Advances in Differential Equations Volume 4, Number 5, September 1999, 755–776.

FUJITA TYPE RESULTS FOR A CLASS OF DEGENERATE PARABOLIC OPERATORS

Andrea Pascucci^{∗}

Dipartimento di Matematica, Università di Bologna Piazza di Porta S. Donato 5, 40127 Bologna, Italy

(Submitted by: Emmanuele DiBenedetto)

Abstract. In this paper we study the global existence of non-negative solutions to the Cauchy problem for Lu = −u^{p} where L belongs to a class L of hypoelliptic operators of degenerate parabolic type and p > 1. Extending some old results by Fujita, we prove the existence and we determine explicitly a critical exponent p^{∗} for the problem. Namely, we prove that if p > p^{∗} then there are global positive solutions to the problem, while if 1 < p < p^{∗} then all non-trivial solutions blow up in finite time. We also study the critical case p = p^{∗} for a remarkable subclass of L.

1. Introduction. In the classical paper [3], Fujita considered the initial value problem

\begin{cases}Lu = −u^{p} & in S_{T} ≡ ℝ^{N}×]0, T [, \\ u(x, 0) = a(x) & x ∈ ℝ^{N}, \end{cases}  (1.1)

where L = △ − ∂_{t} is the heat operator and p > 1. He proved that, if p < 1 + \frac{2}{N} ≡ p^{∗}, then no positive global solutions to (1.1) exist. He also showed that, if p > p^{∗}, then positive global solutions exist for every suitably small initial data a. The critical case p = p^{∗} was first studied by Hayakawa [4] and by Kobayashi, Siaro and Tanaka [8]. It was proved that if p = p^{∗} then u ≡ 0 is the only global solution to (1.1).

756 Andrea Pascucci

Fujita’s results have received relevant interest and they have been extended in several directions. For instance, the case of a domain, bounded or unbounded, replacing ℝ^{N} in (1.1), has been considered. Furthermore, different equations have been studied, involving more general reaction terms instead of u^{p} or non-parabolic operators instead of the heat operator. We refer to the valuable review article [12] where a wide survey of the related literature is presented. We also mention the recent monograph [17].

∗Research supported by University of Bologna, Funds for selected research topics.

Received for publication June 1998.

AMS Subject Classifications: 35K57, 35K65, 35K70.

It is remarkable that in many different settings it turns out that there exists a critical exponent p^{∗} which depends on the geometry of the domain as well as on the operator itself. The existence of a critical exponent was proved by Meier [14] in the case of a parabolic operator of the form

L = ∑_{i,j=1}^{N} a_{ij}∂_{x_{i}}∂_{x_{j}} + ∑_{i=1}^{N} b_{i}∂_{x_{i}} − ∂_{t},

where the principal part of L is supposed uniformly elliptic and the coefficients uniformly bounded. While a critical exponent exists, yet it is not easy to find p^{∗} explicitly. Moreover, Meier’s results make no claim about the critical case p = p^{∗}.

In this paper, we study the existence of global non-negative solutions to problem (1.1) where L is a second order differential operator in ℝ^{N+1} of the form

L = div(A∇) + ⟨x, B∇⟩ − ∂_{t}.  (1.2)

Here A = (a_{ij}) and B = (b_{ij}) are N × N matrices with constant real entries, z = (x, t) is the point in ℝ^{N+1} and ∇ = (∂_{x_{1}}, …, ∂_{x_{N}}), div and ⟨, ⟩ denote, respectively, the gradient, the divergence and the inner product in ℝ^{N}. We suppose A symmetric and positive semidefinite. We assume that the initial value belongs to the class A defined by

A = {φ ∈ C^{1}(ℝ^{N}) ∩ L^{∞}(ℝ^{N}) | φ ≥ 0, ∇φ ∈ L^{∞}(ℝ^{N})}.

Definition 1.1. We denote by B(T) the family of non-negative, continuous and bounded functions in S_{T}. A global solution to (1.1) is a classical solution to (1.1) in B(T) for every T > 0.

Definition 1.2. A critical exponent for the problem (1.1) is a number p^{∗}, 1 ≤ p^{∗} ≤ ∞, such that the following conditions hold

(i) if p > p^{∗}, then there exist non trivial global solutions to (1.1); (ii) if 1 < p < p^{∗} then the only global solution to (1.1) is u ≡ 0.

Fujita type results 757

We assume the following crucial hypothesis:

(H) L is hypoelliptic.

As it is well-known, a differential operator L on ℝ^{m} is said to be hypoelliptic if for any open subset Ω of ℝ^{m} and distributions φ, ψ on Ω satisfying Lφ = ψ, ψ ∈ C^{∞}(Ω) implies φ ∈ C^{∞}(Ω). We denote by L the class of operators (1.2) satisfying (H). Section 2 is devoted to a complete review of basic facts about L. We emphasize that, as we shall see in the sequel, the class L contains strongly degenerate operators like

L = ∂_{x_{1}}^{2} + x_{1}∂_{x_{2}} + … + x_{N−1}∂_{x_{N}} − ∂_{t}.  (1.3)

The main consequence of (H) is the existence of a fundamental solution Γ for L with deep analogies with the Gauss-Weierstrass kernel for the heat operator. In [6] and [9], a fundamental solution for L has been explicitly constructed. This pleasant property of L allows us to generalize and improve in the present degenerate setting previous results established for classical parabolic equations. Indeed, in this paper, we prove the existence of a critical exponent p^{∗} for the problem (1.1) and we give an explicit formula for p^{∗}. Moreover, we study the critical case p = p^{∗} for a remarkable subclass of L. Our main results are stated in Theorem 1.3 below.

Let us introduce some notations:

E(t) = exp(−tB^{T}), C(t) = ∫_{0}^{t} E(s)AE^{T} (s)ds,  t ∈ ℝ,  (1.4)

and

η(t) = \frac{e^{−t tr(B)}}{\sqrt{det C(t)}},  t > 0.  (1.5)

We shall see in Section 2 that C(t) is positive definite (C(t) > 0) for every t > 0. Then, a fundamental solution for L is given by

Γ(x, t; y, s) = Γ(x − E(t − s)y, t − s))

where Γ(x, t) = 0 if t ≤ 0 and

Γ(x, t) = (4π)^{− \frac{N}{2}} η(t) exp(−\frac{1}{4}⟨C^{−1}(t)x, x⟩),  if t > 0.  (1.6)

If L is the heat operator then C(t) = tI_{N} and η(t) = t^{− \frac{N}{2}} , where I_{N} denotes the N × N identity matrix.

Before stating our main results, we recall that L contains a remarkable subclass L_{0} of operators which are, roughly speaking, invariant with respect to a dilations group in ℝ^{N+1}. The precise definition of L_{0} is given in Section 2. In the following statement, Q_{(L)} + 2 denotes the homogeneous dimension of ℝ^{N+1} with respect to L ∈ L_{0} (see (2.9)). The following theorem will be proved.

Theorem 1.3. A critical exponent for the problem (1.1) exists and it is given by

p^{∗} = \begin{cases}+∞ & if lim sup_{t→+∞} η(t) > 0, \\ inf\{ p > 1 | lim sup_{t→+∞} t η^{p−1}(t) < ∞\} & otherwise, \end{cases}

where η(t) is defined in (1.5). In particular

(i) if tr(B) > 0, then p^{∗} = 1 (i.e., for every p > 1, (1.1) has global solutions corresponding to suitably small initial data);

(ii) if L ∈ L_{0}, then p^{∗} = 1 + \frac{2}{Q_{(L)}} .

Moreover, if L ∈ L_{0} and p = 1 + \frac{2}{Q_{(L)}} , then no global solution exists for any non-trivial initial data.

We remark that the non-negativeness of solutions to (1.1) is a crucial hypothesis. Indeed it is proved in [5] that, replacing in (1.1) the non-linear term u^{p} by |u|^{p−1}u, then there are global solutions (of mixed sign) even if 1 < p ≤ p^{∗}.

Local solutions to (1.1) always exist. Indeed, we prove the following

Proposition 1.4. For every a ∈ A there exists T_{0} > 0 such that (1.1) has a unique classical solution u in B(T_{0}).

The proof of Proposition 1.4 is based on a standard fixed point method. We close this section by giving some motivation for the present study.

Problem (1.1) describes the combustion process with convection in a possibly anisotropic stationary medium. The function u represents the temperature, u^{p} is the non-linear heat source, ⟨x, B∇⟩ is the convection term and a_{ij} are the conductivity coefficients. Problem (1.1) has been recently studied in [1] in the case of A = I_{N} positive definite and with bounded convection term. We also quote the recent paper [19]. We wish to emphasize that the fact that the first order part of L ∈ L is unbounded strongly affects the value of the critical exponent even in the case A = I_{N}. In order to clarify this aspect, we consider the following simple example.

Example 1.5. In ℝ × ℝ, we consider the operator in L

L = ∂_{x}^{2} − x∂_{x} − ∂_{t}.

In this case, we have tr(B) = −1 and

E(t) = e^{t}, det C(t) = \frac{e^{2t} − 1}{2}.

From Theorem 1.3, it follows that p^{∗} = +∞. On the other hand, the critical exponent related to the operator

L = ∂_{x}^{2} + x∂_{x} − ∂_{t}

is p^{∗} = 1, since in this case tr(B) > 0. Namely, there exist global non-trivial solutions to (1.1) for every p > 1.

It is interesting to remark that operators in L also arise in the study of the classical semilinear problem for the heat equation through a suitable change of variables. About this kind of questions, we refer to a paper by Kavian [7].

From a merely mathematical point of view, besides the comparison with the classical parabolic case, the interest in Theorem 1.3 relies in giving explicitly the value of p^{∗}. Moreover, Theorem 1.3 discloses the strict analogy between the classical case and the case L ∈ L_{0}. We hope to come back, in a future study, to the analysis of the critical case for a general operator L ∈ L.

The class L contains evolution operators of ultraparabolic type which arise in probabilistic theory of diffusion process. For instance, the operator considered in Example 5.2 is the linearized prototype of the Kolmogorov operator which, under suitable conditions, describes the probability density of a physical system with 2n degree of freedom (see [18], p.167).

The paper is organized as follows. In Section 2 we recall the main properties of the class L and introduce some notations that will be used throughout the paper. In Section 3 we prove a representation formula for solutions to (1.1), Proposition 1.4 and some other preliminary results. Section 4 is devoted to the proof of Theorem 1.3. In Section 5 we examine several examples. Acknowledgments. Some of the results of this paper were announced in [15]. I wish to thank Professor E. Lanconelli for several helpful conversations.

2. The class L. In this section we review some known results about the class L. Given L ∈ L as in (1.2), we denote L(X_{1}, …, X_{N}, Y) the Lie algebra generated by the first order differential operators

X_{i} = ∑_{j=1}^{N} a_{ij}∂_{x_{j}}, i = 1, …, N  and  Y = ⟨x, B∇⟩ − ∂_{t}. (2.1)

It is well-known (see [6]) that the following Hörmander’s condition

rank L(X_{1}, …, X_{N}, Y)(x) = N + 1,  ∀x ∈ ℝ^{N},  (2.2)

is equivalent to the hypoellipticity of L. Condition (2.2) means that the commutators of the operators in (2.1) up to a certain order suffice to span ℝ^{N+1} at each point.

It is not difficult to prove (see, for example, the Appendix of [11]) that Hörmander’s condition is equivalent to each of the following ones:

C(t) > 0,  ∀t > 0;  (2.3)
C(t) < 0,  ∀t < 0;  (2.4)
AB^{k}x = 0, ∀k ∈ ℕ ∪ {0} ⟹ x = 0.  (2.5)

The last condition is well-known in control theory (see, for example, [21]) and it is usually mentioned as Kalman rank condition.

It is proved in [11] that Hörmander’s condition implies that, for some basis on ℝ^{N}, the matrices A and B take the following canonical form:

A = [\begin{matrix}A_{0} & 0 \\ 0 & 0 \end{matrix}] ,  (2.6)

B = [\begin{matrix}∗ & B_{1} & 0 & ⋯ & 0 \\ ∗ & ∗ & B_{2} & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ ∗ & ∗ & ∗ & ⋯ & B_{r} \\ ∗ & ∗ & ∗ & ⋯ & ∗ \end{matrix}] ,  (2.7)

where A_{0} is a square non singular matrix with rank q_{0}, whereas B_{j} is a q_{j−1} × q_{j} block with rank q_{j}, j = 1, …, r. Besides q_{0} ≥ q_{1} ≥ ⋯ ≥ q_{r} ≥ 1 and q_{0}+q_{1}+⋯+q_{r} = N. In general, ∗ blocks are arbitrary. We denote by L_{0} the family of the operators L ∈ L such that all the ∗ blocks in the canonical representation (2.7) of the matrix B are zero matrices. By Proposition 2.2 in [11], L ∈ L_{0} if and only if L is invariant with respect to a dilations group (D(λ))_{λ>0} in ℝ^{N+1} in the sense that

L ∘ D(λ) = λ^{2}(D(λ) ∘ L)  ∀λ > 0.  (2.8)

In this case the dilations are the following ones

D(λ) = diag(λI_{q_{0}}, λ^{3}I_{q_{1}}, ..., λ^{2r+1}I_{q_{r}}, λ^{2}), λ > 0,

where I_{q_{j}} denotes the q_{j} × q_{j} identity matrix. It seems natural to define the homogeneous dimension of ℝ^{N+1} with respect to L the number

Q_{(L)} + 2 = q_{0} + 3q_{1} + ⋯ + (2r + 1)q_{r} + 2.  (2.9)

Every operator L ∈ L corresponds naturally to an operator in L_{0}. Indeed, in [11], the authors define the principal part of L as the operator

L_{0} = div(A∇) + ⟨x, B_{0}∇⟩ − ∂_{t},  (2.10)

where B_{0} is obtained by annihilating every ∗ block of the matrix B of L. Consequently we define E_{0}, C_{0} and D_{0}(λ), λ > 0. The following formula, proved by Kupcov in [9], will be useful in the sequel:

C_{0}(t) = D_{0}(\sqrt{t})C_{0}(1)D_{0}(\sqrt{t}),  ∀t > 0.  (2.11)

From (2.11) we obtain

det C_{0}(t) = det C_{0}(1) t^{Q_{(L_{0})}}.  (2.12)

Next, we shall make use of the following identity proved in [11]:

D_{0}(λ)E_{0}(t)D_{0}(λ^{−1}) = E_{0}(λ^{2}t),  λ, t > 0.  (2.13)

Moreover, the following asymptotic estimates hold (see (3.10) and (3.14) in [11]):

⟨C^{−1}(t)x, x⟩ = ⟨C_{0}^{−1}(t)x, x⟩(1 + O(t))  as t ⟶ 0, ∀x ∈ ℝ^{N}; (2.14)

det C(t) = det C_{0}(t)(1 + O(t))  as t ⟶ 0.  (2.15)

We remark that, from the definition of C(t), it follows immediately that the quadratic form related to C(t) is an increasing function of the variable t.

For reader’s convenience, we also recall some further results that can be proved as in [10]:

a) for every φ ∈ C_{0}(ℝ^{N})

lim_{t→s^{+}} ∫_{ℝ^{N}} Γ(x, t; y, s)φ(y)dy = φ(x), x ∈ ℝ^{N};  (2.16)

b) the fundamental solution has the “reproduction property”:

∫_{ℝ^{N}} Γ(x, t; y, s)Γ(y, s; ξ, τ)dy = Γ(x, t; ξ, τ), x, ξ ∈ ℝ^{N}, τ < s < t; (2.17)

c) if Γ^{∗} denotes the fundamental solution of L^{∗}, formal adjoint of L, then

Γ^{∗}(z; ζ) = Γ(ζ; z).  (2.18)

Finally, a straightforward computation shows that, for every x ∈ ℝ^{N} and t > 0,

∫_{ℝ^{N}} Γ(x, t; y, 0)dy = e^{−t tr(B)}.  (2.19)

3. A representation formula. In this paragraph we prove a representation formula for classical solutions to (1.1). The main result of this section is the following proposition.

Proposition 3.1. A function u ∈ B(T) is a classical solution to (1.1) if and only if

u(z) = u_{0}(z) + Φu(z),  z ∈ ℝ^{N} × [0, T [,  (3.1)

where

u_{0}(z) = ∫_{ℝ^{N}} Γ(z; y, 0)a(y)dy  (3.2)

and

Φu(z) = ∫_{0}^{t} ∫_{ℝ^{N}} Γ(z; ζ)u^{p}(ζ)dζ.  (3.3)

We begin by proving a lemma.

Lemma 3.2. There exist two positive constants ε and c such that for every z = (x, t), ζ = (y, s) ∈ ℝ^{N+1} with 0 < t − s < ε, we have

|∂_{x_{i}}Γ(z; ζ)| ≤ \frac{c}{(t − s)^{\frac{Q_{(L)}+1}{2}}} exp(−\frac{1}{9}|C^{− \frac{1}{2}} (t−s)(x−E(t−s)y)|^{2}−(t−s)tr(B))

for 1 ≤ i ≤ q_{0}, where q_{0} is the rank of the matrix A_{0} appearing in the canonical representation (2.6) of A.

Proof. We have

|∂_{x_{i}}Γ(z; ζ)| = \frac{1}{4} Γ(z; ζ)|∂_{x_{i}}|C^{− \frac{1}{2}} (t − s)(x − E(t − s)y)|^{2}|
= \frac{1}{2} Γ(z; ζ)|⟨C^{−1}(t − s)(x − E(t − s)y), e_{i}⟩|

(by Cauchy-Schwarz inequality)

= \frac{1}{2} Γ(z; ζ)|C^{− \frac{1}{2}} (t − s)(x − E(t − s)y)|\sqrt{(C^{−1}(t − s))_{ii}}

≤ \frac{c_{1}\sqrt{(C^{−1}(t − s))_{ii}}}{\sqrt{det C(t − s)}} exp(−\frac{1}{9}|C^{− \frac{1}{2}} (t − s)(x − E(t − s)y)|^{2} − (t − s) tr(B))

(for some suitable positive constant c_{1})

≤ \frac{c}{(t − s)^{\frac{Q_{(L)}+1}{2}}} exp(−\frac{1}{9}|C^{− \frac{1}{2}} (t − s)(x − E(t − s)y)|^{2} − (t − s)tr(B))

for some positive constant c, whenever 0 < t − s < ε and ε is sufficiently small. In the last inequality we have used (2.15) and the following estimate: (by (2.14))

\sqrt{⟨C^{−1}(t − s)e_{i}, e_{i}⟩} = \sqrt{⟨C_{0}^{−1}(t − s)e_{i}, e_{i}⟩(1 + O(t − s))} as t ⟶ s^{+}

(by (2.11))

= \sqrt{⟨C_{0}^{−1}(1)D_{0}(\frac{1}{\sqrt{t − s}})e_{i}, D_{0}(\frac{1}{\sqrt{t − s}})e_{i}⟩(1 + O(t − s))} as t ⟶ s^{+}

= \sqrt{⟨C_{0}^{−1}(1)\frac{e_{i}}{\sqrt{t − s}}, \frac{e_{i}}{\sqrt{t − s}}⟩(1 + O(t − s))} as t ⟶ s^{+}

(since 1 ≤ i ≤ q_{0})

= \frac{1}{\sqrt{t − s}}\sqrt{(C_{0}^{−1}(1))_{ii}(1 + O(t − s))} ≤ \frac{c_{2}}{\sqrt{t − s}}

for some positive constant c_{2}, whenever 0 < t − s < ε and ε is sufficiently small. □

Now we proceed with the proof of Proposition 3.1. Proof of Proposition 3.1. (Necessity). Let u ∈ B(T) be a solution to (1.1). Let ρ ∈ C_{0}^{∞}(ℝ^{N}, ℝ) be a cut-off function such that 0 ≤ ρ ≤ 1 and ρ(x) = 1 for |x| ≤ 1, ρ(x) = 0 for |x| ≥ 2. We consider the sequence (ρ_{n}) defined by ρ_{n}(x) = ρ(\frac{x}{n}), n ∈ ℕ, and we set v_{n} = uρ_{n}.

For every z ∈ S_{T}, we have

v_{n}(z) = ∫_{ℝ^{N}} Γ(z; y, 0)ρ_{n}(y)a(y)dy − ∫_{0}^{t} ∫_{ℝ^{N}} Γ(z; ζ)L_{ζ}v_{n}(ζ)dζ, n ∈ ℕ, (3.4)

≡ I_{1}^{(n)}(z) − 2I_{2}^{(n)}(z), n ∈ ℕ,

where for n ∈ ℕ,

I_{1}^{(n)}(z) = ∫_{ℝ^{N}} Γ(z; y, 0)ρ_{n}(y)a(y)dy
+ ∫_{0}^{t}∫_{ℝ^{N}} Γ(z; ζ)(ρ_{n}(y)u^{p}(ζ) − u(ζ)(div(A∇ρ_{n}(y)) + ⟨y, B∇ρ_{n}(y)⟩))dζ,
I_{2}^{(n)}(z) = ∫_{0}^{t} ∫_{ℝ^{N}} Γ(z; ζ)⟨A∇ρ_{n}(y), ∇u(ζ)⟩dζ.

Since u and a are bounded functions, it is not difficult to prove that

lim_{n→∞} I_{1}^{(n)} = u_{0} + Φu, in ℝ^{N} × [0, T [.

On the contrary, the study of I_{2}^{(n)}(z) is rather delicate. We claim that I_{2}^{(n)}(z) = O (\frac{1}{n}) as n tends to infinity. We first observe that there exists a positive constant C_{1} such that

|∇ρ_{n}(x)| ≤ \frac{C_{1}}{n} ,  x ∈ ℝ^{N}.  (3.5)

Thus

J(z) ≡ |∫_{0}^{t} ∫_{ℝ^{N}} Γ(z; ζ)u(ζ)div(A∇ρ_{n}(y))dζ| = O (\frac{1}{n^{2}} ) , as n → ∞.

Moreover, integrating I_{2}^{(n)}(z) by parts and denoting

\overline{∇}_{y} = (∂_{y_{1}}, …, ∂_{y_{q_{0}}} , 0, …, 0),  (3.6)

for some positive constants C_{i}, i = 2, …, 6, we have

|I_{2}^{(n)}(z)| ≤ |∫_{0}^{t} ∫_{ℝ^{N}} ⟨A∇_{y}Γ(z; ζ), ∇ρ_{n}(y)⟩u(ζ)dζ| + J(z)

≤ \frac{C_{2}}{n} ∫_{0}^{t} ∫_{ℝ^{N}} |\overline{∇}_{y}Γ(z; ζ)|dζ + J(z)

(by Lemma 3.2, fixed a suitably small ε > 0)

≤ \frac{C_{3}}{n} ∫_{t−ε}^{t} \frac{1}{(t − s)^{\frac{Q_{(L)}+1}{2}}} ∫_{ℝ^{N}}( − \frac{1}{9}|C^{− \frac{1}{2}} (t − s)(x − E(t − s)y)|^{2}

− (t − s) tr(B))dy ds + \frac{C_{4}}{n}

(by the change of variable η = \frac{1}{2}C^{− \frac{1}{2}} (t − s)(x − E(t − s)y))

≤ \frac{C_{5}}{n} ∫_{t−ε}^{t} \frac{\sqrt{det C(t − s)}}{(t − s)^{\frac{Q_{(L)}+1}{2}}}∫_{ℝ^{N}} e^{−|η|^{2}}dηds + \frac{C_{4}}{n} ≤ \frac{C_{6}}{n}

The last inequality is a consequence of estimate (2.15) and formula (2.12). Thus the claim is proved and, as n tends to infinity in (3.4), we obtain the representation formula (3.1). (Sufficiency) Since a ∈ A, then Lu_{0} = 0 in S_{T}. Moreover, by using (2.16), we have

lim_{t→0^{+}} u_{0}(z) = a(x).

Besides, since u is bounded in S_{T}, proceeding as in [13], we obtain that Φu is Hölder continuous. Then, the integral representation (3.1) implies that u is Hölder continuous in S_{T}. Therefore, by the results in [13],

LΦu = −u^{p} in S_{T} .

Finally, it is clear that

lim_{t→0^{+}} Φu(z) = 0.

Proof of Proposition 1.4. We set X = C(\overline{S}_{T_{0}}, [0, M]) where M ≥ 2∥a∥_{∞}, T_{0} is to be determined. We consider

Ku(z) = ∫_{ℝ^{N}} Γ(x, t; y, 0)a(y)dy + ∫_{0}^{t} ∫_{ℝ^{N}} Γ(z; ζ)u^{p}(ζ)dζ.  (3.7)

A straightforward computation shows that it is possible to choose T_{0} = T_{0}(L, M) suitably small so that K is a contraction that maps X in X. In this case K has a unique fixed point u in X. By Proposition 3.1, since u is a solution to the integral equation (3.1), then u is also a solution to (1.1) in S_{T_{0}}.

Proposition 3.3. If u ∈ B(T) is a classical solution to (1.1), then u ∈ C^{∞}(S_{T}) and if the initial data is non-trivial then u is strictly positive in S_{T}.

Proof. If u ∈ B(T) is a solution to (1.1), by using Theorem 16 in [16] and a bootstrap argument, it can be easily seen that u is a smooth function in S_{T}. Moreover, if the initial data is non-trivial, the representation formula of Proposition 3.1 clearly yields u(x, t) > 0 in S_{T}.

4. Proof of the main theorem. In this section we prove the existence and non-existence Theorem 1.3. Let us begin by proving some preliminary results.

Lemma 4.1. Let 1 < p < p^{∗}. If u is a global solution to (1.1) corresponding to a non-trivial initial data a, then there exists T_{p} > 0 such that

T_{p}∥u_{0}(·, T_{p})∥_{∞}^{p−1} ≥ \frac{1}{p − 1}.  (4.1)

where u_{0} is defined in (3.2).

Proof. It is non-restrictive to prove the lemma in the case a(0) > 0. We first observe that, if t ≠ 0 and x ∈ ℝ^{N}, then

⟨C^{−1}(t)E(t)x, E(t)x⟩ = −⟨C^{−1}(−t)x, x⟩.  (4.2)

Indeed,

⟨e^{−tB}C^{−1}(t) e^{−tB^{T}} x, x⟩ = ⟨(e^{tB^{T}} C(t)e^{tB})^{−1} x, x⟩
= ⟨( ∫_{0}^{t} e^{−(s−t)B^{T}} Ae^{−(s−t)B}ds)^{−1}x, x⟩ = ⟨( − ∫_{0}^{−t} e^{−sB^{T}} Ae^{−sB}ds)^{−1}x, x⟩.

We choose δ, a_{0} > 0 such that a(0) > a_{0} if |x| ≤ δ. If t ≥ δ and |x| ≤ δ, then (4.2) yields

⟨C^{−1}(t)E(t)x, E(t)x > = ⟨−C^{−1}(−t)x, x⟩
≤ ⟨−C^{−1}(−δ)x, x⟩ ≤ |C^{−1}(−δ)|δ^{2}.

Hence there exists a positive constant C_{1} such that, if t ≥ δ,

u_{0}(0, t) ≥ ∫_{|x|≤δ} \frac{a_{0}(4π)^{− \frac{N}{2}}}{\sqrt{det C(t)}} exp ( − \frac{1}{4}⟨C^{−1}(t)E(t)x, E(t)x⟩ − t tr(B))dx
≥ C_{1}η(t).  (4.3)

Recalling the definition of p^{∗} it is clear that, if p < p^{∗}, then there exists T_{p} such that

T_{p}η^{p−1}(T_{p}) ≥ \frac{1}{(p − 1)C_{1}} .  (4.4)

Combining (4.3) with (4.4), we obtain (4.1).

Lemma 4.2. Let L ∈ L_{0}. If u ∈ B(T) is a classical solution to (1.1), then

tu_{0}^{p−1}(x, t) ≤ \frac{1}{p − 1},  ∀(x, t) ∈ \overline{S}_{T} ,  (4.5)

where u_{0} is defined in (3.2).

Proof. It suffices to prove inequality (4.5) in the case x = 0 and with non-trivial initial value. For fixed ε > 0 and t ∈ ]0, T], we set

V_{ε}(x, s) = Γ(0, t + ε; x, s), (x, s) ∈ \overline{S}_{t}.

Clearly V_{ε} is a positive, smooth function in S_{t} and since tr(B) = 0, by (2.19), we have

∫_{ℝ^{N}} V_{ε}(x, s)dx = 1  ∀s ∈ [0, t].  (4.6)

We set

P = div(A∇) + ⟨x, B∇⟩

and we denote by P^{∗} the formal adjoint of P. Then, by (2.18),

P^{∗}V_{ε} = −∂_{s}V_{ε}.

We define

J_{ε}(s) = ∫_{ℝ^{N}} V_{ε}(x, s)u(x, s)dx, s ∈ [0, t].

We remark that J_{ε} is strictly positive in [0, t] because u > 0 in ℝ^{N}×]0, t], by Proposition 3.3.

We claim that

\frac{d}{ds}J_{ε}(s) = ∫_{ℝ^{N}} V_{ε}(x, s)u^{p}(x, s)dx,  0 < s < t.  (4.7)

Indeed, let (ρ_{n})_{n∈ℕ} be a sequence of cut-off functions as in the proof of Proposition 3.1. Since u ∈ B(T), clearly

J_{ε}^{(n)}(s) =: ∫_{ℝ^{N}} V_{ε}(x, s)u(x, s)ρ_{n}(x)dx ⟶ J_{ε}(s),  as n → ∞. (4.8)

Moreover, for every s ∈]0, t[, we have

\frac{d}{ds}J_{ε}^{(n)}(s) = ∫_{ℝ^{N}} (−u(x, s)P^{∗}V_{ε}(x, s) + V_{ε}(x, s)P u(x, s))ρ_{n}(x)dx
+ ∫_{ℝ^{N}} V_{ε}(x, s)u^{p}(x, s)ρ_{n}(x)dx ≡ I_{1}^{(n)}(s) + I_{2}^{(n)}(s).

Since u ∈ B(T), we get

I_{2}^{(n)}(s) ⟶ ∫_{ℝ^{N}} V_{ε}(x, s)u^{p}(x, s)dx,  as n → ∞  (4.9)

uniformly in [0, t]. On the other hand, some integrations by parts give

I_{1}^{(n)}(s) = ∫_{ℝ^{N}}(−div(A∇ρ_{n})V_{ε}u − 2⟨A∇u, ∇ρ_{n}⟩V_{ε} − ⟨x, B∇ρ_{n}⟩V_{ε}u)dx.

Arguing as in [3], Proposition A4, and by making use of Lemma 3.2, it can be easily checked that ∂_{x_{i}}u, 1 ≤ i ≤ q_{0}, is bounded in S_{T}. Therefore, since ∇ρ_{n} = O (\frac{1}{n}) as n → ∞, we infer that I_{1}^{(n)}(s) goes to zero, as n → ∞, uniformly on compact subsets of ]0, t[. Thus (4.7) follows from (4.8) and (4.9).

Now, we recall that σ ⟶ σ^{p} is a convex function and V_{ε} is positive and satisfies (4.6). Hence we can use Jensen’s inequality in (4.7), obtaining

\frac{d}{ds}J_{ε}(s) ≥ (J_{ε}(s))^{p},  ∀s ∈ ]0, t[.  (4.10)

Integrating (4.10), we get

∫_{J_{ε}(0)}^{J_{ε}(t)} σ^{−p}dσ ≥ t.  (4.11)

The integral appearing in (4.11) should be read in generalized sense. By (2.16), we have

lim_{ε→0^{+}} J_{ε}(t) = lim_{ε→0^{+}} ∫_{ℝ^{N}} Γ(0, ε; x, 0)u(x, t)dx = u(0, t).

On the other hand

lim_{ε→0^{+}} J_{ε}(0) = lim_{ε→0^{+}} ∫_{ℝ^{N}} Γ(0, t + ε; x, 0)a(x)dx = u_{0}(0, t).

Thus, from (4.11), as ε goes to zero, (4.5) follows. □

Remark 4.3. If tr(B) ≠ 0, then (4.6) does not hold and we do not have an analogous of the Fujita type estimate (4.5) which was crucial in proving the blowing up of solutions in [3]. Essentially due to this fact, in order to study the case p < p^{∗} in the proof of Theorem 1.3, we choose a different approach based on a comparison method. In the following lemma we state a standard comparison result for operators in L.

Lemma 4.4. Let u, v ∈ B(T) be such that

\begin{cases}Lu + u^{p} ≥ Lv + v^{p} & in S_{T} (in classical sense), \\ u(x, 0) ≤ v(x, 0) & x ∈ ℝ^{N}. \end{cases}

Then u ≤ v in S_{T}.

Proof. Let w = v − u. Since u, v ∈ B(T) there exists a constant C_{1} such that

Lw + C_{1}w ≤ 0, in S_{T} .

Then the Lemma can be proved as in [2], Theorem 9, p.43. □

As an immediate consequence, we have the following uniqueness result.

Corollary 4.5. There is at least one solution to (1.1) in B(T).

Now we are in position to prove Theorem 1.3.

Proof of Theorem 1.3. We split the proof in four steps.

I – The case p > p^{∗}. For fixed γ > 0, we set

ρ(x, t) = Γ(x, t + γ), (x, t) ∈ \overline{S}_{∞}.

We prove there exists δ = δ(γ) > 0 such that, if a ∈ C(ℝ^{N}) and 0 ≤ a ≤ δρ(·, 0) in ℝ^{N}, then the Cauchy problem (1.1) has a global solution u satisfying

0 ≤ u(x, t) ≤ Mρ(x, t), ∀(x, t) ∈ \overline{S}_{∞},

for some positive constant M.

As in [3], Theorem 2, by using a fixed point method, we seek a global solution in the class

V(S_{∞}) = \{u ∈ C(\overline{S}_{∞}) | ∃M > 0 s.t. |u(z)| ≤ Mρ(x, t), ∀z ∈ \overline{S}_{∞}\}.

The crucial step in the proof of Theorem 2 in [3] is to show that Φρ ∈ V(S_{∞}). Once we have checked that, it is quite easy to verify that the recurrent sequence (u_{n})_{n∈ℕ}, defined by

u_{n} = u_{0} + Φu_{n−1}, for n = 1, 2, …

converges in V(S_{∞}) to a solution to (3.1) and then of (1.1), provided that 0 ≤ a(x) ≤ δΓ(x, γ) for a suitably small δ > 0.

Obviously Φρ ∈ C(\overline{S}_{∞}) and, by definition,

Φρ(z) = ∫_{0}^{t} ∫_{ℝ^{N}} Γ(z; ζ)Γ(y, s + γ)Γ^{p−1}(y, s + γ)dyds

(by the reproduction property (2.17) of Γ and observing that Γ(y, s + γ) ≤ (4π)^{− \frac{N}{2}} η(s + γ), for all (y, s) ∈ \overline{S}_{∞})

≤ (4π)^{\frac{N(1−p)}{2}} ∫_{0}^{∞} η^{p−1}(σ + γ)dσ ρ(x, t).  (4.12)

Finally, we observe that, since p > p^{∗}, there exist some positive constants C and p_{0}, p^{∗} < p_{0} < p, such that

t^{\frac{p−1}{p_{0}−1}} η^{p−1}(t) ≤ C

for sufficiently large t. Therefore the integral appearing in (4.12) is convergent.

II – The case p < p^{∗}. By contradiction, we suppose that there exists a global solution u to (1.1) corresponding to a non-trivial initial data a. Following the arguments of Meier in [14], we consider the function

\underline{u}(x, t) = (\frac{1}{u_{0}^{1−p}(x, t) − (p − 1)t})^{\frac{1}{p−1}}

defined in ℝ^{N} × [0, T_{1}[, where T_{1} = sup\{t > 0 | ∥\underline{u}∥_{L^{∞}(S_{t})} < ∞\}. Clearly

T_{1} ≥ \frac{1}{p − 1}∥u_{0}∥_{L^{∞}(S_{Tp})}^{1−p} > 0,

for T_{p} fixed as in Lemma 4.1. Proceeding as in [14], we have that \underline{u} is a lower solution of u in the sense that

\begin{cases}−L\underline{u} ≤ \underline{u}^{p} & in S_{T_{1}}, \\ \underline{u}(x, 0) = a(x) & x ∈ ℝ^{N}. \end{cases}

Thus, by Lemma 4.4, \underline{u} ≤ u in S_{T_{1}}. Besides, by construction,

lim sup_{t→T_{1}^{−}} ∥\underline{u}(·, t)∥_{∞} = ∞.

On the other hand, by Lemma 4.1, T_{1} ≤ T_{p}. This yields a contradiction since u was supposed to be global.

III – The case tr(B) > 0. Let L ∈ L be such that tr(B) > 0. Since the quadratic form related to C(t) is an increasing function of the variable t, then, by Courant-Fischer theorem, t ⟶ det C(t) is also increasing. Hence, for every positive γ, we have

η(t) ≤ \frac{e^{−t tr(B)}}{\sqrt{det C(γ)}}, t ≥ γ.

This proves (i), since clearly p^{∗} = 1.

IV – The case L ∈ L_{0} and p = p^{∗}. Let L ∈ L_{0}. Since tr(B) = 0 and, by (2.12), det C(t) = det C(1)t^{Q_{(L)}}, clearly p^{∗} = 1 + \frac{2}{Q_{(L)}} . This proves (ii).

In order to study the critical case, we follow a method by Weissler [20]. Here we only sketch how to adapt the proof of Weissler to our setting. The starting point is the estimate (4.5) of Lemma 4.2. By contradiction, we suppose that there exists a global solution u to (1.1). If p = 1 + \frac{2}{Q_{(L)}} , substituting in (4.5) the explicit expression of u_{0}, we obtain

∫_{ℝ^{N}} exp(−\frac{1}{4}|C^{− \frac{1}{2}} (t)(x − E(t)y))|^{2})a(y)dy ≤ C_{L}, (x, t) ∈ ℝ^{N} × [0, ∞[, (4.13)

where

C_{L} = (\frac{Q_{(L)}}{2} )^{\frac{Q_{(L)}}{2}} (4π)^{\frac{N}{2}} \sqrt{det C(1)}.

Next, we observe that

|C^{− \frac{1}{2}} (t)(x − E(t)y)|^{2} = |C^{− \frac{1}{2}} (1)D(t^{− \frac{1}{2}} )(x − E(t)y)|^{2}

(by (2.13))

= |C^{− \frac{1}{2}} (1)(x^{T} D(t^{− \frac{1}{2}} ) − E(1)D(t^{− \frac{1}{2}} )y)|^{2} = |C^{− \frac{1}{2}} (1)(x^{T} − E(1)y^{T})D(t^{− \frac{1}{2}} )|^{2}

= ⟨C^{−1}(t)(x − yE^{T} (1)), (x − yE^{T} (1))⟩,  (4.14)

and the quadratic form in (4.14) is a decreasing function of the variable t. Hence, by the monotone convergence theorem, as t tends to infinity, (4.13) yields

∥a∥_{1} ≤ C_{L}.

Moreover, since u(·, t) can be regarded as the initial value, we obtain

∥u(·, t)∥_{1} ≤ C_{L},  t ≥ 0.  (4.15)

As in [20], it is non-restrictive to assume that the initial value a dominates a fundamental solution, that is

a(x) ≥ kΓ(x, α), x ∈ ℝ^{N},

for some positive constants k, α. As a consequence, from the representation formula (3.1) and by means of the reproduction property (2.17) of Γ, we obtain

u(x, t) ≥ kΓ(x, t + α),  (x, t) ∈ ℝ^{N} × [0, ∞[.  (4.16)

Therefore, by the integral representation of u, by (4.16), (2.19) and by Fubini’s theorem, we infer that, for some positive constants M_{1}, M_{2},

∥u(·, t)∥_{1} ≥ k^{1+ \frac{2}{Q_{(L)}}} ∫_{0}^{t} ∫_{ℝ^{N}} Γ^{1+ \frac{2}{Q_{(L)}}} (y, s + α)dy ds

= M_{1} ∫_{0}^{t} \frac{1}{(s + α)^{1+ \frac{Q_{(L)}}{2}}} ∫_{ℝ^{N}} exp (−\frac{1}{4} (1 + \frac{2}{Q_{(L)}} ) ⟨C^{−1}(s + α)y, y⟩) dy ds

(by the change of variable η = C^{− \frac{1}{2}} (s + α)y)

= \frac{M_{1}}{\sqrt{det C(1)}} ∫_{0}^{t} \frac{1}{s + α} ∫_{ℝ^{N}} exp (− \frac{1}{4}(1+ \frac{2}{Q_{(L)}} )|η|^{2})dηds = M_{2} log (1+ \frac{t}{α}).

This contradicts (4.15) and completes the proof of the theorem.

5. Some examples. In this section we examine several examples. It is clear that, in virtue of Hörmander’s condition (2.2), each of the following operators is hypoelliptic.

Example 5.1. In ℝ^{N} × ℝ, we consider the operator in (1.3). In this case L ∈ L_{0} since

A = [\begin{matrix}1 & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}]  and  B = [\begin{matrix}0 & 1 & 0 & ⋯ & 0 \\ 0 & 0 & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & 0 & ⋯ & 1 \\ 0 & 0 & 0 & ⋯ & 0 \end{matrix}] .

Therefore, Q_{(L)} = N^{2} and the critical exponent is p^{∗} = 1 + \frac{2}{N^{2}} .

Example 5.2. Let N = 2n. In ℝ^{N} × ℝ, we consider

L = ∑_{i=1}^{n} ∂_{x_{i}}^{2} + ∑_{i=1}^{n} x_{i}∂_{x_{n+i}} − ∂_{t}.

The operator L is the linearized prototype of the Kolmogorov operator which describes the probability density of a physical system with 2n degree of freedom (see [18], p.167). The operator L belongs to L_{0} since

A = [\begin{matrix}I_{n} & 0 \\ 0 & 0 \end{matrix}]  and  B = [\begin{matrix}0 & I_{n} \\ 0 & 0 \end{matrix}] .

Moreover, the homogeneous dimension of ℝ^{N+1} with respect to L is Q_{(L)} + 2 = 2N + 2. Thus, the critical exponent is p^{∗} = 1 + \frac{1}{N}.

Remark 5.3. Operators in Examples 5.1 and 5.2 are particular cases of the following operator. Let N ≥ 2 and 1 ≤ r ≤ N − 1. In ℝ^{N} × ℝ, L is the operator in (1.2) corresponding to the matrices

A = [\begin{matrix}I_{r} & 0 \\ 0 & 0 \end{matrix}]  and  B = [\begin{matrix}0 & I_{N−r} \\ 0 & 0 \end{matrix}] .

Clearly L ∈ L_{0}, thus the critical exponent can be explicitly expressed in terms of N and r.

Example 5.4. In ℝ^{2} × ℝ, we consider

L = ∂_{x_{1}}^{2} + (x_{1} − x_{2})∂_{x_{2}} − ∂_{t}.

In this case, tr(B) = −1 and

det C(t) ∼ te^{2t}, as t ⟶ ∞.

Hence,

η(t) ∼ \frac{1}{\sqrt{t}} as t ⟶ ∞.

By Theorem 1.3, p > 3 corresponds to the existence case, while 1 < p < 3 corresponds to the blow-up case.

Example 5.5. In ℝ^{2} × ℝ, we consider

L = ∂_{x_{1}}^{2} − x_{1}∂_{x_{1}} + (x_{1} + x_{2})∂_{x_{2}} − ∂_{t}.

In this case, tr(B) = 0 and

det C(t) ∼ e^{2t}, as t ⟶ ∞.

Therefore p^{∗} = 1, i.e. for every p > 1 there exist global solutions.

Example 5.6. In ℝ^{2} × ℝ, we consider

L = ∂_{x_{1}}^{2} − x_{2}∂_{x_{1}} + x_{1}∂_{x_{2}} − ∂_{t}.

In this case, tr(B) = 0 and

det C(t) = \frac{1}{4}(t^{2} − sin^{2} t).

By Theorem 1.3, p > 2 corresponds to the existence case, while 1 < p < 2 corresponds to the blow-up case. We remark explicitly that the critical exponent related to L_{0}, the principal part of L defined in (2.10), is equal to \frac{3}{2}.

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