Partial Differential Equations of Parabolic Type

By Avner Friedman

This accessible and self-contained treatment provides even readers previously unacquainted with parabolic and elliptic equations with sufficient background to understand research literature. Author Avner Friedman — Director of the Mathematical Biosciences Institute at The Ohio State University — offers a systematic and thorough approach that begins with the main facts of the general theory of second order linear parabolic equations.
Subsequent chapters explore asymptotic behavior of solutions, semi-linear equations and free boundary problems, and the extension of results concerning fundamental solutions and the Cauchy problem to systems of parabolic equations. The final chapter concerns questions of existence and uniqueness for the first boundary value problem and the differentiability of solutions, in terms of both elliptic and parabolic equations. The text concludes with an appendix on nonlinear equations and bibliographies of related works.

• Language: English
• Publisher: Dover Publications
• Release date: Aug 16, 2013
• ISBN: 9780486318264

Book preview

TYPE

CHAPTER 1

FUNDAMENTAL SOLUTIONS AND THE

CAUCHY PROBLEM

Introduction. Consider the heat equation

The function

satisfies (0.1) for every fixed (ξ, τ). Furthermore, for any continuous function f(xfor some h > 0, the integral

exists for 0 < t ≤ T if T < 1/4h and

Γ(x, t; ξ, τ) is called a fundamental solution of the heat equation. The function u(x, t) is a solution of the Cauchy problem, i.e., the problem of finding a solution to (0.1) for 0 < t ≤ T which satisfies the initial condition u(x, 0) = f(x).

The adjoint equation of the heat equation is defined as

One easily verifies that for each fixed (x, t) Γ(x, t; ξ, τ) satisfies (0.5) as a function of (ξ, τ). This fact may be used in order to prove the uniqueness of solutions of the Cauchy problem (for (0.1)) under the condition that

for some k > 0.

In this chapter we construct a fundamental solution Γ(x, t; ξ, τ) for second-order parabolic equations with Hölder continuous coefficients. It is then used to solve the Cauchy problem. Under some additional differentiability assumptions on the coefficients of the equation we construct a fundamental solution Γ*(x, t; ξ, τ) for the adjoint equation such that Γ(x, t; ξ, τ) = Γ*(x, t; ξ, τ). We use this relation in order to prove the uniqueness of the solution of the Cauchy problem under the condition (0.6).

1. Definitions

We denote by Rn the real n-dimensional euclidean space. The distance of a point x = (x1, . . . , xn) of Rn to the origin (i.e., the norm of x. A function f(x) defined on a bounded closed set S of Rn is said to be Hölder continuous of exponent α (0 < α < 1) in S if there exists a constant A such that

for all x, y in S. The smallest A for which (1.1) holds is called the Hölder coefficient. If S is an unbounded set whose intersection with every bounded closed set B is closed, then f(x) is said to be Hölder continuous of exponent α in S if for every bounded closed set B (1.1) holds in S ∩ B with some A which may depend on B. If A can be taken independently of B, then we say that f(x) is uniformly Hölder continuous (of exponent α).

If S is an open set then f(x) is said to be locally Hölder continuous (exponent α) in S if (1.1) holds in every bounded closed set B ⊂ S, with A which may depend on B. If A is independent of B then f(x) is uniformly Hölder continuous (exponent α).

If f depends on a parameter λ, i.e., f = f(x, λ), and if the Hölder coefficient is independent of λ, then we say that f(c, λ) is Hölder continuous in x, uniformly with respect to λ.

Sometimes it is convenient to define Hölder continuity with respect to different norms |x|; see, for instance, (1.4) below; also Chaps. 4, 5.

The reader may easily verify that if f1, . . . , fm are Hölder continuous (exponent α), then any rational function of f1, . . . ,fm with nonvanishing denominator is also Hölder continuous (exponent α).

If in (1.1) α = 1 then we say that f(x) is Lipschitz continuous.

Consider the differential equation

where the coefficients aij, bi, c are defined in a cylinder

is the closure of a bounded domain D ⊂ Rn We always take (aij(x, t)) to be a symmetric matrix, i.e., aij = aji. If the matrix (aij(x, t)) is positive definite, i.e., if for every real vector ξ = (ξ1, . . . , ξn) ≠ 0, Σ aij(x, t)ξiξi > 0 then we say that the operator L is of parabolic type (or that L is parabolic) at the point (x, t). If L is parabolic at all the points of Ω then we say that L such that, for any real vector ξ,

for all (x, t) ∈ Ω then we say that L is uniformly parabolic in Ω.

Throughout this chapter we shall assume:

(A1) L is parabolic in Ω;

(A2) the coefficients of L are continuous functions in Ω and, in addition, for all (x, t) ∈ Ω, (x⁰, t⁰) ∈ Ω,

Since the aij(x, t) are continuous functions in the bounded set Ω, (A1) implies that L is also uniformly parabolic in Ω, i.e., (independent of (x, t) ∈ Ω.

Definition. We say that u = u(x, t) is a solution of Lu = 0 in some region Δ if all the derivatives of u which occur in Lu (i.e., ∂u/∂xi, ∂²u/∂xi ∂xj, ∂u/∂t) are continuous functions in Δ and Lu(x, t) = 0 at each point (x, t) of Δ. A similar definition holds for any differential operator L.

Definition. A fundamental solution of Lu = 0 (in Ω) is a function Γ(x, t; ξ, τ) defined for all (x, t) ∈ Ω, (ξ, τ) ∈ Ω, t > τ, which satisfies the following conditions:

(i) for fixed (ξ, τ) ∈ Ω it satisfies, as a function of (x, t) (x ∈ D, τ < t ≤ T1) the equation Lu = 0;

(ii) for every continuous function f(x, if x ∈ D then

Here dξ = dξ1 ⋅⋅⋅ dξn. The domain D is always assumed to be Lebesgue measurable; since this is an obvious assumption, we shall usually not mention it in what follows.

Sections 2–4 are devoted to the construction of a fundamental solution. In Sec. 5 we shall study some of its properties. The results of Secs. 2–5 are extended in Sec. 6 to the case of unbounded domains D.

2. The Parametrix Method

Let (aij(x, t)) be the inverse matrix to (aij(x, t)) and set

where y = (y1, . . . , yn). From (1.3), (1.4) it follows that

where λ0, λ1, A, A.

We introduce the functions, for t > τ,

where

For each fixed (ξ, τ) the function Z(x, t; ξ, τ) satisfies the equation with constant coefficients

It also follows from Theorem 1 below that (1.7) is also satisfied for Γ = Z. Thus, Z(x, t; ξ, τ) is a fundamental solution of L0u = 0. In order to construct a fundamental solution for Lu = 0 we look upon L0 as a first approximation to L and we view Z as a principal part of the fundamental solution Γ of Lu = 0. We then try to find Γ in the form

where Φ is to be determined by the condition that Γ satisfies the equation Lu = 0.

This procedure is called the parametrix method (of E. E. Levi). Z is called the parametrix. In Chap. 9 we shall use this method also to construct fundamental solutions for systems of parabolic equations of any order.

Theorem 1. Let f(x, t) be a continuous function in Ω. Then

is a continuous function in

and

uniformly with respect to (x, t), x ∈ S, T0 < t ≤ T1 where S is any closed subset of D.

Proof. Consider first the case where f and the aij are all constants. The linear substitution ζ = P(x – ξ) reduces Σ aij(xi – ξi)(xj – ξjprovided P*P = (aij), where P* is the transpose of the matrix P. Denoting by D* the image of D by this substitution and noting that

we get

where JR is the part of the integral taken over a ball |ζ| ≤ R with radius R sufficiently small (so that the ball is contained in Dis the complementary part.

Introducing polar coordinates, we get

where

is the surface area of the unit hypersphere. It follows that

Since R tends to 0 as τ → tas τ → t. Combining this remark with (2.13) it follows, by (2.11), that J(x, t, τ) → f as τ → t.

Consider now the general case where f and the aij are not constants. Writing

we may treat the integral of J1 as J in the previous case of constant coefficients. Hence,

As for J2,

Writing the integrand of the integral of J2 in the form

> 0 there exist R and δ sufficiently small such that the expression I is bounded by

if |ξ – x| ≤ R and t – τ < δ. Dividing the integral of J2 into two parts, say I1 + I2 where for I1 the integration is taken over |ξ – x| ≤ R, and estimating I1 by using the bound (2.16) for its integrand and then using polar coordinates, as in (2.12), and substituting σ = r²/(t – τ), we find that |I1| ≤ C if t – τ < δ, where C . Now, if R is fixed and r → t then I2 → 0 since its integrand tends to zero. It follows that |J2| < Cif τ is sufficiently close to t, where C, τ. Hence

J3 can be treated in a similar way. We break it into a sum J31 + J32 where the integration in J31 is taken over a ball |ξ – x| ≤ R. Since f(x, t> 0 the integrand of J31 is bounded by

provided R and t – τ are sufficiently small. Introducing polar coordinates, as in (2.12), and substituting σ = r2/(t – τ) we find that |J31| ≤ C , C . For R fixed, J32 → 0 as τ → t. We thus obtain: J3 → 0 if τ → t. Combining this with (2.17), (2.15) and recalling (2.14), we get (2.10).

The assertion concerning the uniform convergence follows from the previous proof.

3. Volume Potentials

Given a function f(x, t) in Ω we consider the function

and call it the volume potential of f (with respect to the parametrix Z). We shall study in this section some differentiability properties of V. These will be used in the construction of a fundamental solution in the following section.

Note that the volume potential is an improper integral, the integrand having a singularity at ξ = x, τ = t. The singularity, however, is integrate. Indeed, writing wy, τ in the form

where ϑy, τ = ϑy, τ(x, ξ), we get

and the right-hand side is integrable.

Before proceeding to the actual study of V, we give a useful elementary lemma.

Lemma 1. Let f(x, y) be a continuous function of (x, y) when x, y vary in a compact domain S of Rm and x ≠ y, and let

uniformly with respect to x in S, where S(x) is the intersection of S with the ball with center x and radius . Then, for any bounded measurable function g(y) in S, the (improper) integral

is a continuous function in S.

Proof> 0 choose δ1 such that

for all x ∈ S, δ ≤ δ1. Since f(x, y) is a uniformly continuous (and hence a bounded) function of (x, y) when x ∈ S, y ∈ S, |x – y| ≥ δ1/2, there exists a δ2 < δ1/4 such that

for all x ∈ S, z ∈ S. Denoting the complement in S of S(z, δ2) by S2(z) we obtain, from (3.4),

Using the uniform continuity of f(x, y) for x ∈ S, y ∈ S, |x – y| ≥ δ2/2 we have

provided |x′ – x″| < δ for some δ sufficiently small (δ < δ2/2). Hence, together with (3.5) (for z = x′, z⁰ = x″, x = x″) we get

Combining this with (3.4) (for z = x = x′ and z = x = x″) we derive the inequality |φ(x′) – φ(x, and the proof is completed.

If we define Z(x, t; ξ, τ) = 0 for t < τ then we can apply Lemma 1 to the volume potential, and thus conclude:

Theorem 2. If f(x, t) is a bounded measurable function in Ω then the volume potential V(x, t) is a continuous function in Ω.

We next prove:

Theorem 3. If f(x, t) is a continuous function in Ω then V(x, t) has first continuous derivatives with respect to x for x ∈ D, T0 < t ≤ T1 and

Proof. Writing ∂wy, τ(x, t; ξ, τ)/∂xi analogously to (3.2) we find that

Thus, the singularity of the integrand in (3.6) is integrable. If we define ∂Z(x, t; ξ, τ)/∂xi = 0 for t < τ then we can apply Lemma 1 and thus conclude that the integral of (3.6) is a continuous function. It remains to verify (3.6). Set

Then

By Theorem 1, Sec. 2, J(x, t, τ) is a continuous function in (x, t, τ) when x ∈ D, T0 ≤ τ ≤ t ≤ T1, provided we define

Clearly, if t > τ

Using (3.7) it follows that

Consequently, the improper integral

is absolutely uniformly convergent.

Let xh = (x1, . . . , xi –1, xi + h, xi+1, . . . , xn) and consider

where x* is some point in the interval connecting xh to x. Using (> 0,

provided t is sufficiently small (independently of x< t) such that if τ , x ∈ D,

provided |h| < δ.

Writing I in the form

and using (3.15), (3.16), we get |Iif |h| < δ. Thus ∂V/∂xi , i.e., (3.6) holds.

Theorem 4. Let f(x, t) be a continuous function in Ω and locally Hölder continuous (exponent β) in x ∈ D, uniformly with respect to t. Then V(x, t) has second continuous derivatives with respect to x, for x ∈ D, T0 < t ≤ T1, and

Observe that whereas in (3.6) the order of integration is immaterial since the singularity of the integrand is absolutely integrable in each variable separately (by (3.7)), the integral in (3.17) is a repeated integral and thus only the τ-integral is taken as an improper integral. In contrast to (3.7) we now have the inequality (whose proof is similar to that of (3.7)),

so that we cannot assert that the singularity of the integrand of (3.17) is absolutely integrable in Ω.

Proof. By Theorem 3,

where J is defined by (3.8). Write ∂J/∂xi in the form

where y is a fixed point. Let K be a fixed ball contained in D and denote by ∂K the boundary of K and by K* the complement of K in D. Breaking the first integral on the right-hand side of (and applying the divergence theorem to the first integral, we obtain

where v is the outwardly directed normal to ∂K and dSη is the surface element on ∂K. Substituting (3.21) into (3.20), then differentiating (3.20) once with respect to xj and choosing y = x, we get

Let x be a fixed point lying in the interior of K. Then each of the first two integrals on the right-hand side of (3.22) is a bounded function of its variables which tends uniformly to zero if τ → t. To estimate the last integral I4 on the right-hand side of (3.22), we use (3.18) and the Hölder continuity of f. We find that

if 1 – (β/2) < μ < 1, where the constant is independent of (x, t, τ) provided x is restricted to a closed subset of D.

To evaluate the third integral I3 on the right-hand side of (3.22) we need the explicit formula

Using (2.2), (2.3), and the mean value theorem we find that

Using this inequality and employing once more (2.2), (2.3) we find from the formula (3.23) that

, from which it follows (as in the proof of (3.3), (3.7)) that

Since |C(x, τ) – C(ξ, τ)| ≤ const. |x – ξ|α, using (3.18) we conclude that the integrand of I3 is also bounded by the right-hand side of (3.25) (with a different constant). Hence |I3| ≤ const, (t – τ)–μ if 1 – (α/2) < μ < 1. Since the same bound was established for I4, and since, as has already been noted, the first two terms on the right-hand side of (3.22) are bounded functions of (t, τ), we conclude that

where δ = min (α, β).

We can now proceed as in the proof of Theorem 3; namely, we set

and prove that the integral (3.19) satisfies

i.e., (3.17) holds. The continuity of ∂²V/∂xi ∂xj follows by applying the method of proof of Lemma 1 to the right-hand side of (3.17).

Theorem 5. Let f(x, t) be as in Theorem 4. Then ∂V(x, t)/∂t exists and is continuous for x ∈ D, T0 < t ≤ T1, and

Proof. Since Z(x, t; ξ, τ) is a solution of (2.7), if t > τ

Each term on the right-hand side can be treated similarly to ∂²J/∂xi ∂xj in the proof of Theorem 4. Hence we get (compare (3.26))

We shall prove that ∂V(x, t)/∂t exists and

Taking h > 0 we consider the finite difference

where t < t* < t + h. As h → 0 the first term on the right-hand side converges to J(x, t, t). To evaluate the second term, consider

Using (> 0,

< t provided h is sufficiently small, say h , there exists a δ ≤ δ1 such that if h < δ then

for all τ .

Breaking each of the integrals in (3.32) into two integrals by

and using (3.33), (3.34), it follows that |Hif h < δ. Using this in (3.31), we obtain (3.30) for the right-hand t-derivative. The considerations for h < 0 are similar.

In view of (2.10) and the fact that J(x, t, τ) is a solution of (2.7), (3.30) reduces to (3.27). The continuity of ∂V(x, t)/∂t follows by applying the method of proof of Lemma 1.

Combining Theorems 4, 5 we obtain:

Theorem 6. Let f(x, t) be as in Theorem 4. Then V(x, t) satisfies the equation (x ∈ D, T0 < t ≤ T1)

Note that the singularity of the integrand on the right-hand side of (3.35) is absolutely and separately integrable in ξ and τ.

4. Construction of Fundamental Solutions

We shall construct a fundamental solution Γ(x, t; ξ, τ) of Lu = 0 in the form (2.8). If Φ is such that Theorem 6 applies for f(x, t) = Φ(x, t; ξ, τ) then the equation LΓ = 0 becomes

Thus, for each fixed (ξ, τ) Φ(x, t; ξ, τ) is a solution of a Volterra integral equation with a singular kernel LZ(x, t; y, σ).

From

we obtain the inequality

Hence the singularity is integrable.

We shall prove that there exists a solution Φ of (4.1) of the form

where (LZ)1 = LZ and

We shall prove the convergence of the series in (4.4). The following elementary lemma will be needed.

Lemma 2. If G is a bounded domain in Rn and 0 < α < n, 0 β < n, then for any x ∈ G, z ∈ G, x ≠ z,

The proof is obtained by breaking G into three sets corresponding to |y – z| < |x – z|/2, |y – x| < |x – z|/2 and the complementary set, and estimating the corresponding integrals separately.

Using Lemma 2 and (4.3) we get, if 2μ < 1 and 2(n + 2 – 2μ – α) < n,

Since μ < 1, 2 < 2μ + α, the singularity of (LZ)2 is weaker than that of LZ. Proceeding similarly to evaluate (LZ)3, (LZ)4, etc., we arrive at some v0 for which

We proceed to prove by induction on m that

where K0, K are some constants and Γ(t) is the gamma function. For m = 0 this follows from (4.6). Assuming now that (4.7) holds for some integer m ≥ 0 and using (4.3) we get

Substituting ρ = (σ – τ)/(t – τ) and using the formula

(4.7) follows for m + 1, provided the constant K is appropriately chosen.

From (4.7) it follows that the series expansion of Φ(x, t; ξ, τ) is convergent and that the integral in (4.1) is equal to

Hence Φ is a solution of (4.1). We also have

In order to study Φ in more detail we shall need the following lemma.

Lemma 3. If

, then

Proof. Substitute

and note that

If we proceed as in (3.2) but then write

and use the inequality σn/2–μe–eσ ≤ const. for 0 ≤ σ < ∞, then we obtain a bound from which the following inequality for Z follows:

. In a similar way it can be proved that

; in (4.10) 0 ≤ μ ≤ (n + 1)/2, in (4.11) 0 ≤ μ ≤ (n + 2)/2, and in (4.13) 0 ≤ μ ≤ (n + 2 – α)/2.

Using (4.13) with μ = (n + 2 – α)/2 and applying Lemma 3, we find that

Proceeding by induction one easily establishes, with the aid of Lemma 3, the inequalities

where H0, H are some positive constants. From the definition of Φ we then conclude that

From this inequality it also follows (compare the derivation of (4.9)) that

for 0 ≤ μ ≤ (n + 2 – αappearing in (4.13) and, consequently, it can be taken to be any number < λ0. (4.16) is an improvement of (4.8).

By using (4.14) and employing the method of proof of Lemma 1, Sec. 3, one can show that the (LZ)v(x, t; ξ, τ) are continuous functions of (x, t), uniformly with respect to (ξ, τ) if t – τ ≥ const. > 0, and are continuous functions of (ξ, τ), uniformly with respect to (x, t) if t – τ ≥ const. > 0. It follows that the (LZ)v(x, t; ξ, τ) are continuous functions of (x, t; ξ, τ). From (4.4), (4.14) one then concludes that Φ(x, t; ξ, τ) is also a continuous function of (x, t; ξ, τ).

Theorem 7. Φ(x, t; ξ, τ) is Hölder continuous in x; more precisely, for any 0 < β < α,

where γ = α – β and where λ* is a positive constant.

Proof. We first prove the inequality

where k is a positive constant. Consider first the case where

and take the term

of LZ(x, t; ξ, τ) (see (4.2)). We have

Using (4.11) with μ = (n + 2)/2 we get

Using the mean value theorem and (4.12), and noting that, because of (4.19), for any point ζ in the interval (x, y)

we get

where ki are used to denote appropriate positive constants. Hence,

Combining (4.24), (4.22) and using (4.19), we get

The estimation of the Hölder exponent and coefficient for the lower-order terms in LZ is similar to the estimation for F. Combining these estimates, the inequality (4.18) follows.

If (4.19) is not satisfied then, by (4.13) (with μ = (n + 2 – α)/2),

A similar inequality holds for LZ(y, t; ξ, τ). Since (t – τ)β/2 ≤ |x – y|β (4.18) follows.

Denoting the integral on the right-hand side of (4.1) by Ψ(x, t; ξ, τ), it remains to prove that (4.18) holds with LZ replaced by Ψ (with a possibly different k). Writing

and using (4.18), (4.15), we get

where

By Lemma 3,

A similar inequality holds for I(y, t; ξ, τ). Substituting these inequalities into (4.25) we find that (4.18) is satisfied if LZ is replaced by Ψ and if k is replaced by k4. This completes the proof of Theorem 7.

By reading carefully the proof of Theorem 7 one deduces:

Corollary. The inequality (4.17) holds for any λ* < λ0/4.

Theorem 8. The function Γ(x, t; ξ, τ), defined by (2.8), is a fundamental solution of Lu = 0 in Ω.

Proof. We first prove that for each fixed (ξ, τ) LΓ = 0. Write Γ in the form

where t0 is a fixed number satisfying τ < t0 < t. For the first integral, the first two x-derivatives of Z(x, t; η, σ) are continuous functions in (x, t; η, σ) whereas Φ(η, σ; ξ, τ) is absolutely integrable in (η, σ) (by (4.8) or (4.16)). Hence, by a standard theorem of calculus, the first two x-derivatives of the integral exist, and the order of any x-differentiation (up to the second order) and of the integration may be changed.

As for the second integral on the right-hand side of (4.26), Φ(η, σ; ξ, τ) (for (ξ, τ) fixed) is uniformly Hölder continuous in η with any exponent <α, as follows by Theorem 7. By Theorems 3, 4 of Sec. 3 it follows that the first two x-derivatives of the integral exist, and the order of any x-differentiation (up to the second order) and of the integration may be changed. We conclude that the first two x-derivatives of Γ exist, and

a similar formula holds for ∂Γ/∂xi.

The existence of ∂Γ(x, t; ξ, τ)/∂t can be established by the same considerations as for ∂Γ/∂xi, ∂²Γ/∂xi ∂xj, making use of Theorem 5, Sec. 3. The formula

is valid.

Combining (4.28), (4.27) and the analogue of (4.27) for ∂Γ/∂xi, we get

Since Φ satisfies the integral equation (4.1), LΓ = 0.

Using (2.8) and the method of proof of Lemma 1, Sec. 3, one can show that Γ(x, t; ξ, τ) is a continuous function of (x, t), uniformly with respect to (ξ, τ) if t – τ ≥ const. > 0, and it is a continuous function of (ξ, τ) uniformly with respect to (x, t) if t – τ ≥ const. Hence Γ(x, t; ξ, τ) is a continuous function of (x, t; ξ, τ). Here x and T0 ≤ τ < t ≤ T1.

Using (4.28), (4.27) and the analogue of (4.27) for ∂Γ/∂xi, one can similarly show that

are continuous functions of (x, t; ξ, τ), where x, ξ vary in D and T0 ≤ τ < t ≤ T1. This fact, however, will not be needed in the present chapter.

It remains to prove that for any continuous function f(x,

In view of Theorem 1, Sec. 2, it suffices to show that

Using (4.9) (with μ = n/2), (4.15), and Lemma 3, we get

Substituting ρ = |x – ξ|(t – τ)–1/2 we find that

from which (4.30) follows.

5. Properties of Fundamental Solutions

In Sec. 3 we have introduced the concept of volume potentials with respect to the parametrix Z(x, t; ξ, τ) and established some differentiability properties (Theorems 2–6). The purpose of the present section is to make a similar study of volume potentials with respect to the fundamental solution Γ(x, t; ξ, τ). Thus, we shall consider functions of the form

For simplicity, f .

If we substitute for Γ its expression from (2.8), then we find that

where V(x, t) is the potential (3.1) and U(x, t) can be written (after changing the order of integration) in the form

where

Defining Φ(x, t; ξ, τ) = 0 if t < τ and recalling (is also uniformly Hölder continuous in x with any exponent β < α. Indeed, using (4.17) we get

where

Substituting (for τ fixed) ρ = |x – ξ|(t – τ)–1/2 we find that

A similar inequality holds for A(y, t). Substituting these inequalities into (5.5), the uniform Hölder continuity (exponent βin x follows.

Theorem 9. If f(x, t) is a continuous function in Ω then W(x, t) is a continuous function in Ω and ∂W/∂xi are continuous functions for x ∈ D, T0 < t ≤ T1. If f(x, t) is also locally Hölder continuous in x ∈ D, uniformly with respect to t, then ∂²W/∂xi ∂xj and ∂W/∂t are continuous functions for x ∈ D, T0 < t ≤ T1, and

Proof. All the assertions of the theorem, except for (in x, by applying Theorems 2–5 of Sec. 3.

(5.6) is a consequence of Theorem 6, Sec. 3 and (4.1). Indeed,

6. Fundamental Solutions in Unbounded Domains

In this section we shall extend the results of Secs. 1–5 to the case where D is an unbounded domain in Rn. The special case D = Rn is of particular importance. Since most of the arguments are similar to those for the case where D is bounded, we shall only describe the necessary modifications.

If D is unbounded we always assume in this chapter that L satisfies the following assumptions (which coincide with (A1), (A2) of Sec. 1 if D is bounded):

(A1)′ L ;

(A2)′ the coefficients of L are bounded continuous functions in Ω and (1.4), (1.5), (1.6) hold throughout Ω.

The definition of a fundamental solution Γ(x, t; ξ, τ) is the same as in Sec. 1 except that in (ii) we require that f(x) satisfies the inequality

for some positive constant h.