In this connection we investigate the problem of the existence of non negative solutions defined for all positive times

NEW RESULTS ON THE CAUCHY PROBLEM FOR PARABOLIC SYSTEMS AND EQUATIONS

WITH STRONGLY NON LINEAR SOURCES Daniele Andreucci¹

We consider both degenerate and uniformly parabolic systems and equations, containing a forcing term (a “source”) depending on the solution itself (see (1), (5), (6) below). The source is such that the solution may become unbounded in a finite time, even if the initial data are bounded. In this connection we investigate the problem of the existence of non negative solutions defined for all positive times. Moreover, even the problem of the existence of local in time solutions is not trivial, owing to the effect of nonlinear sources of this kind. In fact (as a marked difference with the correspond- ing homogeneous problems), local solutions may exist only under certain restrictions on the local regularity of the initial data.

1. Introduction

In this paper we consider non negative solutions to certain para- bolic systems or equations, defined in a strip S_T = IR^N × (0, T ), 0 < T ≤ ∞, N ≥ 1.

1 Member of GNFM of Italian CNR. Work supported by MURST project “Equazioni di evoluzione e applicazioni fisico-matemati- che”.

Appeared in Manuscripta Mathematica, 77 (1992), pp.127–159.

Copyright c Springer-Verlag 1992.

Namely, we look first at the Cauchy problem in S_T for the system

 u_t − ∆ u^m= u^av^b , in S_T ,

v_t− ∆ v^µ = u^αv^β , in S_T , (1) with m, µ ≥ 1, a, β > 1, b, α ≥ 0. The case of the single equation zt− ∆ z^m = z^a , in ST , (2) m ≥ 1, a > 1, has been studied in a large number of papers (for a bibliography we refer to [2], [14]). Especially, the question of existence of local in time solutions under optimal assumptions on the initial datum, and the question of the existence of global solutions have been investigated. We mention e.g., [2], [4], [10], [19], [20] for the case m = 1, and [2], [11] for the case m > 1.

It has been shown in [2] that a solution to (2) with T < ∞ exists, provided

sup

x∈IR^N

kz(·, 0)k_Lp(B1(x)) < ∞ , (3) where p ≥ 1 is such that p > N (a − m)/2; (3) is in some sense optimal: we refer to Section 2 below for further comments on this point.

Here we investigate first the minimal local regularity of the initial data uo(x) = u(x, 0), vo(x) = v(x, 0), x ∈ IR^N, needed to guarantee existence of local solutions to (1). We find that, if u_o and vo satisfy

sup

x∈IR^N

ku_ok_Lp(B1(x))+ kv_ok_Lq(B1(x))
< ∞ , (4) p, q ≥ 1, a solution to the Cauchy problem for (1) exists, provided p, q are connected by an explicit relation involving also N and the parameters in (1) (see Theorem 2.1). In some cases, u_o and v_o can be allowed to be just σ finite Borel measures (such that an analogue of (4) holds). We also remark that (4) reduces to (3) in the case of the completely uncoupled system b = α = 0. Moreover we show that if p, q ≥ 1 violate the condition mentioned above, two functions u_o ∈ L^p(IR^N), v_o ∈ L^q(IR^N) can be given so that no solution to the Cauchy problem for (1) exists in any strip S_T, provided some additional assumptions are fulfilled.

We also investigate the existence of solutions to (1) defined in IR^N × (0, ∞). We give a condition for N , m, a, b, µ, α, β, discriminating between existence and non existence of global non negative non trivial solutions to (1). Again, this condition coincides with the known one ([10], [11]) in the case of the uncoupled system b = α = 0.

To the best of our knowledge, these results on system (1) are new in the literature (the reference [9], appeared after this paper was submitted for publication, is concerned with global solutions to the semilinear system (1) with m = µ = 1; our results in this case agree with those given there). Our approach allows us to con- sider both the degenerate case m > 1, or µ > 1, and the strictly parabolic case m = 1, or µ = 1, even if (1) is replaced by a system of more general form (see subsection 2.iv). Finally, we note that the proofs and the results given here can be immediately extended to systems with more than 2 equations and 2 unknowns.

Next we consider the case of the single equation

u_t − ∆ u^m= t^−σu^a , in S_T , (5) m ≥ 1, a > 1, 1 > σ > −∞. Conditions in the spirit of the ones discussed above are found for the existence, or non existence, of local and global non negative solutions to the Cauchy problem for (5). We also consider solutions of (5) defined in S_T for t > 0, with no reference to initial data, proving the existence, and studying the behaviour, of initial traces.

When σ = 0 our results about (5) coincide with those given in [2]. When m = 1, the conditions we find for the existence of global solutions to (5) coincide with those given in [18] (where domains different from IR^N are also considered).

We remark that a strongly space-dependent right hand side in (5) could be considered, as in [2], but for the sake of brevity we restrict ourselves to the case above.

Finally, we investigate the existence of local and global solu- tions to the Cauchy problem in ST for

 u_t− ∆ u^m = v^b , in S_T ,

v_t − ∆ v^µ= u^α , in S_T , (6)

with m, µ ≥ 1, b, α > 0, bα > 1. In particular we prove existence of global solutions under conditions that, in the case m = µ = 1, reduce to the sufficient and necessary conditions first given in [8]

(system (6) has been also considered in [14]; for the case of domains different from IR^N, we mention [15], [12], [17]).

Our approach is based on a priori L^∞-estimates valid for non negative subsolutions to (1) (or (5), (6)), in the spirit of [2] (see propositions 2.4, 2.5, 2.8); we give also estimates for the space gradients of the solutions up to time t = 0 (see subsection 2.iv).

In Section 2, we formulate exactly the problems, and state our main results; some extensions of the existence results to sys- tems and equations of more general structure are given in subsec- tion 2.iv. The proofs are collected in sections 3–6.

2. Statements of the problems and main results

2.i The system (1)

Let us consider the problem

u_t− ∆ u^m = u^av^b , in S_T , (7) vt− ∆ v^µ = u^αv^β , in ST , (8) u(x, 0) = uo(x) , v(x, 0) = vo(x) , x ∈ IR^N , (9)

where (unless otherwise specified) we assume m, µ ≥ 1, a, β > 1, b, α ≥ 0, and S_T = IR^N × (0, T ), 0 < T < ∞, u_o, v_o ∈ L¹_loc(IR^N).

Throughout this paper we stipulate the convention s^ε ≡ 1 for all s ≥ 0 if ε = 0.

Definition 2.1. A pair (u, v) of non negative functions u, v ∈ L^∞_loc(S_T) is a (weak) solution to (7)–(9) if for every bounded open set Ω ⊂ IR^N with smooth boundary ∂Ω, Ω_T = Ω × (0, T ),

u , v ∈ C_loc(0, T ; L¹(Ω)) ; u^m, v^µ ∈ L²_loc(0, T ; W_loc^1,2(Ω)) ; (10)

∀η ∈ W^1,2(Ω_T) , η compactly supported in Ω_T , (11) Z

Ω

u(x, t)η(x, t) dx + Zt

0

Z

Ω

{−uη_τ + Du^m· Dη} dx dτ

≤ Zt

0

Z

Ω

u^av^bη dx dτ ,

∀0 < t < T , and the obvious analogue for v ;

u(·, t) → u_o(·), v(·, t) → v_o(·) in L¹(Ω), as t → 0. (12) Remark 2.1. If in (9) we substitute u_o ∈ L¹_loc(IR^N) with a σ finite Borel measure ν, we define a solution (u, v) to (7)–(9) as above, with the only difference that in (12) we require u(·, t) → ν as t → 0 in the sense of measures. In the following we say that a Borel measure λ is an initial trace for a measurable function u : ST −→ IR⁺, if u(·, tj) → λ as j → +∞, in the sense of measures, for some sequence {t_j} & 0.

Definition 2.2. A local non negative subsolution [supersolution]

to (7)–(8) in ST is a pair (u, v) of non negative functions u, v ∈ L^∞_loc(S_T) satisfying (10)–(11), with the restriction η ≥ 0 [η ≤ 0].

A pair (u, v) is a local solution if it is both a subsolution and a supersolution.

Local solutions to (7)–(8) are continuous (see [6]).

Remark 2.2. (Notation) We write |A| for the Lebesgue measure of any measurable set A ⊂ IR^N, and denote A(t) = A × {t} ⊂ R^{N +1}; for f ∈ L¹(A) and |A| < ∞ we set

Z

A

f (x) dx = 1

|A|

Z

A

f (x) dx ; χ

A(x) =

 1 , x ∈ A,

0 , x /∈ A ; (13) k·k_p,A = k·k_Lp(A) ; B_d(x) = {y ∈ IR^N | |x − y| < d}, (14)

for p ≥ 1, d > 0 and x ∈ IR^N. We also denote by γ(c₁, . . . , c_n) a positive constant depending on the quantities c₁, . . . , c_n. We make use of the norms (introduced in [2])

|||f |||_p ≡ sup

x∈IR^N

kf k_p,B1(x) ; γ⁻¹ sup

x∈IR^N

kf k_p,Bd(x) ≤ |||f |||_p ≤ γ sup

x∈IR^N

kf k_p,Bd(x) (15) p ≥ 1, f ∈ L^p_loc(IR^N); (15) holds for any d > 0, with γ = γ(N, d, p) > 1. We define w as the first eigenfunction of the prob- lem

∆ w = −λ₁w , in B_ρ ⊂ IR^N, w = 0 , on ∂B_ρ, (16) where B_ρ = {|x| < ρ}, ρ > 0. We may assume w ≥ 0, kwk_1,Bρ = 1. It is known by classical results that λ1 = γ(N )ρ⁻², and that kwk_∞,Bρ ≤ γ(N )ρ^−N, w(x) ≥ γ_o(N )ρ^−N for |x| < ρ/2. Finally, we denote K_p = N (m − 1) + 2p, H_q = N (µ − 1) + 2q for p, q ≥ 1, and K = K1 = N (m − 1) + 2, H = H1 = N (µ − 1) + 2.

We first prove existence of local in time solutions to (7)–(9) under sufficient conditions on u_o, v_o. The optimality of such con- ditions is discussed below.

Theorem 2.1. Let u_o, v_o ∈ L¹_loc(IR^N), u_o, v_o ≥ 0, |||u_o|||_p < ∞,

|||vo|||_q < ∞, with p, q ≥ 1 such that (for Kp = N (m − 1) + 2p, H_q = N (µ − 1) + 2q)

N

Kp(a − 1) + _H^N

qb < 1 , _K^N

pα + _H^N

q(β − 1) < 1 . (17) Then a solution to (7)–(9) exists in STo, where To > 0 is defined by

T¹⁻

N

Kp(a−1)−_Hq^N b o |||u_o|||^p

2 Kp(a−1) p |||v_o|||^q

2 Hqb q

+ T¹⁻

N

Kpα−_Hq^N (β−1) o |||u_o|||^p

2 Kpα p |||v_o|||^q

2 Hq(β−1) q

+ T^p

2

oKp|||uo|||^p

2 Kp(m−1)

p + T^q

2

oHq|||vo|||^q

2 Hq(µ−1)

q = γ⁻¹ (18)

Moreover for 0 < t < To

|||u(·, t)|||_p ≤ γ|||uo|||_p, ku(·, t)k_∞,IRN ≤ γt^−KpN |||uo|||^p

2

pKp , (19)

|||v(·, t)|||_q ≤ γ|||v_o|||_q, kv(·, t)k_∞,IRN ≤ γt^−HqN |||v_o|||^q

2

qHq . (20) Here γ = γ(N, m, a, b, µ, α, β, p, q) > 1.

For estimates of the gradients Du and Dv we refer to subsection 2.iv.

Remark 2.3. If the choice p = 1 is admissible in (17) (for a suitable q), Theorem 2.1 still holds if u_o is replaced by a σ finite Borel measure ν in IR^N, satisfying

|||ν||| = sup

x∈IR^N

ν(B₁(x)) < +∞ (21) (see Remark 2.1). Indeed, both uo and vo may be replaced by σ finite Borel measures satisfying (21), provided (17) holds for p = q = 1. This extension of Theorem 2.1 follows from the proof given in subsection 3.i, and by a suitable approximation of ν with smooth functions. We refer to subsection 2.iv for further extensions.

If b = α = 0, i.e., if the system is completely uncoupled, (17) reduces to the assumptions on local regularity of initial data known to be optimal (in the sense specified below) in the case of the single equation, see [4] for the case m = 1, and [2] for the case m > 1.

Indeed in [2] it is proved that any non negative solution to (2) with m ≥ 1, a > 1, has as initial trace a measure ν such that

|||ν||| ≤ γ(T, N, m, a) < +∞ ; (22) moreover, if a > m, the trace is unique and for all x ∈ IR^N

Z

Bρ(x)

dν ≤ γρ^−a−m2 , 0 < ρ < γo(T, N, m, a) . (23)

It is easily seen that if N (a − 1)/K_p > 1, there exist functions u_o ∈ L^p(IR^N) not satisfying (23) (so that no solution to (2) may exist assuming u_o as initial datum); assumption N (a − 1)/K_p < 1 is optimal in this sense. (We refer to [4] for more information on the case N (a − 1)/2p = 1, m = 1.)

For solutions (u, v) of the system at hand, in general neither (22) nor (23) may be expected. Assume for definiteness m > 1.

The existence of an initial trace ν of, say, u, satisfying sup

ρ>1

ρ^{−m−12 −N} ν(B_ρ(0)) < +∞ , (24) is an immediate consequence of the Harnack inequality

Z

Bρ(0)

u(x, τ ) dx ≤ γ ( ρ²

t

_m−1¹

+ t ρ²

^N₂ 

|x|<ρinf u(x, t)

^K₂) , (25) 0 < 2τ < t < T , ρ > 0, that is in force because u is a continu- ous non negative supersolution of the homogeneous porous media equation

z_t− ∆ z^m= 0 , in S_T (26)

(see [3], [1]). But assume b > 0, and v_o ≡ 0. Then we may take v ≡ 0, and (7) reduces to (26) (for any a > 1), which is known to admit a non negative solution if the initial datum is any σ finite Borel measure ν satisfying (24) [5]. Thus, for a suitable choice of ν, both (22) and (23) are contradicted (as well as any other requirement on u(·, 0) stronger than (24)). Anyway, we are able to prove the following result in the direction of optimality of (17).

Proposition 2.1. Let ¯p, ¯q ≥ 1 be such that 1 < _K^N

¯

p(a − 1) + _H^N

¯

qb . (27)

Then there exist uo ∈ L^p¯(IR^N), vo ∈ L^q¯(IR^N), uo, vo ≥ 0 supported in B₁(0) ⊂ IR^N, such that no non negative solution to (7)–(9) may exist in S_T for any T > 0, provided the additional requirements a > m and ¯q(m − 1) ≤ ¯p(µ − 1) are satisfied.

Remark 2.4. It follows from the proof of Proposition 2.1 that, if N b/H_q¯ > 1 (in the notation of Proposition 2.1), then there exist v_o ∈ L^q¯(IR^N), u_o ∈ L^∞(IR^N) such that no non negative solution to (7)–(9) may exist (roughly speaking, a solution may exist only if uo vanishes where vo becomes unbounded; see Remark 4.1).

We should note that both (22) and (23) still hold for any initial trace ν of u, if, e.g., we know a priori v ≥ 1, so that u is a supersolution of (2) in S_T.

The proof of Proposition 2.1 will be based on the following results:

Lemma 2.1. Let u ∈ C^o(S_T) be a non negative supersolution of (26) with m ≥ 1, and let u(·, t) → ν as t → 0 in the sense of measures, with

Z

B_ρo(xo)

dν ≥ ε_oρ^−ϑ_o , (28)

for some given x_o ∈ IR^N, ε_o, ρ_o > 0, ϑ ≤ N , and ϑ > −2/(m − 1) if m > 1. Then

|x−xinfo|<ρo

u(x, to) ≥ γoε

2 2+ϑ(m−1)

o t⁻

ϑ 2+ϑ(m−1)

o , (29)

where γ_ot_o = ε^−(m−1)o ρ^2+ϑ(m−1)o , γ_o = γ_o(N, m, ϑ), provided t_o <

T .

Proof. If m > 1 we only need take ρ = ρ_o, t = t_o in (25) and choose γo suitably small (after a translation in space). If m = 1, (29) follows directly from the Harnack inequality for non negative solutions to the heat equation in ST [21].

Proposition 2.2. Let u ∈ C^o(IR^N × (τ_o, T )), T > τ_o ≥ 0, be a non negative supersolution of

z_t− ∆ z^m = c t^−σχ_Σz^a, in IR^N × (τ_o, T ) , (30) where Σ = {(x, t) | λ|x − x_o| < t^s, τ_o< t < T }, and a > m ≥ 1, s, c > 0, σ, λ ≥ 0, x_o ∈ IR^N are given. Then for all τ_o < t_o < t/2 <

t < T , ρ = ρ(t_o) > 0 such that 2λρ < t^s_o, we have for all σ ≥ 0 Z

Bρ(xo)

u(x, t_o) dx ≤ γ max

t^{−1−σa−1} , ρ^−a−m2 t^a−mσ 

. (31)

Moreover in the case σ = 1 we can replace the term t^{−1−σa−1} in (31) with (ln(t/t_o))^−a−11 . Here γ = γ(N, m, a, c, σ).

Concerning the existence of global non negative solutions we prove

Theorem 2.2. Assume (denoting K = N (m − 1) + 2, H = N (µ − 1) + 2)

N

K(a − 1) + ^N_Hb > 1 , ^N_Kα + ^N_H(β − 1) > 1 , (32) and let p, q ≥ 1 satisfy (17). Assume uo, vo ∈ L¹(IR^N), uo, vo ≥ 0,

|||u_o|||_p + |||v_o|||_q ≤ C < +∞. Then a constant γ_o > 0 exists such that if

kuok_1,IR^K2(a−1)N kvok_1,IR^H2bN + kuok_1,IR^K2αN kvok_1,IR^H2(β−1)N ≤ γo , (33)

problem (7)–(9) has a solution defined in S_∞ = IR^N × (0, ∞).

Here γo = γo(N, m, a, b, µ, α, β, p, q, C).

Remark 2.5. Note that Theorem 2.2 guarantees existence of global solutions even if one between the initial data u_o and v_o is “large”, provided the other one is chosen accordingly “small” and b, α > 0.

See subsection 2.iv for extensions of Theorem 2.2.

Again, if b = α = 0, (32) gives the known threshold values of a and β discriminating between existence and non existence of non trivial global non negative solutions to the resulting pair of uncoupled equations (7)–(8) (see [10] for m = 1, [11] for m > 1). In the general case we prove, about non existence of global solutions, Proposition 2.3. Let

N

K(a − 1) + ^N_Hb < 1 , (34) and assume also a ≥ m ≥ µ. Let (u, v) be a non negative solution to (7)–(8) defined for all t > 0. Then u ≡ 0, or b > 0 and v ≡ 0.

Remark 2.6. If b = 0, Proposition 2.3 has been proved in [2]

(in this case the system is at least partially uncoupled, and the requirements a ≥ m ≥ µ are not necessary).

The main tool in proving theorems 2.1 and 2.2 is the following estimate, proved in [2], that we state in a form suitable to our purposes.

Proposition 2.4. ([2] Section 7) Let (u, v) be any non negative local subsolution to (7)–(8) in S_T, and let u, v ∈ C^o(S_T), 0 < T <

∞. Then for every ball B_2ρ = B_2ρ(x_o), ρ > 0, x_o ∈ IR^N, and for all 0 < t < T satisfying for all 0 < τ < t

ρ⁻²

u^m−1(·, τ )

∞,B2ρ+ ρ⁻²

v^µ−1(·, τ ) ∞,B2ρ

+

u^a−1(·, τ )v^b(·, τ )

∞,B2ρ +

u^α(·, τ )v^β−1(·, τ )

∞,B2ρ ≤ τ⁻¹, (35) we have

kw(·, t)k_∞,Bρ(xo) ≤ γt^−Nr





 Zt

0

Z

B2ρ(xo)

w^p(x, t) dx dτ







2 r

, (36)

γ = γ(N, m, a, b, µ, α, β, p), for all p ≥ 1, and either w = u, r = K_p, or w = v, r = Hp.

Theorems 2.1 and 2.2 are proved in Section 3, propositions 2.1 and 2.2 in Section 4, and Proposition 2.3 in Section 5.

2.ii The equation (5)

The techniques used here to study system (7)–(9) yield some new results on non negative solutions to

u_t− ∆ u^m = t^−σu^a , in S_T = IR^N × (0, T ) , (37) u(x, 0) = u_o(x) , x ∈ IR^N, (38) where m ≥ 1, a > 1, 1 > σ > −∞, uo ∈ L¹_loc(IR^N), uo ≥ 0.

Definitions 2.1–2.2 carry over trivially to the case of problem (37)–

(38).

Theorem 2.3. Let u_o ∈ L¹_loc(IR^N), u_o ≥ 0, |||u_o|||_p < +∞, with p ≥ 1 such that

σ + _K^N

p(a − 1) < 1 , (39)

K_p = N (m − 1) + 2p. Then a non negative solution to (37)–(38) exists in S_To, where T_o > 0 is defined by

T^p

2

oKp|||u_o|||^p

2 Kp(m−1)

p + T^1−σ−

N Kp(a−1) o |||u_o|||^p

2 Kp(a−1)

p = γ⁻¹. (40) Moreover the estimates of u stated in (19) hold for 0 < t < T_o, with γ = γ(N, m, a, σ, p).

Initial data measures are admissible if one can take p = 1 in (39), in the spirit of Remark 2.3.

Theorem 2.4. Let u be a local non negative solution to (37) in S_T, 0 < T < ∞ (with no reference to initial data). Then

|||u(·, t)|||₁ ≤ γ(N, m, a, σ, T ) , 0 < t < T /2 ; (41) more explicitly, for a, m > 1, 0 < t < T /2,

|||u(·, t)|||₁ ≤ γ

T^−m−11 + T^−2(a−1)K (^1−σ−NK(a−1)) , (42) where γ = γ(N, m, a, σ). It follows that u has an initial trace ν; ν is a σ finite Borel measure in IR^N satisfying the estimate provided

by (41) or (42). If, moreover, 1 > σ ≥ 0, σ + ^N_K(a − 1) > 1 and a > m ≥ 1, then for all x ∈ IR^N,

Z

Bρ(x)

dν ≤ γρ^{−a−m+σ(m−1)2(1−σ)} , 0 < ρ < γo(N, m, a, σ, T ) . (43)

Proof. Estimate (41) is derived straightforwardly from Proposi- tion 2.2 if σ ≥ 0. If σ < 0, m = 1, we first employ the well known Harnack inequality for non negative supersolutions of the heat equation in S_T [21] to find for all 0 < t < T /2,

|||u(·, t)|||₁ ≤ γ(N, T )|||u(·, 3T /4)|||₁ .

Then we apply Proposition 2.2, with σ = 0, in IR^N × (³₄T, T ).

Estimate (42) can be proved by comparing u with the subso- lution to (37)

z(x, t) = A T

σ a−1

1

(T_o− t)^a−11







1 − B|x − x_o|² T^σ

m−1 a−1

1

(T_o− t)^m−aa−1







1 m−1

+

, (44) where T₁ = T_o, (x, t) ∈ S_To for σ ≥ 0, and T₁ = T , (x, t) ∈ IR^N × (T /2, T_o) for σ < 0. Here A, B > 0 depend on N , m, a, while x_o ∈ IR^N, T_o > 0 are arbitrarily chosen. We should remark that z is a modification of the subsolution to (2) introduced in [11].

Employing the subsolution z above, one can follow the argument given in [2] subsection 12.ii for the case of equation (2); we do not reproduce the details here. To prove (43), we note that (31) implies for all x ∈ IR^N, ρ > 0, 0 < t < T ,

Z

Bρ(x)

dν ≤ γ

t^{−1−σa−1} + ρ^−a−m2 t^a−mσ  .

Minimizing the right hand side with respect to t we get (43).

Remark 2.7. A simple calculation shows that if σ +_K^N_p(a − 1) > 1, there exist functions u_o ∈ L^p(IR^N) not satisfying (43), so that for this choice of the initial datum u_o, (37)–(38) does not have any non negative solution. In this sense (39) is optimal.

Remark 2.8. If a > m, the uniqueness of the initial trace of u could be proved as in [2].

Next we state two results about existence and non existence of global solutions.

Theorem 2.5. Let

σ + ^N_K(a − 1) > 1 , (45) and let uo ≥ 0, |||uo|||_p ≤ C < ∞, for a p > 1 such that (39) holds. Then a constant γ_o > 0 exists such that if ku_ok_1,IRN < γ_o, a solution to (37)–(38) exists in S_∞ = IR^N × (0, ∞). Here γ_o = γ_o(N, m, a, σ, p, C).

Theorem 2.6. Let

σ + ^N_K(a − 1) < 1 . (46) If u is a non negative solution to (37) defined for all t > 0, then u ≡ 0.

Proof of Theorem 2.6. In the case a > m ≥ 1 we refer to Section 5, where Proposition 2.3 is proved by means of arguments that extend to the present case. If m > 1, we only have to let T → +∞ in (42).

Remark 2.9. Let u be any non negative local solution to (37) in ST, with σ ≥ 1. Then Proposition 2.2 implies u(·, t) → 0 in L¹_loc(IR^N) as t → 0, so that problem (37)–(38) has no meaning for σ ≥ 1.

In the case m = 1, conditions (45), (46) for the existence or non existence of non trivial global non negative solutions to (37)–(38) are due to Meier, see [18].

The proof of Theorem 2.3 [2.5] is in all similar to the proof of Theorem 2.1 [2.2], when we use the estimates of Proposition 2.5 below: see subsection 3.i [3.ii] for more details.

Proposition 2.5. Let u be any non negative local solution to (37) in S_T, and let u ∈ C^o(S_T), 0 < T < ∞. Then for every ball B2ρ(xo), ρ > 0, xo ∈ IR^N, and for all 0 < t < T satisfying for all 0 < τ < t

ρ⁻²

u^m−1(·, τ )

∞,B2ρ(xo)+ τ^−σ

u^a−1(·, τ )

∞,B2ρ(xo)≤ τ⁻¹ ,

estimate (36) with w = u, r = K_p, holds for all p ≥ 1.

Proposition 2.5 is proved exactly like Proposition 2.4.

2.iii The system (6)

Let us look at the problem

u_t − ∆ u^m= v^b, in S_T , (47) v_t− ∆ v^µ= u^α, in S_T , (48) u(x, 0) = u_o(x) , v(x, 0) = v_o(x) , x ∈ IR^N, (49) u_o, v_o ∈ L¹_loc(IR^N), u_o, v_o ≥ 0. Here we assume m, µ ≥ 1, b, α > 0, bα > 1. Definitions 2.1–2.2 carry over to problem (47)–(49), if we set formally a = β = 0.

System (47)–(49) with m = µ = 1 has been studied in [8], where a sufficient and necessary condition for the existence of global solutions has been found. More precisely, a solution defined in IR^N × (0, ∞) to (47)–(49) with m = µ = 1 exists if and only if bα ≤ 1, or bα > 1 and

N (bα − 1)/2(1 + max(b, α)) > 1 (50) (see also [15] for a different proof in the special case b, α ≥ 1).

Here we prove the following results about local and global solv- ability of (47)–(49).

Proposition 2.6. Assume (without loss of generality) b ≥ α.

There exists a constant p_o = p_o(N, m, b, µ, α) ≥ 1 such that if u_o, v_o ∈ L¹_loc(IR^N), u_o, v_o ≥ 0 and |||u_o|||_p, |||v_o|||_q < ∞ with p ≥ po, p/q = ^α+1_b+1, a solution to (47)–(49) exists in STo, where 0 < T_o = T_o(N, m, b, µ, α, p, q, |||u_o|||_p, |||v_o|||_q) < ∞. Moreover for 0 < t < To we have

ku(·, t)k_∞,IRN + v

b+1 α+1(·, t)

∞,IR^N ≤ γW

2

oKp

t^KpN + γW

2 Hq

b+1 α+1

o

t^{HqN α+1b+1} ,

(51)

|||u(·, t)|||^p_p + |||v(·, t)|||^q_q ≤ W_o, (52) where K_p = N (m − 1) + 2p, H_q = N (µ − 1) + 2q, γ = γ(N, m, b, µ, α, p, q) and Wo = |||uo|||^p_p+ |||vo|||^q_q.

Theorem 2.7. Let b ≥ α, and assume

N K

bα−1

b+1 > 1 , _H^N

o

bα−1

α+1 > 1 , (53) K = N (m − 1) + 2, H_o = N (µ − 1) + 2_α+1^b+1. Assume also |||u_o|||_p+

|||v_o|||_q ≤ C < ∞, where u_o, v_o, p, q are chosen as in Proposition 2.6.

Then a constant γo > 0 exists such that if kuok_1,IRN + kvok^b+1

α+1,IR^N ≤ γo, (54) problem (47)–(49) has a solution defined in IR^N × (0, ∞).

Here γ_o = γ_o(N, m, b, µ, α, p, q, C).

Remark 2.10. The dependence of the constants p_o and T_o on the quantities specified in the statement of Proposition 2.6 can be made explicit; we refer to the proof in subsection 3.iii. Here we note that (53) reduces to (50) when m = µ = 1.

Some extensions of Proposition 2.6 and Theorem 2.7 are given in subsection 2.iv.

The following proposition gives information on the behaviour of solutions to (47)–(48) as |x| → ∞ or t → 0, as well as on the behaviour of admissible initial data.

Proposition 2.7. Let (u, v) be a non negative local solution to (47)–(48) in S_T, 0 < T < ∞, and let b ≥ µ, α ≥ m, and b, α > 1.

Then

|||u(·, t)|||₁ , |||v(·, t)|||₁ ≤ γ(N, m, b, µ, α, T ) , 0 < t < T /2 . The proofs of both Proposition 2.6 and Theorem 2.7 rely on the estimates provided by

Proposition 2.8. Let (u, v) be any non negative local subso- lution to (47)–(48) in ST, and let u, v ∈ C^o(ST), 0 < T < ∞.

Then for every ball B_2ρ = B_2ρ(x_o), ρ > 0, x_o ∈ IR^N, and for all 0 < t < T satisfying for all 0 < τ < t

ρ⁻²

u^m−1(·, τ )

∞,B2ρ+ ρ⁻²

v^µ−1(·, τ ) ∞,B2ρ

+ u

bα−1 b+1 (·, τ )

∞,B2ρ

+ v

bα−1 α+1 (·, τ )

∞,B2ρ

≤ τ⁻¹ , (55) we have for all p, q ≥ 1, p/q = ^α+1_b+1, x ∈ B_ρ(x_o),

u(x, t) + v^α+1b+1(x, t) ≤ γt^−KpN W^Kp2 (t) + γt^{−HqN α+1b+1}W^{Hq2 α+1b+1}(t) , (56) where K_p = N (m − 1) + 2p, H_q = N (µ − 1) + 2q, γ = γ(N, m, b, µ, α, p, q), and

W (t) = sup

0<τ <t

R

B2ρ(xo)(u^p(x, τ ) + v^q(x, τ )) dx .

Proposition 2.6 is proved in subsection 3.iii, Theorem 2.7 in subsection 3.iv, Proposition 2.7 in subsection 4.iii, and Proposi- tion 2.8 in subsection 6.i.

2.iv Generalisations Consider the system

u_t− div a(x, t, u, Du^m) = f (x, t, u, v) , in S_T , (57) v_t− div b(x, t, v, Dv^µ) = g(x, t, u, v) , in S_T . (58) We assume for all p, q ∈ IR^N, and a.e. ξ ≡ (x, t, z) ∈ S_T × IR,

a(ξ, p) · p ≥ Λ⁻¹|p|² , |a(ξ, p)| ≤ Λ|p| , a(ξ, p) − a(ξ, q)
· (p − q) ≥ 0 ,

for a given Λ > 1. We also require for all p ∈ IR^N, a.e. (x, t) ∈ S_T, z 7→ a(x, t, z, p) ∈ C^o(IR)
N

, (z, w) 7→ f (x, t, z, w) ∈ C^o(IR²) . The functions b and g are assumed to fulfill similar requirements.

Moreover let

0 ≤ f (x, t, z, w) ≤ Λ|z|^a|w|^b , 0 ≤ g(x, t, z, w) ≤ Λ|z|^α|w|^β , a.e. (x, t, z, w) ∈ ST × IR². Here m, µ ≥ 1, a, β > 1, α, b ≥ 0, unless otherwise specified.

Since the proofs of the local and global existence theorems are based on L^∞-estimates (i.e., propositions 2.4, 2.5 and 2.8) that hold for non negative local subsolutions to (57)–(58) (see, e.g., the proof of Proposition 2.8 in Section 6 below), we have that theorems 2.1, 2.2 hold if (7)–(8) are replaced by (57)–(58), even in the case of initial data measures. (In this connection, note that definitions

2.1–2.2 immediately extend to the present case.) Similar extensions hold for theorems 2.3, 2.5, 2.7, and for Proposition 2.6.

We refer to remarks 3.1, 3.2 for more comments on the proofs of these results. Here we only note that the following gradient estimates are needed.

Proposition 2.9. ([2] Section 7) Let (u, v) be a local non nega- tive subsolution of (57)–(58) in S_T. Let the assumptions of Propo- sition 2.4 be fulfilled. Then for all p, q ≥ 1, and for all t, ρ > 0 satisfying (35) we have

Zt

0

Z

Bρ(xo)

u^p−1|Du^m| dx dτ

≤ γ ρ² t

ϑm

t^Kpp





 sup

0<τ <t

Z

B2ρ(xo)

u^p(x, τ ) dx







1+^m−1_Kp

, (59)

where if m > 1 we take ϑ_m = 0; in the non degenerate case m = 1, ϑ_m is an arbitrary number in (0, 1/2). An obviously analogue estimate for |Dv^µ| also holds. Here γ = γ(N, m, a, b, µ, α, β, p, ϑ_m).

Remark 2.11. Estimate (59) holds for local non negative subsolu- tions u of

u_t − div a(x, t, u, Du^m) = t^−σu^a , σ < 1 , a > 1 , provided the assumptions of Proposition 2.5 are satisfied.

Proposition 2.10. Let (u, v) be a local non negative subsolution to (57)–(58) in S_T, with a = β = 0, bα > 1. Let the assumptions of Proposition 2.8 be fulfilled. Then for p, q > p_o(N, m, b, µ, α) ≥ 1, p/q = ^α+1_b+1, we have for all t, ρ > 0 satisfying (55)

Zt

0

Z

Bρ(xo)

u^p−1|Du^m| dx dτ ≤ γ Ξ(t; (m, p, Kp, b), (µ, q, Hq, α)) , (60) Zt

0

Z

Bρ(xo)

v^q−1|Dv^µ| dx dτ ≤ γ Ξ(t; (µ, q, H_q, α), (m, p, K_p, b)) , (61)

where for r_i, s_i > 0, i = 1, 2, 3, 4, and W (t) defined as in Propo- sition 2.8 we let

Ξ(t; (r₁, r₂, r₃, r₄), (s₁, s₂, s₃, s₄))

= t^r2r3W^1+r1−1r3 (t) + t^{12 1−r4+1s4+1s3N(r1−1)}

W^{1+r1−1s3 r4+1s4+1}(t) . It follows from the proof of Proposition 2.10, that is sketched in subsection 6.ii, that p_o must be chosen so that all the exponents in (60)–(61) are positive.

Remark 2.12. Note that the gradient estimates given here are actually valid for the solutions we find, due to inequalities (19)–

(20), (51)–(52).

Remark 2.13. The existence theorems hold for solutions of vari- able sign to (57)–(58), where we stipulate the convention u^p =

|u|^p−1u, u ∈ IR, p > 0. Indeed all the sup and gradient estimates above continue to hold for such solutions, since they may be proved separately for the negative and positive part of any solution.

Remark 2.14. Let a, b in (57)–(58) take the special form a = A · Du^m, b = B · Dv^µ, where A and B are two (constant) N × N positive definite symmetric real matrices. Then all the results in this paper hold for non negative solutions to (57)–(58). Indeed, the proofs can be reproduced by changing the space variables in each equation separately.

3. Proofs of the existence results 3.i Proof of Theorem 2.1

Consider for n = 1, 2, 3, . . . the sequence of approximating prob- lems

(u_n)_t− ∆ u^m_n = min(n, u^av^b) , in Q_n, (62) (v_n)_t− ∆ v^µ_n = min(n, u^αv^β) , in Q_n, (63) un(x, t) = 0 , vn(x, t) = 0 , |x| = n , t > 0 , (64) un(x, 0) = uon(x) , vn(x, 0) = von(x) , |x| < n ; (65) here Q_n = B_n× (0, ∞), B_n = {|x| < n} ⊂ IR^N, and we define for u_o as in Theorem 2.1

u_on(x) = min(u_o(x), n) , |x| < n , u_on = 0 , |x| ≥ n ; von is defined similarly. A solution to (62)–(65) is a pair (un, vn) of non negative measurable functions u_n, v_n: Q_n −→ IR⁺ satisfy- ing (10)–(12) with Ω = Bn, and u^m, v^µ ∈ L²_loc(0, T ; W_o^1,2(Bn)).

In order to show that a solution (u_n, v_n) exists, we first approxi- mate (62)–(65) by uniformly parabolic problems with smooth data, which can be solved, e.g., by means of Galerkin’s method (see [16]

Chapter V). Then, existence of a solution to (62)–(65) follows from the compactness results of [6]. Moreover u_n, v_n ∈ L^∞(Q_n), and u_n, v_n are continuous in B_n× [ε, ∞), ∀ε > 0.

Existence of solutions to (7)–(9) follows from a priori L^∞ bounds on (u_n, v_n) uniform on n, via a standard reasoning based on the quoted above compactness results ((12) can be proved as in [7]). In this subsection we derive such bounds; for the sake of no- tational simplicity we define u_n ≡ 0, v_n≡ 0 in [IR^N× (0, ∞)] \ Q_n, and drop the subscript n in the following.

Let us define ˜t as the largest time such that

u^m−1(·, t)

∞,IR^N +

v^µ−1(·, t) ∞,IR^N

+

u^a−1(·, t)v^b(·, t)

∞,IR^N +

u^α(·, t)v^β−1(·, t)

∞,IR^N ≤ t⁻¹, (66) for all 0 < t < ˜t. It follows from Proposition 2.4 and (15) that for 0 < t < ˜t, x ∈ IR^N, p, q ≥ 1,

u(x, t) ≤ γt^−KpN Φ^Kp2 (t) , v(x, t) ≤ γt^−HqN Ψ^Hq2 (t) , (67) where Kp = N (m − 1) + 2p, Hq = N (µ − 1) + 2q, and

Φ(t) = sup

0<τ <t

sup

x∈IR^N

ku(·, τ )k^p_p,B1(x) , Ψ(t) = sup

0<τ <t sup

x∈IR^N

kv(·, τ )k^q_q,B1(x) .

Indeed Proposition 2.4 holds for solutions to problem (62)–(65);

similar remarks apply to propositions 2.5 and 2.8–2.10 (this follows from the proofs in [2] and in Section 6 below).

Let ζ ∈ C_o^∞(IR^N) be a cutoff function in B₂(x), ζ ≡ 1 in B₁(x), |Dζ|, | ∆ ζ| ≤ γ(N ), x ∈ IR^N fixed. We use u^p−1ζ² as a testing function in (62), to get for 0 < t < ˜t,

Z

B1(x)

u^p(y, t) dy ≤ γ Z

B2(x)

u^p_ody + γ Zt

0

Z

B2(x)

u^m−1u^pdy dτ

+ γ Zt

0

Z

B2(x)

u^a−1v^bu^pdy dτ

≤ γ|||uo|||^p_p+ γΦ(t)

Z^t

0

τ^{−KpN (m−1)}Φ^{Kp2 (m−1)}(t) dτ

+ Zt

0

τ^{−KpN (a−1)−HqN b}Φ^{Kp2 (a−1)}(t)Ψ^{Hq2 b}(t) dτ



≤ γ|||uo|||^p_p+ γΦ(t)L(t) , (68) where, denoting Φ = Φ(t), Ψ = Ψ(t),

L(t) ≡ t^{1−KpN (m−1)}Φ^{Kp2 (m−1)}+ t^{1−KpN (a−1)−HqN b}Φ^{Kp2 (a−1)}Ψ^{Hq2 b} ; here we use (17). Since x ∈ IR^N in (68) can be chosen arbitrarily, we have in fact

Φ(t) ≤ γ|||u_o|||^p_p+ γΦ(t)L(t) , 0 < t < ˜t . (69) A similar calculation gives

Ψ(t) ≤ γ|||v_o|||^q_q+ γΨ(t)M (t) , 0 < t < ˜t , (70) M (t) ≡ t^{1−HqN (µ−1)}Ψ^{Hq2 (µ−1)}+ t^{1−KpNα−HqN (β−1)}Φ^{Kp2 α}Ψ^{Hq2 (β−1)} (Φ = Φ(t), Ψ = Ψ(t)). Next we define t^∗ = sup{t > 0 | L(t) + M (t) ≤ δ}, for δ > 0 to be chosen.

Note that for t ≤ t^∗, we have for all x ∈ IR^N (denote u = u(x, t), v = v(x, t))

tu^m−1+ tv^µ−1+ tu^a−1v^b+ tu^αv^β−1 ≤ γL(t) + γM (t) ≤ γδ , implying, if δ = δ(N, m, a, b, µ, α, β) is small enough, that t^∗ < ˜t.

Also, (69) and (70) yield, possibly by choosing δ even smaller, Φ(t) ≤ γ|||uo|||^p_p , Ψ(t) ≤ γ|||vo|||^q_q , 0 < t < t^∗. (71)

Estimate (71), together with (67), gives the sought after L^∞- estimates, up to time t^∗. Finally, t^∗ is bounded below by the time T_o defined in (18): indeed this follows by using (71) in the definition of t^∗.

Remark 3.1. The proof given in this subsection covers, with minor changes, the case of the more general system (57)–(58). We still let (u_n, v_n) be a solution to a problem similar to (62)–(65), with the obvious modifications. The only new difficulty is that the second term on the right hand side of (68) is now replaced by an integral containing |Du^m_n| (indeed, a double integration by parts was only made possible in (68) by the special form of (7)). Using (59) to es- timate this integral, the rest of the proof is in practice unchanged.

We remark that the limiting process n → ∞ still yields a solu- tion, by standard arguments, in view of the structure assumptions stipulated on (57)–(58), and of the compactness results of [6].

3.ii Proof of Theorem 2.2

In this section, we still denote by (u, v) a solution to (62)–(65). If we prove a priori L^∞-estimates for all t > 0, existence of global solutions will follow at once, reasoning as in the previous subsec- tion.

Note that in Proposition 2.4 we may let ρ → ∞ to prove that for all ˜t > 0, such that for 0 < t < ˜t

u^a−1(·, t)v^b(·, t)

∞,IR^N +

u^α(·, t)v^β−1(·, t)

∞,IR^N ≤ t⁻¹, (72) it holds for all s ≥ 1, 0 < t < ˜t (here γ = γ(N, m, a, b, µ, α, β, s)),

ku(·, t)k_∞,IRN ≤ γt^−KsN sup

0<ϑ<t

ku(·, ϑ)k

2s Ks

s,IR^N , kv(·, t)k_∞,IRN ≤ γt^−HsN sup

0<ϑ<t

kv(·, ϑ)k

2s Hs

s,IR^N .

(73)

It follows from Theorem 2.1 and its proof that estimates (19)–(20) hold for (u, v) in the strip STo, To> 0 given by (18).

Performing a standard procedure of integration by parts in (62), and using the estimates provided by Theorem 2.1, we find for 0 < t < T_o

ku(·, t)k_1,IRN ≤ ku_ok_1,IRN + γ Zt

0

Z

IR^N

u^a−1v^bu dx dτ

≤ kuok_1,IRN + γU (t)t^{1−KpN (a−1)−HqN b}|||uo|||^p

2 Kp(a−1) p |||vo|||^q

2 Hqb

q ,

(74) where we set

U (t) = sup

0<ϑ<t

ku(·, ϑ)k_1,IRN , V (t) = sup

0<ϑ<t

kv(·, ϑ)k_1,IRN .

Owing to (74) and to a similar estimate of v, and choosing if nec- essary a smaller time T_o, we get for 0 < t < T_o

ku(·, t)k_1,IRN ≤ γ ku_ok_1,IRN , kv(·, t)k_1,IRN ≤ γ kv_ok_1,IRN . (75) Let T be the largest time ˜t for which (72) holds. Of course T ≥ T_o > 0. The L^∞-estimates will follow from (73) and (75), if we show that T = +∞ and that (75) holds for all t > 0. Assume T < +∞ and (75) holds for 0 < t < T . We have by virtue of (73), for all x ∈ IR^N,

u^a−1(x, T )v^b(x, T ) ≤ γT^{−NK(a−1)−NHb}ku_ok_1,IR^K2(a−1)N kv_ok_1,IR^H2bN

≤ ¹₄

T To

−^NK(a−1)−^NHb

T_o⁻¹ ≤ ¹₄

T To

−1

T_o⁻¹ = ¹₄T⁻¹ , provided γ_o in (33) is chosen suitably small (in particular γ_o de- pends on T_o); indeed T ≥ T_o and −^N_K(a − 1) − ^N_Hb < −1.

In the same way, if γ_oin (33) is small enough, u^α(x, T )v^β−1(x, T ) ≤ 1/(4T ), x ∈ IR^N, contradicting the definition of T . Thus, in order to prove T = +∞, it will suffice to show that (75) actually holds for all 0 < t < T .

For any T_o < t < T it follows from (73) ku(·, t)k_1,IRN ≤ ku(·, T_o)k_1,IRN

+ γ Zt

To

τ^{−NK(a−1)−NHb}U (τ )^1+K2(a−1)V (τ )^H2bdτ . (76)