

(a) Case of an interval of ℝ. Let Ω_{1} =]1, ∞[, then Ω_{1} is regular and for all ϕ ∈ ℝ and all c ∈ ℝ, there exists a unique classical solution of (1.2) satisfying the condition at infinity:

lim_{x→+∞} \frac{u(x)}{x} = c.

(b) Case of the exterior of a ball of ℝ^{n}, n ≥ 2. Let Ω_{n} = ℝ^{n} \ \overline{B(0, 1)}, then Ω_{n} is regular and given u a bounded classical solution of (1.2), then c = lim_{|x|→∞} u(x) exists and

u(x) = (1 − \frac{1}{|x|^{n−2}} )c + \frac{1}{σ_{n}} ∫_{∂B} \frac{|x|^{2} − 1}{|t − x|^{n}}ϕ(t)dγ(t)

is a classical solution of (1.2), with σ_{n} being the total surface area of the unit sphere in ℝ^{n}. Conversely, the function u given by the last formula is a classical solution of (1.2). Moreover, one has: If n = 2, then for all ϕ ∈ C(∂B), there exists a unique classical solution of (1.2) satisfying the condition at infinity:

u is bounded on Ω.

This solution will have a limit at infinity which is imposed by the giving ϕ:

lim_{|x|→∞} u(x) = \frac{1}{2π} ∫_{∂B} ϕ(t)dγ(t).  (2.1)

If n ≥ 3, then for all ϕ ∈ C(∂B) and all c ∈ ℝ, there exists a unique classical solution of (1.2) satisfying the condition at infinity:

lim_{|x|→∞} u(x) = c.

Note that a solution u of (1.2) satisfying the null condition at infinity (NC) does not necessarily satisfy:

lim_{|x|→∞} u(x) = 0.

This remains true for the exterior of a compact set of ℝ^{n}, n ≥ 3.

Proposition 2.4 ([5]). Let K be a compact set of ℝ^{n}, n ≥ 3 with boundary Γ. If Ω = ℝ^{n} \K is regular, then for all ϕ ∈ C(Γ), there exists a unique classical solution of (1.2) satisfying the condition at infinity:

lim_{|x|→∞} u(x) = 0.

Cauchy Problem. Let X be a Banach space and consider the inhomogeneous Cauchy Problem:

\begin{matrix}u′(t) = Au(t) + f(t), t ∈ [0, τ] \\ u(0) = u_{0}, \end{matrix}  (2.2)

where u_{0} ∈ X and f ∈ C([0, τ]; X).

Definition 2.5. A mild solution of (ACP_{f}) is a function u ∈ C([0, τ]; X) such that ∫_{0}^{t} u(s)ds ∈ D(A) and for all t ∈ [0, τ],

u(t) = u_{0} + A ∫_{0}^{t} u(s)ds + ∫_{0}^{t} f(s)ds .

We recall now some results on resolvent positive operators and Cauchy problems, we refer to [2, Chapter 3], for more details.

Theorem 2.6 ([2]). Let A be a resolvent positive operator on a Banach lattice X, that means, there exists w ∈ ℝ such that (w, ∞) ⊂ ρ(A) and R(λ, A) ≥ 0 for all λ > w.

(i) Let u_{0} ∈ D(A), f_{0} ∈ X such that Au_{0} + f_{0} ∈ \overline{D(A)}. Let f(t) = f_{0} + ∫_{0}^{t} f′(s)ds where f′ ∈ L^{1}((0, τ); X). Then (ACP_{f}) has a unique mild solution. (ii) Let f ∈ C([0, τ]; X_{+}), u_{0} ∈ X_{+} and let u be a mild solution of (ACP_{f}). Then u(t) ≥ 0 for all t ∈ [0, τ].

Define now the Gaussian semigroup (G(t))_{t≥0} on the space C_{0}(ℝ^{n}) of all continuous functions vanishing at infinity by:

G(t)f(x) = (4πt)^{−n/2} ∫_{ℝ^{n}} f(x − y)e^{−|y|^{2}/(4t)}dy, t > 0, x ∈ ℝ^{n}, f ∈ C_{0}(ℝ^{n}).

Theorem 2.7 ([2]). The family (G(t))_{t≥0} defines a bounded holomorphic C_{0}−semigroup of angle \frac{π}{2} on C_{0}(ℝ^{n}). Its generator is the Laplacian Δ_{G} on C_{0}(ℝ^{n}) with maximal domain; i.e.,

D(Δ_{G}) = {f ∈ C_{0}(ℝ^{n}), Δf ∈ C_{0}(ℝ^{n})},
Δ_{G}f = Δf,

here one identifies C_{0}(ℝ^{n}) with a subspace of D(ℝ^{n})′.

Proposition 2.8 ([2]). Let A be the generator of a bounded C_{0}-group (U(t))_{t∈ℝ} on X. Then A^{2} generates a bounded holomorphic C_{0}-semigroup (T (t))_{t≥0} of angle \frac{π}{2} on X. Moreover, for t > 0,

T (t) = (4πt)^{−1/2} ∫_{ℝ} e^{−|y|^{2}/(4t)}U(y) dy.

3. Main result

We consider the problem (1.1) with Ω presenting the cases in Example 2.3 and Proposition 2.4. Since in the case n = 2, the condition at infinity (2.1) is imposed by the boundary function, we restrict our study of (1.1) for n = 2 to the case where φ = 0.

Theorem 3.1. Let n ∈ ℕ. Case n = 1: Let Ω_{1} =]1, +∞[ with boundary Γ_{1} = {1} and denote by (C_{∞}(\overline{Ω}_{1}); ∥.∥_{C_{∞}(\overline{Ω}_{1})}) the Banach space

C_{∞}(\overline{Ω}_{1}) := \{u ∈ C([1, +∞[), lim_{x→+∞} \frac{u(x)}{x} exists\}

with the norm ∥u∥_{C_{∞}(\overline{Ω}_{1})} = max_{x∈[1,∞[} |u(x)/x|.

Then for all u_{0} ∈ C_{∞}(\overline{Ω}_{1}) and all φ ∈ C([0, τ]) such that u_{0}(1) = φ(0), there exists a unique mild solution u ∈ C([0, τ]; C_{∞}(\overline{Ω}_{1})) of the problem

\begin{matrix}u_{t}(t, x) = u′′(t, x), t ∈ [0, τ], x ∈]1, +∞[ \\ u(t, 1) = φ(t), t ∈ [0, τ] \\ u(0, x) = u_{0}(x). \end{matrix}  (3.1)

Case n = 2: Let Ω_{2} = ℝ^{2} \ B(0, 1) with boundary Γ_{2} = ∂B and set

C_{∞}(\overline{Ω}_{2}) := \{u ∈ C(\overline{Ω}_{2}), u|_{Γ_{2}} = 0 and lim_{|x|→+∞} u(x) = 0\}

with the supremum norm∥u∥_{C_{∞}(\overline{Ω}_{2})} = max_{x∈Ω_{2}} |u(x)|. Then for all u_{0} ∈ C_{∞}(\overline{Ω}_{2}), there exists a unique mild solution u ∈ C([0, τ]; C_{∞}(\overline{Ω}_{2})) of the problem

\begin{matrix}u′(t) = Δu(t), t ∈ [0, τ] \\ u|_{Γ_{2}} = 0, \\ u(0) = u_{0}. \end{matrix}  (3.2)

Case n ≥ 3: Let Ω_{n} = ℝ^{n} \ B(0, 1) or more generally Ω_{n} = ℝ^{n} \ K with K being a compact set of ℝ^{n} with boundary Γ_{n}, and set

C_{∞}(\overline{Ω}_{n}) := \{u ∈ C(\overline{Ω}_{n}), lim_{|x|→+∞} u(x) = 0\}

with the supremum norm. If Ω_{n} is regular, then for all u_{0} ∈ C_{∞}(\overline{Ω}_{n}) and all φ ∈ C([0, τ]; C(Γ_{n})) such that u_{0}|_{Γ_{n}} = φ(0), there exists a unique mild solution u ∈ C([0, τ]; C_{∞}(\overline{Ω}_{n})) of the problem

\begin{matrix}u′(t) = Δu(t), t ∈ [0, τ] \\ u(t)|_{Γ_{n}} = φ(t), t ∈ [0, τ] \\ u(0) = u_{0}. \end{matrix}  (3.3)

Let Ω_{n}, n ≥ 1 be defined as in Theorem 3.1 and define the operator Δ_{max}^{n} on C_{∞}(\overline{Ω}_{n}) as follows

D(Δ_{max}^{n}) = \{u ∈ C_{∞}(\overline{Ω}_{n}), Δu ∈ C_{∞}(\overline{Ω}_{n})\}
Δ_{max}^{n}u = Δu in D(Ω_{n})′.

We mean by mild solution of (3.3) a function u ∈ C([0, τ]; C_{∞}(\overline{Ω}_{n})) such that ∫_{0}^{t} u(s)ds ∈ D(Δ_{max}^{n}) and for all t ∈ [0, τ],

u(t) = u_{0} + Δ ∫_{0}^{t} u(s)ds in D(Ω_{n})′
u(t)|_{Γ_{n}} = φ(t).

To prove Theorem 3.1, we will reformulate the problem (3.3) as an inhomogeneous Cauchy problem with resolvent positive operator.

4. Inhomogeneous Cauchy Problem

Define for n ≥ 1 the Poisson operators A_{n} with domain D(A_{n}) = D(Δ_{max}^{n})×{0} by

A_{1}(u, 0) = (Δu, −u(1)),
A_{2}(u, 0) = (Δu, 0),
A_{n}(u, 0) = (Δu, −u|_{Γ_{n}}), n ≥ 3,

and consider the Cauchy problem

\begin{matrix}U′(t) = A_{n}U(t) + Φ_{n}(t), t ∈ [0, τ] \\ U(0) = U_{0}, \end{matrix}  (4.1)

where U_{0} = (u_{0}, 0), u_{0} ∈ C_{∞}(\overline{Ω}_{n}) is the initial data , Φ_{2} = (0, 0) and for n ≠ 2, Φ_{n}(t) = (0, φ(t)), φ ∈ C([0, τ]; C(Γ_{n})) is the boundary condition.

Proposition 4.1. Let n ≥ 1 and U ∈ C([0, τ]; C_{∞}(\overline{Ω}_{n}) × C(Γ_{n})). Then U is a mild solution of (4.1) if and only if U(t) = (u(t), 0) where u ∈ C([0, τ]; C_{∞}(\overline{Ω}_{n})) is the mild solution of (1.1).

The proof is immediate from the definition of A_{n} and the fact that \overline{D(A_{n})} = C_{∞}(\overline{Ω}_{n}) × \{0\}.

To show the well-posedness of (4.1), we first prove that A_{n} is a resolvent positive operator.

Theorem 4.2. Let λ > 0, if n ≠ 2 then for all (f, ϕ) ∈ C_{∞}(\overline{Ω}_{n}) × C(Γ_{n}) there exists a unique function u ∈ D(Δ_{max}^{n}) such that

\begin{matrix}(λ − Δ)u = f in D(Ω_{n})′ \\ u|_{Γ_{n}} = ϕ. \end{matrix}  (4.2)

Moreover, if f ≤ 0, ϕ ≤ 0, then u ≤ 0. If n = 2, then for all f ∈ C_{∞}(\overline{Ω}_{2}), there exists a unique function u ∈ D(Δ_{max}^{2}) such that

\begin{matrix}(λ − Δ)u = f in D(Ω_{2})′ \\ u|_{Γ_{2}} = 0. \end{matrix}  (4.3)

Moreover, if f ≤ 0, then u ≤ 0.

Proof. (1) Existence. (a) Case n = 1 : Set C_{∞}(ℝ) = \{f ∈ C(ℝ), lim_{x→−∞} f(x) = 0 and lim_{x→+∞} \frac{f(x)}{x} exists\} and define on C_{∞}(ℝ) the translation group

T (t)f(x) = f(x − t), t ∈ ℝ, x ∈ ℝ.

Then (T (t))_{t∈ℝ} is a C_{0}−group with generator A_{T} defined by

D(A_{T}) = {f ∈ C_{∞}(ℝ), f′ ∈ C_{∞}(ℝ)}
A_{T} f = f′.

It follows from Proposition 2.8 that A_{T}^{2} generates a C_{0}−semigroup (G(t))_{t≥0} which is the Gaussian semigroup:

G(t)f(x) = (4πt)^{−1/2} ∫_{ℝ} e^{−|y|^{2}/(4t)}f(x − y)dy.

Moreover, G(t)C_{∞}(\overline{Ω}_{1}) ⊂ C_{∞}(\overline{Ω}_{1}) for all t ≥ 0. Let λ > 0 and (f, ϕ) ∈ C_{∞}(\overline{Ω}_{1})×ℝ, and take

v_{0}(x) = ∫_{0}^{+∞} e^{−λt}G(t)f(x)dt,
v(x) = (ϕ − v_{0}(1))e^{−\sqrt{λ}(x−1)},

Then u = v + v_{0} is a solution of (4.2).

(b) Case n ≥ 2: Let f ∈ C_{∞}(\overline{Ω}_{n}). Then f can be extended to C_{0}(ℝ^{n}). Since the Gaussian semigroup generates an holomorphic C_{0}−semigroup on C_{0}(ℝ^{n}), we get that

v_{0}(x) = ∫_{0}^{+∞} e^{−λt}G(t)f(x)dt,

is a solution of

(λ − Δ)v = f for all λ > 0.  (4.4)

Moreover, if f ∈ C_{∞}(\overline{Ω}_{n}), then v_{0} ∈ C_{∞}(\overline{Ω}_{n}). If n = 2, then v_{0} is a solution of (λ − A_{2})(u_{0}, 0) = (f, 0). If n ≥ 3, it remains to show that there exists a solution of

\begin{matrix}(λ − Δ)v = 0, in D(Ω_{n})′ \\ v|_{Γ_{n}} = ϕ − v_{0}|_{Γ_{n}} =: ψ. \end{matrix}  (4.5)

Let Ω_{nk} = Ω_{n} ∩ B(0, R_{k}) where (R_{k})_{k≥1} is an increasing sequence of positif reals such that R_{k} → ∞ as k → ∞ and consider the following problem on C(\overline{Ω}_{nk}).

(λ − Δ)v_{k} = 0 in D(Ω_{nk})′
v_{k}|_{Γ_{k}} = 0 on Γ_{k} = ∂B(0, R_{k})
v_{k}|_{Γ_{n}} = ψ.

(4.6)

Since Ω_{n} is regular, Ω_{nk} is regular and it follows from [10], [13] that (4.6) has a solutionv_{k} ∈ C(\overline{Ω}_{nk}). Our aim now is to show that the sequence (v_{k})_{k≥1} converges to the solution of (4.5), for that, we use the following maximum principle due to [2].

Theorem 4.3 (Maximum Principle for distributional solutions). Let Ω_{0} be a bounded open set of ℝ^{n} with boundary Γ. Let M ≥ 0, λ ≥ 0, u ∈ C(\overline{Ω}_{0}) such that

(i) λu − Δu ≤ 0, in D(Ω_{0})′ (ii) u|_{Γ} ≤ M, Then u ≤ M on \overline{Ω}_{0}.

Without loss of generality, we can assume that ψ ≥ 0. Claim 1: (v_{k})_{k≥1} is an increasing bounded sequence. Indeed, by applying the Maximum principle in Ω_{nk} to v_{k} and v_{k} − v_{k+1} respectively, we obtain:

0 ≤ v_{k} ≤ ∥ψ∥,

and

(λ − Δ)(v_{k} − v_{k+1}) = 0, in D(Ω_{nk})′
(v_{k} − v_{k+1})|_{Γ_{k}} = −v_{k+1} ≤ 0,
(v_{k} − v_{k+1})|_{Γ_{n}} = 0.

Hence v_{k} ≤ v_{k+1} in Ω_{nk}. Claim 2: Let v = lim_{k→∞} v_{k}, then v ∈ C_{∞}(\overline{Ω}_{n}). Indeed, denote by w_{k} the solution of the problem

Δw_{k} = 0, in D(Ω_{nk})′
w_{k}|_{Γ_{k}} = 0,
w_{k}|_{Γ_{n}} = ψ.

Then w_{k} ≥ 0. Define the Poisson operator B_{k} on C(\overline{Ω}_{nk}) × C(Γ_{n} ∪ Γ_{k}) by

D(B_{k}) = \{w ∈ C(\overline{Ω}_{nk}), Δw ∈ C(\overline{Ω}_{nk})\} × \{0\},
B_{k}(w, 0) = (Δw, −(w|_{Γ_{n}}, w|_{Γ_{k}})).

Since Ω_{nk} is regular, we deduce from [2, Chapter 6], that B_{k} is a resolvent positive operator and then

(w_{k}, 0) = R(λ, B_{k})(λw_{k}, (ψ, 0)) ≥ R(λ, B_{k})(0, (ψ, 0)) = (v_{k}, 0).  (4.7)

On the other hand, it follows from Proposition 2.4 that for all Φ ∈ C(Γ_{n}), the Dirichlet problem (1.2)(with ϕ = Φ) has a unique solution w satisfying the condition at infinity lim_{|x|→∞} w(x) = 0. Moreover, if Φ ≤ 0, then w ≤ 0. Indeed, let ε > 0, since w ∈ C^{1}(Ω_{n}), there exists Ω_{0} ⊂⊂ Ω_{n} such that supp(w − ε)^{+} ⊂ Ω_{0}, Thus (w − ε)^{+} ∈ H_{0}^{1}(Ω_{0}) and ∫_{{w>ε}} |∇w|^{2} = 0. Hence,

w ≤ ε.

Denote by w_{0} the solution of (1.2) (with ϕ = ψ) vanishing to zero at infinity, then

Δ(w_{k} − w_{0}) = 0, in D(Ω_{nk})′
(w_{k} − w_{0})|_{Γ_{k}} = −w_{0|Γ_{k}} ≤ 0,
(w_{k} − w_{0})|_{Γ_{n}} = 0.

Theorem 4.3 and (4.7) imply that 0 ≤ v_{k} ≤ w_{k} ≤ w_{0}. Hence

lim_{|x|→∞} v(x) = lim_{|x|→∞} w_{0}(x) = 0.

Finally, u = v_{0} + v is a solution of (4.2). (2)Positivity and Uniqueness. Let (f, ϕ) ∈ C_{∞}(\overline{Ω}_{n}) × C(Γ_{n}) such that f ≤ 0, ϕ ≤ 0 and u a solution of (4.2). Case n = 1: Since in that case u ∈ C^{2}(Ω_{1}) ∩ C(\overline{Ω}_{1}), we apply the Phragmèn-Lindelöf principle to deduce that u ≤ 0 whenever f ≤ 0, ϕ ≤ 0. (See [12, Chapter 2]). By applying this maximum principle to u and −u respectively when f = 0, we get uniqueness. Case n ≥ 2: Since u ∈ D(Δ_{max}^{n}), we get u ∈ C^{1}(Ω_{n}). Let Ω_{0} ⊂⊂ Ω_{n} such that supp(u − ε)^{+} ⊂ Ω_{0}, ε > 0. then (u − ε)^{+} ∈ H_{0}^{1}(Ω_{0}) and

∫ f(u − ε)^{+} = λ ∫ u(u − ε)^{+} + ∫ ∇u∇(u − ε)^{+}
= λ ∫ (u − ε)(u − ε)^{+} + ελ ∫ (u − ε)^{+} + ∫_{{u>ε}} |∇u|^{2}
≤ 0.

Hence u ≤ ε. □

We are now in position to show the well-posedness of the Cauchy problem (4.1). If n ≠ 2, let φ ∈ W^{1,1}((0, τ); C(Γ_{n})) and U_{0} = (u_{0}, 0) ∈ D(A_{n}) = D(Δ_{max}^{n}) × {0}, then

A_{n}U_{0} + Φ_{n}(0) = (Δu_{0}, −u_{0|Γ_{n}} + φ(0)).

Hence A_{n}U_{0} + Φ_{n}(0) ∈ \overline{D(A_{n})} = C_{∞}(\overline{Ω}_{n}) × \{0\} if and only if

u_{0}|_{Γ_{n}} = φ(0).  (4.8)

Assumption (4.8) becomes trivial in the case n = 2 since we have assumed φ = 0. On the other hand, it follows from Theorem 4.2 that A_{n} is a resolvent positive operator. Hence, by applying Theorems 2.6 we obtain the following result.

Proposition 4.4. Let n ∈ ℕ. Case n = 1: Let Ω_{1} =]1, +∞[. Then for all u_{0} ∈ D(Δ_{max}^{1}) and all φ ∈ W^{1,1}((0, τ)) such that u_{0}(1) = φ(0), there exists a unique mild solution of (4.1) with n = 1. Case n = 2: Let Ω_{2} = ℝ^{2} \ B(0, 1). Then for all u_{0} ∈ D(Δ_{max}^{2}), there exists a unique mild solution of (4.1) with n = 2. Case n ≥ 3: Let Ω_{n} = ℝ^{n} \ K with boundary Γ_{n}, K being a compact set of ℝ^{n}. If Ω_{n} is regular, then for all u_{0} ∈ D(Δ_{max}^{n}) and all φ ∈ W^{1,1}((0, τ); C(Γ_{n})) such that u_{0}|_{Γ_{n}} = φ(0), there exists a unique mild solution of (4.1).

The proof of Theorem 3.1 will be complete by combining Theorem 4.1 and the following result.

Proposition 4.5. (i) Let u_{0} ∈ C_{∞}(\overline{Ω}_{1}) and φ ∈ C([0, τ]) such that u_{0}(1) = φ(0), then there exists a unique mild solution of (4.1) with n = 1. (ii) Let u_{0} ∈ C_{∞}(\overline{Ω}_{2}), then there exists a unique mild solution of (4.1) with n = 2. (iii) Assume that Ω_{n} = ℝ^{n}\K is regular and let u_{0} ∈ C_{∞}(\overline{Ω}_{n}) and φ ∈ C([0, τ]; C(Γ_{n})) such that u_{0}|_{Γn} = φ(0), then there exists a unique mild solution of (4.1).

Proof. Choose u_{0_{k}} ∈ D(Δ_{max}^{n}) such that u_{0_{k}} → u_{0} as k → ∞ in C_{∞}(\overline{Ω}_{n}). Choose φ_{k} ∈ W^{1,1}((0, τ); C(Γ_{n})) such that φ_{k}(0) = u_{0_{k}} |_{Γ_{n}} and φ_{k} → φas k → ∞ in C([0, τ]; C(Γ_{n})). By applying Proposition 4.4 and Theorem 4.1, we deduce that there exists a unique mild solution u_{k} ∈ C_{∞}(\overline{Ω}_{n}) of P_{τ} (u_{0_{k}}, φ_{k}). We can show that

∥u_{k}∥_{C([0,τ];C_{∞}(\overline{Ω}_{n}))} ≤ max\{∥φ_{k}∥_{C([0,τ];C(Γ_{n}))}, ∥u_{0k}∥_{C_{∞}(\overline{Ω}_{n})}\}.

where

∥φ_{k}∥_{C([0,τ];C(Γ_{n}))} = sup_{0≤t≤τ} ∥φ_{k}(t)∥_{C(Γ_{n})}

∥u_{k}∥_{E_{∞}(\overline{Ω}_{n})} = sup_{0≤t≤τ} ∥u_{k}(t)∥_{E_{∞}(\overline{Ω}_{n})} .

Hence (u_{k})_{k≥1} is a Cauchy sequence in C([0, τ]; C_{∞}(\overline{Ω}_{n})). Let u = lim_{k→∞} u_{k}, then ∫_{0}^{t} u(s)ds = lim_{k→∞} ∫_{0}^{t} u_{k}(s)ds ∈ D(Δ_{max}^{n}) and

u(t) = u_{0} + Δ ∫_{0}^{t} u(s)ds in D(Ω_{n})′

u(t)|_{Γ_{n}} = lim_{k→∞} φ_{k}(t) = φ(t).

for all t ∈ [0, τ]. □

References

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[12] M.H. Protter, H.F. Weinberger: Maximum Principle in Differential Equations. Prentice-Hall Partial Differential Equations Series. 1967.

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[14] H.R. Thieme: Remarks on resolvent positive operators and their perturbation. Discrete and Continous Dynamical Systems 4 (1998) 73-90.

Wolfgang Arendt Universität Ulm, Angewandte Analysis, D-89069 Ulm, Germany

   E-mail address: arendt@mathematik.uni-ulm.de

Soumia Lalaoui Rhali Faculté Polydisciplinaire de Taza, Université Sidi Mohamed Ben Abdellah, B.P: 1223 Taza, Morocco

   E-mail address: slalaoui@ucam.ac.ma