Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 25, 2000, 151–160

A DECOMPOSITION THEOREM FOR SOLUTIONS OF PARABOLIC EQUATIONS

N.A. Watson

University of Canterbury, Department of Mathematics Private Bag 4800, Christchurch, New Zealand; N.Watson@math.canterbury.ac.nz

Abstract. Let L be a second order linear parabolic partial differential operator, with smooth and bounded coefficients defined on X = R^{n}×]0, a[. Let E be an open subset of X, and let K be a compact subset of E. If u is a solution of Lu = 0 on E\K, we prove that there is a unique decomposition u = v + w, where Lv = 0 on E, Lw = 0 on X\K, and w is zero both at infinity and on R^{n}×]0, k[, where k = inf{t : K ∩ (R^{n} × {t}) ≠ ∅}. A more detailed decomposition is given for the case where K ⊆ R^{n} × {d}.

1. Introduction

Let L be a second order linear uniformly parabolic partial differential operator, in divergence form and with bounded smooth coefficients, defined on the closure of X = R^{n}×]0, a[. The central theorem of this paper states that, if E is an open subset of X, K is a compact subset of E, and u is a solution of Lu = 0 on E\K, then u can be written uniquely in the form u = v + w, where v is a solution of Lv = 0 on the whole of E, w is a solution on X\K, and w vanishes at infinity and on R^{n}×]0, k[, where k = inf{t : K ∩ (R^{n} × {t}) ≠ ∅}. This is analogous to a classical result for harmonic functions [3, p. 172], and is new even for the heat equation.

To prove the decomposition theorem, we require a representation theorem for an arbitrary C^{2,1} function, in terms of the fundamental solution of Lu = 0. Such a formula in terms of the fundamental solution of Laplace’s equation is classical [7, p. 11]. The result below was proved for the heat equation by Smyrnélis [11]; furthermore Doob gave a less natural version for that case [7, p. 271].

Once the decomposition theorem is established, it permits an easy deduction of a general analogue of Bôcher’s theorem from the particular case where E is an infinite strip.

Finally, we consider in detail the case of the decomposition where K is a subset of a characteristic hyperplane. Under minimal conditions on u, we establish that w is the integral of a signed measure against the fundamental solution of Lu = 0.

A typical element of X is written p = (x, t) or q = (y, s). Therefore the element of Lebesgue measure in n + 1 dimensions is written as dq, that in n dimensions as dx or dy, and that in 1 dimension as dt or ds. Where an integral is taken over the boundary of a piecewise smooth domain, the element of surface area is written as dσ, and the outward unit normal as (ν_{x}, ν_{t}). The gradient in the spatial variables is written ∇_{x}, and the inner product in R^{n} as ⟨· , · ⟩. All measures appearing below are Radon measures.

1991 Mathematics Subject Classification: Primary 35K10; Secondary 31B35, 35C15, 35K05.

Let A = (a_{ij}) be a C^{∞}, symmetric, n × n matrix-valued function on X such that, for some λ ∈]0, 1[,

λ∥ξ∥^{2} ≤ ⟨A(x, t)ξ, ξ⟩ ≤ λ^{−1}∥ξ∥^{2}

whenever (x, t) ∈ X and ξ ∈ R^{n}. Let

(1)  Lu = ∑_{i=1}^{n} D_{i}(a_{ij}D_{j}u) − D_{t}u = 0

be the corresponding parabolic partial differential equation with divergence form, and let

L^{∗}u = ∑_{i=1}^{n} D_{i}(a_{ij}D_{j}u) + D_{t}u = 0

be its adjoint.

Under these hypotheses on A, the fundamental solution Γ exists and satisfies

Γ(x, t; y, s) ≤ (\frac{β}{4π(t − s)})^{n/2} exp(−\frac{∥x − y∥^{2}}{2α(t − s)}),
∥∇_{x}Γ(x, t; y, s)∥ ≤ β(t − s)^{−(n+1)/2} exp(−\frac{∥x − y∥^{2}}{2α(t − s)})

for all (x, t), (y, s) ∈ \overline{X} such that s < t, where α and β are positive constants. Furthermore, for each fixed (x, t),

L^{∗}Γ(x, t; · , ·) = 0

on R^{n}×]0, t[. Details are given in [9]. We adopt the convention that Γ(x, t; y, s) = 0 whenever t ≤ s.

Given (x_{0}, t_{0}) ∈ X and c > 0, the identity

(\frac{β}{4π(t_{0} − s)})^{n/2} exp(−\frac{∥x_{0} − y∥^{2}}{2α(t_{0} − s)}) = (4πc)^{−n/2}

holds if and only if

∥x_{0} − y∥^{2} = nα(t_{0} − s) log(\frac{cβ}{t_{0} − s}).

Therefore, if c < t_{0}/β, the sets

Ω(x_{0}, t_{0}; c) = {(y, s) : Γ(x_{0}, t_{0}; y, s) > (4πc)^{−n/2}}

and

Ψ(x_{0}, t_{0}; c) = {(y, s) : Γ(x_{0}, t_{0}; y, s) = (4πc)^{−n/2}}

have their closures in X. Since Γ(x_{0}, t_{0}; · , ·) ∈ C^{∞}(R^{n}×]0, t_{0}[ ), for almost every such c the set Ψ(x_{0}, t_{0}; c) is a smooth regular n-dimensional manifold, by Sard’s theorem [12, p. 45]. Fabes and Garofalo [8] have studied mean values of solutions of (1) over the sets Ω and Ψ. In particular, they have shown that, if

ℳ (u; x_{0}, t_{0}; c) = ∫_{Ψ(x_{0},t_{0};c)} u⟨A∇_{x}Γ(x_{0}, t_{0}; · , ·), ν_{x}⟩ dσ,

then u(x_{0}, t_{0}) = ℳ (u; x_{0}, t_{0}; c) whenever u is a solution of (1). In the sequel, we shall use only the form of ℳ and the fact that ℳ (1; x_{0}, t_{0}; c) = 1.

2. The decomposition theorem

To prove the decomposition theorem, we require the following representation theorem, for an arbitrary sufficiently smooth function, in terms of the fundamental solution Γ.

Theorem 1. Let E be a bounded open subset of X with a piecewise smooth boundary, and let u ∈ C^{2,1}(\overline{E}). Then

u(x_{0}, t_{0}) = − ∫_{E} Γ_{0}(Lu) dq − ∫_{∂E}(⟨A(u∇_{x}Γ_{0} − Γ_{0}∇_{x}u), ν_{x}⟩ + uΓ_{0}ν_{t}) dσ

for each (x_{0}, t_{0}) ∈ E, where Γ_{0} = Γ(x_{0}, t_{0}; · , ·).

Proof. Given (x_{0}, t_{0}) ∈ E, and any c ∈]0, t_{0}/β[ such that Ψ(x_{0}, t_{0}; c) is a smooth surface, for all γ ∈]0, cβ/e] we put

S(γ) = {(y, s) : ∥x_{0} − y∥^{2} < αnγ log(cβ/γ), t_{0} − γ < s < t_{0}}

and

Ω_{γ} = Ω(x_{0}, t_{0}; c) ∪ S(γ).

We choose c and γ such that \overline{Ω}_{γ} ⊆ E. Then, by Green’s formula for L,

(2)  ∫_{E\\overline{Ω}_{γ}} Γ_{0}(Lu) dq = ∫_{∂(E\\overline{Ω}_{γ})}(⟨A(Γ_{0}∇_{x}u − u∇_{x}Γ_{0}), ν_{x}⟩ − uΓ_{0}ν_{t}) dσ

since L^{∗}Γ_{0} = 0 on E\\overline{Ω}_{γ}.

We consider the integral on the right-hand side of (2), splitting the range of integration into five pieces. First, on ∂Ω_{γ} ∩ (R^{n} × {t_{0}}) we have Γ_{0} = 0 and ∇_{x}Γ_{0} = 0, so that this piece contributes nothing. Second, if

Λ(γ) = {(y, s) : ∥x_{0} − y∥^{2} = αnγ log(cβ/γ), t_{0} − γ < s < t_{0}}

denotes the lateral boundary of S(γ), then ν_{t} = 0 on Λ(γ), and

∫_{Λ(γ)}⟨AΓ_{0}∇_{x}u, ν_{x}⟩ dσ → 0 as γ → 0

because A∇_{x}u and Γ_{0} are bounded on S(cβ/e)\Ω(x_{0}, t_{0}; c). Furthermore, if σ_{n} denotes the surface area of the unit sphere in R^{n}, then for any r > 0 we have

∫_{∂B(0,r)×]0,γ[} t^{−(n+1)/2} exp(−\frac{∥x∥^{2}}{2αt} ) dσ = σ_{n}(2α)^{(n−1)/2} ∫_{r^{2}/2αγ}^{∞} s^{(n−3)/2}e^{−s} ds,

so that

∫_{Λ(γ)} ∥∇_{x}Γ_{0}∥ dσ ≤ κ ∫_{(n/2) log(cβ/γ)}^{∞} s^{(n−3)/2}e^{−s} ds → 0

as γ → 0, where κ = βσ_{n}(2α)^{(n−1)/2} . It follows that

∫_{Λ(γ)}⟨uA∇_{x}Γ_{0}, ν_{x}⟩ dσ → 0 as γ → 0,

so that the entire integral over Λ(γ) tends to zero. Third, if

F (γ) = ∂Ω_{γ} ∩ (R^{n} × {t_{0} − γ}),

then the measure of F (γ) tends to zero as γ → 0, and uΓ_{0} is bounded on the union over all γ ∈]0, cβ/e] of the sets F (γ), so that

∫_{F (γ)} uΓ_{0}ν_{t} dσ → 0 as γ → 0.

Fourth, let B(γ) = (R^{n}×] − ∞, t_{0} − γ[ ) ∩ ∂Ω_{γ}, so that Γ_{0} = (4πc)^{−n/2} on B(γ). Therefore, as γ → 0,

\begin{aligned}− ∫_{B(γ)}(⟨A∇_{x}u, ν_{x}⟩ − uν_{t})Γ_{0} dσ & → (4πc)^{−n/2} ∫_{∂Ω(x_{0},t_{0};c)}(⟨A∇_{x}u, ν_{x}⟩ − uν_{t}) dσ \\ & = (4πc)^{−n/2} ∫_{Ω(x_{0},t_{0};c)} Lu dq \end{aligned}

by Green’s formula. Furthermore, as γ → 0,

∫_{B(γ)} u⟨A∇_{x}Γ_{0}, ν_{x}⟩ dσ → ℳ (u; x_{0}, t_{0}; c).

Fifth, the integral over ∂E is left unchanged. It now follows from (2) that

\begin{aligned}∫_{E\\overline{Ω}(x_{0},t_{0};c)} Γ_{0}(Lu) dq & = ∫_{∂E}(⟨A(Γ_{0}∇_{x}u − u∇_{x}Γ_{0}), ν_{x}⟩ − uΓ_{0}ν_{t}) dσ \\ & − (4πc)^{−n/2} ∫_{Ω(x_{0},t_{0};c)} Lu dq − ℳ (u; x_{0}, t_{0}; c). \end{aligned}

We now make c → 0, so that ℳ (u; x_{0}, t_{0}; c) → u(x_{0}, t_{0}) because u is continuous and ℳ (1; x_{0}, t_{0}; c) = 1, and

(4πc)^{−n/2} ∫_{Ω(x_{0},t_{0};c)} Lu dq → 0

because the integrand is bounded and the measure of Ω(x_{0}, t_{0}; c) is dominated by c^{(n+2)/2} . This proves the theorem.

For our present purpose, we do not need the full generality of Theorem 1, just the following consequence.

Corollary. If u ∈ C^{2,1}(X) and has compact support in X, then

u(x_{0}, t_{0}) = − ∫_{X} Γ(x_{0}, t_{0}; · , ·)Lu dq

for each (x_{0}, t_{0}) ∈ X.

Proof. In Theorem 1, choose E to contain both (x_{0}, t_{0}) and the support of u.

We can now prove the decomposition theorem, using the method employed in [3, p. 172] to prove the corresponding result for Laplace’s equation.

Theorem 2. Let K be a compact subset of an open subset E of X, and let u satisfy Lu = 0 on E\K. Then u can be written uniquely as u = v + w, where Lv = 0 on E, Lw = 0 on X\K, and w is zero both at infinity and on R^{n}×]0, k[ for k = inf{t : K ∩ (R^{n} × {t}) ≠ ∅}.

Proof. Suppose first that E is bounded. For any set S and r > 0, we denote by S_{r} the set of all points with distance less than r from S. We choose r such that K_{r} ∩ (∂E)_{r} = ∅, and put

E(ρ) = E\(K_{ρ} ∪ (∂E)_{ρ})

for each ρ ≤ r. Let ϕ_{r} ∈ C^{∞}(X), have compact support in E\K, and be equal to 1 throughout E(r). Given (x_{0}, t_{0}) ∈ E(r), we can apply the above corollary to uϕ_{r} and obtain

−u(x_{0}, t_{0}) = v_{r}(x_{0}, t_{0}) + w_{r}(x_{0}, t_{0}),

where v_{r} and w_{r} are defined on X by

v_{r}(x, t) = ∫_{(∂E)_{r}} Γ(x, t; · , ·)L(uϕ_{r}) dq

and

w_{r}(x, t) = ∫_{K_{r}} Γ(x, t; · , ·)L(uϕ_{r}) dq.

Differentiation under the integral sign shows that Lv_{r} = 0 and Lw_{r} = 0 outside their respective ranges of integration. Furthermore, w_{r} is zero at infinity and on R^{n}×]0, k[.

If 0 < s < r, then on E(r) we have v_{r} + w_{r} = u = v_{s} + w_{s}, so that w_{r} − w_{s} is a solution of (1) on X\K_{r} that equals v_{s} − v_{r} on E(r), and can therefore be extended to a solution of (1) on X. Since w_{r} − w_{s} is zero at infinity and on R^{n}×]0, k[, it is zero everywhere. Hence w_{r} = w_{s} and v_{r} = v_{s} on E(r). Therefore, given (ξ, τ) ∈ E we can choose r such that (ξ, τ) ∈ E\(∂E)_{r}, and define v(ξ, τ) = v_{r}(ξ, τ) unambiguously. Similarly, if (ξ′, τ′) ∈ X\K we can define w(ξ′, τ′) = w_{r}(ξ′, τ′) for small r. Then u = v + w as asserted, and the uniqueness follows by similar reasoning to that used to show that w_{r} = w_{s} and v_{r} = v_{s} above.

Now suppose that E is unbounded. For any bounded open set D such that K ⊆ D ⊆ E, we have the unique decomposition u = v + w on D\K, as above. Then u − w is a solution of (1) on E\K that can be extended by v to a solution h on E. Hence u = h + w as required, and the uniqueness follows as before.

3. Some consequences of the decomposition theorem

Theorem 2 enables us to easily deduce a general analogue of Bôcher’s theorem from the particular case first considered by Krzyżański [10]. Subtler analogues were given by Aronson [1]. Isolated singularities of nonnegative solutions of the heat equation were characterized by Widder [16, p. 119], and those of arbitrary solutions by Chung and Kim [5].

Theorem 3. Let E be an open subset of X, let (y_{0}, s_{0}) ∈ E, and let u be a solution of (1) on E\{(y_{0}, s_{0})} such that u is bounded below on some cylinder B(y_{0}, r)×]s_{0}, t_{0}[. Then u can be written uniquely in the form

u = v + κΓ(· , ·; y_{0}, s_{0}),

where v is a solution of (1) on E, and κ ∈ [0, ∞[.

Proof. By Theorem 2, there is a unique decomposition u = v + w on E\{(y_{0}, s_{0})}, where Lv = 0 on E, Lw = 0 on X\{(y_{0}, s_{0})}, and w is zero both at infinity and on R^{n}×]0, s_{0}[. If B(y_{0}, r)×]s_{0}, t_{0}[ is chosen to have its closure in E, then w = u − v is bounded below on that set. Hence, if ε > 0 and

h(x, t) = w(x, t) + ε ∫_{B(y_{0},r)} Γ(x, t; y, s_{0})∥y − y_{0}∥^{−n/2} dy,

then

lim inf_{(x,t)→(z,s_{0}+)} h(x, t) ≥ 0

for all z ∈ R^{n}. Since h is a solution of (1) that vanishes at infinity, it follows from the minimum principle that h ≥ 0. Making ε → 0, we deduce that w ≥ 0. It now follows that w = κΓ(· , ·; y_{0}, s_{0}), by [4, Theorem 3].

Using more sophisticated techniques, we can improve Theorem 3 in several directions. This requires the following result on the uniqueness of parts of a representing measure. The result holds in the more general context of [14], but here we keep to the present one.

We shall use the following terminology. A family ℱ of closed balls in R^{n} is called an abundant Vitali covering of R^{n} if, given any x ∈ R^{n} and ε > 0, ℱ contains uncountably many balls centred at x with radius less than ε.

We also use the following notation. Given any open subset D of R^{n+1} such that D ∩(R^{n} ×{0}) ≠ ∅, we put D(0) = {x ∈ R^{n} : (x, 0) ∈ D} and D_{+} = D ∩X.

Theorem 4. Let u be a solution of (1) such that

u(x, t) = ∫_{R^{n}} Γ(x, t; y, 0) dμ(y) + v(x, t)

for all (x, t) ∈ D_{+} , where μ is a signed measure concentrated on D(0) and v is a solution of (1) on D_{+} with a continuous extension to 0 on D(0) × {0}. Let ℱ be an abundant Vitali covering of R^{n}. If there is a signed measure ν concentrated on D(0) such that

(3)  lim_{t→0+} ∫_{A∩V} u(x, t) dx = ν(A ∩ V)

whenever A, V ∈ ℱ, V ⊆ D(0), and A ∩ V ≠ ∅, then μ = ν.

Proof. By [14, Theorem 3(i)], there is an abundant Vitali covering ℱ_{0} ⊆ ℱ such that |μ|(∂A) = 0 for all A ∈ ℱ_{0} . Given V ∈ ℱ_{0} such that V ⊆ D(0), put

w_{V} (x, t) = ∫_{V} Γ(x, t; y, 0) dμ(y)

and

w_{D\V} (x, t) = ∫_{D(0)\V} Γ(x, t; y, 0) dμ(y)

for all (x, t) ∈ X. Then w_{V} = u − v − w_{D\V} on D_{+} .

If A ∈ ℱ_{0} and A ∩ V ≠ ∅, then A ∩ V is a compact subset of D(0), so that

(4)  lim_{t→0+} ∫_{A∩V} v(x, t) dx = 0.

Furthermore, because the boundaries of A∩V and A\V are both μ-null, it follows from [14, Theorem 1(i)] that

(5)  lim_{t→0+} ∫_{A∩V} w_{D\V} (x, t) dx = 0 = lim_{t→0+} ∫_{A\V} w_{V} (x, t) dx.

Combining (3), (4) and (5), we obtain

lim_{t→0+} ∫_{A} w_{V} (x, t) dx = lim_{t→0+} ∫_{A∩V} w_{V} (x, t) dx = ν(A ∩ V).

On the other hand, if A ∈ ℱ_{0} and A ∩ V = ∅, then it follows from [14, Theorem 1(i)] that

lim_{t→0+} ∫_{A} w_{V} (x, t) dx = 0 = ν(A ∩ V).

Therefore the restrictions of μ and ν to V are identical, by [14, Theorem 3(ii)].

Given any open subset U of D(0), choose a sequence {V_{k}} in ℱ_{0} with union U, and put W_{1} = V_{1} , W_{j} = V_{j}\ ⋃_{k=1}^{j−1} V_{k} for all j ≥ 2. Then, by the above,

μ(U) = ∑_{j=1}^{∞} μ(W_{j}) = ∑_{j=1}^{∞} ν(W_{j}) = ν(U).

The result now follows from the regularity of Radon measures.

Theorem 4 enables us to consider the uniqueness of just a part of the representing measure, because we can vary D(0) without altering D_{+} . For example, if D_{+} = X, G is any relatively open subset of R^{n}, and u has the representation

u(x, t) = ∫_{R^{n}} Γ(x, t; y, 0) dλ(y)

for all (x, t) ∈ X, we can take D = X ∪ (G × {0}) ∪ (R^{n}×] − ∞, 0[ ), μ the restriction of λ to G, and

v(x, t) = ∫_{R^{n}\G} Γ(x, t; y, 0) dλ(y).

This technique is used in the proof of our next theorem, which generalizes [15, Theorem 5], where only the heat equation was considered and the method of proof was very different. The case of lower bounded solutions of the heat equation was discovered independently by Chung [6], who used yet another approach.

Theorem 5. Let E be an open subset of X, and let K be a nonempty compact subset of E ∩ (R^{n} × {b}). If u is a solution of (1) on E\K such that

lim inf_{t→b+} ∫_{U} u^{+}(x, t) dx < ∞

for some relatively open subset U of R^{n} such that K ⊆ U × {b}, then there exist a unique solution v of (1) on E, and a unique signed measure μ supported in K(b) = {x ∈ R^{n} : (x, b) ∈ K}, such that

u(x, t) = v(x, t) + ∫_{K(b)} Γ(x, t; y, b) dμ(y)

for all (x, t) ∈ E\K.

Proof. We may assume that \overline{U} × [b, d] is a compact subset of E, for some d > b. By Theorem 2, u can be written uniquely as the sum of a solution v of (1) on E, and a solution w of (1) on X\K that is zero both at infinity and on R^{n}×]0, b[. If M = max\{|v(x, t)| : x ∈ \overline{U}, b ≤ t ≤ d\}, and m_{n} denotes n-dimensional Lebesgue measure, then

lim inf_{t→b+} ∫_{U} w^{+}(x, t) dx ≤ lim inf_{t→b+} ∫_{U} u^{+}(x, t) dx + Mm_{n}(U) < ∞.

By the maximum principle, w is bounded outside U × [b, ε] for any ε ∈]b, d[. Therefore, for any α > 0, the function

∫_{R^{n}} exp(−α∥x∥^{2})w^{+}(x, ·) dx

is bounded on ]ε, a[, and there is a number N such that

lim inf_{t→b+} ∫_{R^{n}} exp(−α∥x∥^{2})w^{+}(x, t) dx ≤ N ∫_{R^{n}\U} exp(−α∥x∥^{2}) dx

+ lim inf_{t→b+} ∫_{U} w^{+}(x, t) dx < ∞.

It now follows from [13, Theorem 13, Corollary] that there is a signed measure μ on R^{n} such that

w(x, t) = ∫_{R^{n}} Γ(x, t; y, b) dμ(y)

for all (x, t) ∈ X\K. The uniqueness of such a representation is proved in [2, p. 688]. Finally, since w is continuous and zero on (R^{n} × {b})\K, it follows from Theorem 4 (with D(0) corresponding to R^{n}\K(b)) that R^{n}\K(b) is μ-null.

160 N.A. Watson

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Received 21 April 1998