1Introduction JúliusPaˇcuta Aprioriestimatesofglobalsolutionsofsuperlinearparabolicsystems ElectronicJournalofQualitativeTheoryofDifferentialEquations

A priori estimates of global solutions of superlinear parabolic systems

Július Paˇcuta

Comenius University, Mlynská dolina, 84248 Bratislava, Slovakia Received 22 February 2016, appeared 20 April 2016

Communicated by Michal Feˇckan

Abstract. We consider the parabolic system ut−_∆u = u^rv^p, vt−_∆v = u^qv^s in Ω×(0,∞), complemented by the homogeneous Dirichlet boundary conditions and the initial conditions(u,v)(·, 0) = (u0,v0)inΩ, whereΩis a smooth bounded domain in R^N and u0,v0 ∈ L^∞(Ω) are nonnegative functions. We find conditions on p,q,r,s guaranteeing a priori estimates of nonnegative classical global solutions. More pre- cisely every such solution is bounded by a constant depending on suitable norm of the initial data. Our proofs are based on bootstrap in weighted Lebesgue spaces, universal estimates of auxiliary functions and estimates of the Dirichlet heat kernel.

Keywords: parabolic system, a priori estimates, bootstrap.

2010 Mathematics Subject Classification: 35K58, 35B45.

1 Introduction

Superlinear parabolic problems represent important mathematical models for various phe- nomena occurring in physics, chemistry or biology. Therefore such problems have been in- tensively studied by many authors. Beside solving the question of existence, uniqueness, regularity etc. significant effort has been made to obtain a priori estimates of solutions. A priori estimates are important in the study of global solutions (i.e. solutions which exist for all positive times) or blow-up solutions (i.e. solutions whose L^∞-norm becomes unbounded in finite time); superlinear parabolic problems may possess both of these types of solutions.

Uniform a priori estimates also play a crucial role in the study of so-called threshold solutions, i.e. solutions lying on the borderline between global existence and blow-up.

Stationary solutions of parabolic problems are particular global solutions and their a priori estimates are of independent interest since they can be used to prove the existence and/or multiplicity of steady states, for example. The proofs of such estimates are usually much easier than the proofs of estimates of time-dependent solutions. On the other hand, the methods of the proofs of a priori estimates of stationary solutions can often be modified to yield a priori estimates of global time-dependent solutions.

BEmail: julius.pacuta@fmph.uniba.sk

In this paper we study global classical positive solutions of the model problem u_t−_∆u=u^rv^p, (x,t)∈_Ω×(0,∞),

vt−_∆v=u^qv^s, (x,t)∈_Ω×(0,∞), u(x,t) =v(x,t) = 0, (x,t)∈_∂Ω×(0,∞), u(x, 0) =u₀(x)_, x∈_Ω,

v(x, 0) =v₀(x), x∈_Ω,



















(1.1)

where p,q,r,s≥0 and

Ω⊂_R^N is smooth and bounded,u0,v0 ∈ L^∞(_Ω)are nonnegative. (1.2) In this case, sufficient conditions on the exponentsp,q,r,sguaranteeing a priori estimates and existence of positive stationary solutions have been obtained in [3,13,16–19]. In particular, the conditions in [10] are valid for a large class of so-called very weak solutions, and they are optimal in this class. We find sufficient conditions on the exponents guaranteeing uniform a priori estimates of global classical solutions. Our method is in some sense similar to that used in [10] (both methods are based on bootstrap in weighted Lebesgue spaces and estimates of auxiliary functions of the formu^av^1−a; the idea of using such auxiliary functions for elliptic systems seems to go back to a paper [12]) but our proofs are much more involved. In partic- ular, we have to use precise estimates of the Dirichlet heat semigroup and several additional ad-hoc arguments. These difficulties cause that our sufficient conditions are quite technical and probably not optimal. On the other hand, our results are new and our approach is also new in the parabolic setting: Although the bootstrap in weighted Lebesgue spaces has been used many times in the case of superlinear elliptic problems (see the references in [10], for example), it has not yet been used to prove a priori estimates of global solutions of super- linear parabolic problems. In fact, the known methods for obtaining such estimates always require some special structure of the problem and cannot be used for system (1.1) in general.

In addition, our method is quite robust: It can also be used if the problem is perturbed or if we replace the Dirichlet boundary conditions by the Neumann ones, for example.

Next we present our main results concerning problem (1.1). Beside (1.2), we will further assume that

p,q,r,s ≥0; if q=0 then eitherr>1 ors ≤1, (1.3) and we denote byk · k_1,δ the norm in the weighted Lebesgue spaceL¹(_{Ω; dist}(x,∂Ω)dx). Theorem 1.1. Assume(1.2),(1.3)and pq>(r−1)(s−1). Assume also that either

r>1, p>0, p+r< ^N+3

N+1, s+ ² N+1

r−1

p+r−1 < ^N+3 N+1 or

r ≤1, 0< p< ²

N+1, s< ^N+3 N+1.

Let(u,v)be a global solution of problem(1.1). Then there exists C = C(p,q,r,s,Ω,ku₀k_∞,kv₀k_∞) such that

ku(t)k_∞+kv(t)k_∞≤ C, t ≥0.

Theorem 1.2. Assume(1.2),(1.3)and eithermax{r,s}>1or pq >(r−1)(s−1). Assume also p≥1, p+r< ^N+3

N+1, s≤1, (p+r)

p− ² N+1

+r<1 and

0< q< ¹−r p− _N²₊₁

1− ^N−1 N+1s

.

Let(u,v)be a global solution of problem(1.1). Then, givenτ >0, there exists C= C(p,q,r,s,Ω,τ, ku(τ)k_1,δ,kv(τ)k_1,δ)such that

ku(t)k_∞+kv(t)k_∞ ≤C, t ≥τ. (1.4) The constant C may explode ifτ→₀⁺, and is bounded forku(τ)k_1,δ,kv(τ)k_1,δ bounded.

One of the main applications of uniform a priori estimates of global positive solutions of (1.1) is the proof of global existence and boundedness of threshold solutions lying on the borderline between global existence and blow-up. Let us mention that our conditions on p,q,r,s from Theorems 1.1 and1.2 guarantee that both global and blow-up solutions (hence also threshold solutions) of (1.1) exist; see [1,14]. See also [2,15,20] for other results on blow-up of positive solutions of (1.1).

As already mentioned, our approach is quite robust. It can also be used, for example, for the following problem with Neumann boundary conditions

u_t−_∆u=u^rv^p−λu, (x,t)∈ _Ω×(_0,∞)_, v_t−_∆v=u^qv^s−λv, (x,t)∈ _Ω×(0,∞), u_ν(x,t) =v_ν(x,t) = 0, (x,t)∈ _∂Ω×(0,∞),

u(x, 0) =u₀(x), x∈ _Ω, v(x, 0) =v₀(x), x∈ _Ω,



















(1.5)

whereΩ, p,q,r,sandu₀,v₀are as above,λ>0 andνis the outer unit normal on the boundary

∂Ω. The terms −λu,−λv with λ > 0 are needed in (1.5), since otherwise (1.5) cannot admit both global and blow-up positive solutions. Let us also note that in this case one has to work in standard (and not weighted) Lebesgue spaces and that the restrictions on the exponents p,q,r,s are less severe than in the case of Dirichlet boundary conditions: roughly speaking, one can replace N with N−1 in those restrictions (in particular, the condition p+r < ^N_N⁺₊³₁ becomes p+r < ^N_N⁺² in this case). As other particular application of our method, we present the following theorem.

Theorem 1.3. Consider problem

u_t−_∆u=uv−b₁u, (x,t)∈_Ω×(0,∞), v_t−_∆v =b₂u, (x,t)∈_Ω×(0,∞), u(x,t) =v(x,t) = 0, (x,t)∈_∂Ω×(0,∞), u(x, 0) =u₀(x), x∈_Ω,

v(x, 0) =v0(x), x∈_Ω,



















(1.6)

where Ωis a bounded domain with smooth boundary, N ≤ 2, b₁ ≥ 0, b₂ > 0and u₀,v₀ ∈ L^∞(_Ω). Then there exists C= C(_Ω,b₁,b₂)such that

lim sup

t→_∞

(ku(t)k_∞+kv(t)k_∞)≤C for every global nonnegative solution(u,v)of problem(1.6).

More detailed proofs of Theorems1.1–1.3can be found in [9].

If r = s = 0 and p,q > 1, then a very easy argument in [6] yields a universal estimate of ku(τ)k_1,δ,kv(τ)k_1,δ for all τ ≥ 0, hence Theorem 1.2 guarantees estimate (1.4) with C = C(p,q,Ω,τ). The same estimate was obtained in [6] under the assumption p,q ∈ 1,^N_N⁺₊³₁ which is different from that in Theorem 1.2 (we do not require q < ^N_N⁺₊³₁, for example). Of course, ifr=s =0, then one could also use different methods for obtaining a priori estimates, e.g. the parabolic Liouville theorems in [5] together with scaling and doubling arguments to prove qualitative universal estimates. The main advantage of our results and proofs is the fact that we do not need the assumptionr=s =0.

2 Preliminaries

We introduce some notation we will use frequently. Denote δ(x) = dist(x,∂Ω) for x ∈ _Ω, and for 1 ≤ p ≤ _∞ define the weighted Lebesgue spaces L_δ^p = L^p_δ(_Ω) := L^p(_Ω;δ(x) dx). If 1 ≤ p < ∞, then the norm in L_δ^p is defined by kuk_p,δ = R

Ω|u(x)|^pδ(x)dx1/p

. Recall that L^∞_δ = L^∞(_Ω;δ(x) dx) _with kuk_∞,δ = kuk_∞. We will use the notation k · k_p for the norm in L^p(_Ω)for p∈[1,∞), as well.

Letλ₁ be the first eigenvalue of the problem

−_∆φ=λφ, x ∈_Ω, φ=0, x ∈_∂Ω,

)

andϕ₁to be the corresponding positive eigenfunction satisfyingkϕ₁k₂ =1. There holds C(_Ω)δ(x)≤ ϕ₁(x)≤ C⁰(_Ω)δ(x) _{for all}x ∈_{Ω. (2.1)} Therefore the norm kuk_p,ϕ1 = R

Ω|u(x)|^pϕ₁(x)dx1/p

is equivalent to the norm kuk_p,δ in L^p_δ(_Ω)for 1≤ p<_∞.

Let(u,v)be a solution of system (1.1). Then (u,v)solves the system of integral equations u(t) =e^t∆u0+

Z _t

0 e^(t−s0)∆u^rv^p(s⁰)ds⁰, v(t) =e^t∆v0+

Z _t

0 e^(t−s0)∆u^qv^s(s⁰)ds⁰ (2.2) where t ≥ 0 and e^t∆

t≥0 is the Dirichlet heat semigroup in Ω. In the following lemma we recall some basic properties of the semigroup e^t∆

t≥0, which we will use often. The corresponding proofs can be found e.g. in [6].

Lemma 2.1. LetΩbe arbitrary bounded domain.

(i) Ifφ∈ L¹_δ(_Ω), φ≥0, then e^t∆φ≥0.

(ii) ke^t∆φk_1,ϕ1 =e^−λ1tkφk_1,ϕ1 for t≥0,φ∈L¹_δ(_Ω).

(iii) If p∈(_1,∞], thenke^t∆φk_p,δ ≤C(_Ω)e^−λ1tkφk_p,δ for t≥0,φ∈ L_δ^p(_Ω).

(iv) LetΩbe of the class C². For1≤ p< q≤ ∞, there exists constant C =C(_Ω)such that, for all φ∈ L_δ^p(_Ω), it holds

ke^t∆φk_q,δ ≤C(_Ω)t^−N

+1 2

1 p−¹_q

kφk_p,δ, t >0.

Assertions (iii) and (iv) from Lemma2.1for 1≤ p <q≤_∞,t>0 andε∈(0, 1)imply ke^t∆φk_q,δ ≤C(_Ω)e^−λ1εt((1−ε)t)^−N

+1 2

1 p−¹_q

kφk_p,δ, φ∈ L^p_δ(_Ω).

If we multiply the equations in (2.2) by ϕ₁ and integrate onΩ, then assertions (i) and (ii) from Lemma2.1imply

ku(t)k_1,ϕ1 ≥e^−λ1tku₀k_1,ϕ1, kv(t)k_1,ϕ1 ≥e^−λ1tkv₀k_1,ϕ1. (2.3) We will also use the following estimate of the semigroup e^t∆

t≥0; see e.g. [11].

Lemma 2.2. LetΩbe smooth bounded domain. For every f ∈ L¹_δ(_Ω), f ≥0, there holds (e^t∆f)(x)≥C(t)δ(x)kfk_1,δ, x∈_Ω,

where the constant C may be arbitrarily small if t→0⁺, and is positive for t bounded.

Let (u,v) be a solution of system (1.5). Then (u,v) solves the system of integral equa- tions similar to (2.2) with e^tL instead of e^t∆, where e^tL := e^−λte^t∆N, t ≥ 0 is the semigroup corresponding to operator L:=_∆−λwith homogeneous Neumann boundary condition and

e^t∆N

t≥0 is the Neumann heat semigroup inΩ. For the Neumann semigroup, estimates sim- ilar to those from Lemma 2.1 are true; see [4,8]. One can also obtain inequalities similar to (2.3) with ϕ₁ replaced by 1 andλ₁replaced byλ.

In the following we will use the notation from [10]. We set A:=

([ar,as]∩(0, 1) if pq≥(r−1)(s−1)or min{r,s} ≤1, [a_s,a_r]∩(0, 1) if pq<(r−1)(s−1)andr,s >1, where

ar:= ( _r−1

p+r−1 ifr>1,

0 ifr≤1, as:= ( _q

q+s−1 ifs >1,

1 ifs ≤1.

Note that the set Ais nonempty provided there holds

if p=0, then eithers >1 orr ≤1, ifq=0, then eitherr >1 ors≤1.

)

(2.4) The following lemma is an adaptation of [10, Lemma 7] to systems (1.1) and (1.5):

Lemma 2.3. Assume p,q,r,s ≥ 0, pq 6= (1−r)(1−s) and(2.4). For given a ∈ A, there exists κ ≥0and C =C(p,q,r,s,a)such that any global nonnegative solution of (1.1)satisfies

(u^av^1−a)_t−_∆(u^av^1−a)≥F_a(u,v)≥C(u^av^1−a)^κ, t∈ (0,∞), (2.5) where

F_a(u,v)_:=au^a−1v^1−a(u_t−_∆u) + (₁−a)u^av^−a(v_t−_∆v)

=au^r+a−1v^p+1−a+ (1−a)u^q+av^s−a, t ∈(0,∞). Similarly, for any global nonnegative solution of (1.5), there holds

(u^av^1−a)_t−_∆(u^av^1−a) +λ(u^av^1−a)≥C(u^av^1−a)^κ, t ∈(0,∞). (2.6) If

max{r,s}>1 or pq>(r−1)(s−1), (2.7) thenκ>1.

Let (u,v) be a global nonnegative solution of system (1.1). Denote w = w(t) := R

Ωu^av^1−a(t)ϕ₁ dx. The following estimates are based on ideas from [7]. Let a ∈ A and condition (2.7) be true (thenκ >1). Then due to Lemma2.3and due to Jensen’s inequality, it holds

w_t+λ₁w≥C Z

Ωu^aκv^(1−a)κ(t)ϕ₁ dx≥ Cw^κ, t ∈(0,∞), (2.8) whereC= C(_Ω,p,q,r,s,a)is independent ofw. Sincewis global and satisfies the inequality (2.8) for allt>0, it holds

w(t) =

Z

Ωu^av^1−a(t)ϕ₁dx ≤ λ₁

C _κ−¹₁

for allt ≥0 anda∈ A. (2.9) Lemma2.3also implies

w_t(s⁰) +λ₁w(s⁰)≥C Z

Ωu^r+a−1v^p+1−a(s⁰)ϕ₁dx, s⁰ ∈ (0,∞). (2.10) Multiplying inequality (2.10) bye^λ1s0, integrating on interval[0,t]with respect to s⁰ and using 0≤w≤C, we deduce that

Z _t

0

e^{−λ1(t−s0)} Z

Ωu^r+a−1v^p+1−a(s⁰)ϕ₁dxds⁰ ≤C. (2.11) Since there holdse^{−λ1(t−s0)} ≥e^−λ1tfors⁰ ∈[0,t], the ineqality (2.11) implies

Z _t

0

Z

Ωu^r+a−1v^p+1−a(s⁰)ϕ₁ dxds⁰ ≤Ce^λ1t ≤C⁰, (2.12) whereC⁰ =C⁰(_Ω,p,q,r,s,a,t).

Let (u,v) be a global nonnegative solution of system (1.5). Since (u,v) satisfies homo- geneous Neumann boundary conditions, so does u^av^1−a and hence Green’s formula implies R

Ω∆(u^av^1−a(t))dx =0 fort ≥0 anda∈ A. We obtain estimates similar to (2.9), (2.11), (2.12) with ϕ₁,λ₁ replaced by 1,λ, respectively, in (2.9), (2.11), (2.12) if (2.7) is true.

3 Proofs of Theorems 1.1–1.3

In the following proofs, every constant may depend onΩ,p,q,r,s, however we do not denote this dependence. The constants may vary from step to step.

For 0< p< _N²₊₁, r≤1 denote

Kb:

1, p+1 p

−→_R∪ {_∞},

Kb(M) =







M(p+1)(N+1)

(p+1)(N+1)−2M, M ∈^h1,^(p+1)(₂^N+1) ,

∞, M ∈^h(p+1)(₂^N+1), ^p+_p¹ , bk:

1, p+1 p

−→_R, bk(M) = ^M(p+r)

M−(M−1)(p+1)^.

(3.1)

For r>1, p+r < ^N_N⁺₊³₁ denote K⁰ :

1, p+r p+r−1

−→_R∪ {_∞}, K⁰(M) =







M(p+r)(N+1)

(p+r)(N+1)−2M, M∈^h1,^(p+r)(₂^N+1) ,

∞, M∈^h(p+r)(₂^N+1),_p+^p+_r−^r₁ , k⁰ :

1, p+r p+r−1

−→_R, k⁰(M) = ^M(p+r)

M−(M−1)(p+r)^.

(3.2)

Observe that

Kb(M)>max{M,bk(M)} for all M ∈

1, p+1 p

, (3.3)

since p < _N²₊₁ and

K⁰(M)>k⁰(M)> M for all M∈

1, p+r p+r−1

, (3.4)

since p+r< ^N_N⁺₊³₁.

Lemma 3.1. Let p+r < ^N_N⁺₊³₁, p > 0 and conditions (2.4), (2.7) be true. Let (u,v) be a global nonnegative solution of problem(1.1).

(i) Assume r>1. Then forγ∈ [p+r,∞]and T≥0, there exists C =C(p,q,r,s,Ω,T)such that sup

s⁰∈[0,T]

ku(s⁰)k_γ,δ ≤Cku0k_γ,δ.

(ii) Assume r > _1, pq> (r−1)(s−1)or r≤1, p< _N²₊₁. Then for γ∈ max{1,p+r},^N_N⁺₊³₁ , there exists C=C(p,q,r,s,Ω)such that

sup

s⁰∈[0,T]

ku(s⁰)k_γ,δ ≤C(1+ku₀k_γ,δ), T≥0.

(iii) Assume r ≤ 1, p < _N²₊₁. Then for γ ∈ [max{1,p+r},∞] and T ≥ 0, there exists C = C(p,q,r,s,Ω,T)such that

sup

s⁰∈[0,T]

ku(s⁰)k_γ,δ ≤C(1+ku₀k_γ,δ).

Remark 3.2. In the assertion (i) of Lemma3.1, the constantCis bounded for Tbounded.

Proof. Let γ ∈ max{1,p+r},^N_N⁺₊³₁

, a ∈ A and ε ∈ 0, 1− _p+^p_1−a. Denote κ := ^(p+_p+^r)(₁₋¹⁻_a^a). For T≥0,t∈[0,T]we estimate

ku(t)k_γ,δ ≤C

ku₀k_γ,δ+

Z _t

0 e^−λ1

p

p+1−a+ε

(t−s⁰)

(t−s⁰)^−N2+1(1−¹

γ)ku^rv^p(s⁰)k_1,δ ds⁰

≤C

ku₀k_γ,δ+

Z _t

0

Z

Ω

e^−λ1

_p

p+1−a

(t−s⁰)

u^r−κv^p(s⁰)

f u^κ(s⁰)ϕ₁dxds⁰

where f = f(s⁰):= e^{−λ1ε(t−s0)}(t−s⁰)^−N+21(1−¹_γ). Now, using Hölder’s inequality we obtain ku(t)k_γ,δ ≤C

"

ku₀k_γ,δ+ _{Z t}

0 gds⁰

_p+^p_{1−a Z t}

0 f^p+1−1−aaku^p+r(s⁰)k_1,δ ds⁰

_p+¹⁻₁₋^a_a# , whereg= g(s⁰):=e^{−λ1(t−s0)}ku^r+a−1v^p+1−a(s⁰)k_1,δ. We use (2.11) to estimate

ku(t)k_γ,δ ≤C

"

ku₀k_γ,δ+I^p+1−1−aa sup

s⁰∈[0,T]

ku(s⁰)k_γ,δ

!κ#

, (3.5)

where I = I(t):=Rt

0e^{−λ1εp+1−1−aa(t−s0)}(t−s⁰)^−N+21(1−¹_γ)^p+₁₋¹⁻_a^a _ds0.

We prove that the functionI is finite in[_0,∞), i.e. due to our assumptions onp,q,r,s, there holds

N+1 2

1− ¹

γ

p+1−a

1−a <1 (3.6)

for somea∈ A.

In fact, in the following proof we will choose a= ^r−1

p+r−1 in case (i), (3.7)

a> ^r−1

p+r−1 sufficiently close to r−1

p+r−1 in case (ii) forr >1, (3.8) a>0 sufficiently small in case (iii) or (ii) forr ≤1. (3.9) The choice (3.8) is possible, since due to the assumptions pq > (r−1)(s−1) and p > 0, we have a ∈ A. If a is defined by (3.7) or (3.8) then ^p+₁₋¹⁻_a^a is close to p+r and condition p+r < ^N_N⁺₊³₁ implies the inequality (3.6). Ifa is defined by (3.9), then ^p+₁₋¹⁻_a^a is close to p+₁ and condition p< _N²₊₁ implies the inequality (3.6). Note that the function I is bounded by a constant independent ofT.

First we prove (ii). In the estimate (3.5) we chooseadefined by (3.8), ifr>1, or by (3.9), if r≤1. In both cases we haveκ<1, hence the assertion (ii) follows from Young’s inequality.

Assertion (iii) forγ∈max{1,p+r},^N_N⁺₊³₁

follows from assertion (ii).

To prove (i) forγ∈p+r,^N_N⁺₊³₁

we chooseadefined by (3.7) in estimate (3.5). Thenκ =1 and the assertion (i) forγ∈ p+r, ^N_N⁺₊³₁

andT small enough follows from the estimate (3.5).

The assertion (i) forγ∈p+r,^N_N⁺₊³₁

actually holds for everyT ≥0.

Now we prove the assertion (i) for γ ∈ ^N_N⁺₊³₁,∞

. Fix K ∈ ^N_N⁺₊³₁,∞

. Then there exists M∈ ^h1,^(p+r)(₂^N+1)i

such thatK⁰(M)>K >k =k⁰(M)(where functionsK⁰, k⁰ are defined by (3.2)). Fort∈ [0,T]anda defined by (3.7) we estimate

ku(t)k_K,δ ≤C

ku₀k_K,δ+

Z _t

0

(t−s⁰)^−N+21(_M¹⁻_K¹)ku^rv^p(s⁰)k_M,δ ds⁰

. (3.10)

Observe that M < ^p+_p^1−a, since M ≤ ^(p+r)(₂^N+1) < _p+^p+_r−^r₁ (the last inequality is true due to the assumptionp+r< ^N_N⁺₊³₁). Hence Hölder’s inequality yields

ku^rv^p(s⁰)k_M,δ = _Z

Ω

h

u^{pMpr++a1−−1a}v^pM(s⁰)^{i h}u^Mκ(s⁰)ⁱϕ₁dx _M¹

≤ _Z

Ωu^r+a−1v^p+1−a(s⁰)ϕ₁ dx

_p+^p_1−aZ

Ωu^k(s⁰)ϕ₁dx

^p_M⁺₍¹_p⁻₊^a₁⁻₋^pM_a) ,

(3.11)

since k= M_p⁽₊^p+₁₋^r)(_a¹₋⁻_pM^a) due to our choice ofa. We use (3.11) and Hölder’s inequality to obtain ku(t)k_K,δ ≤C

"

ku₀k_K,δ+ sup

s⁰∈[0,T]

ku(s⁰)k_k,δ

!κ_{Z t}

0

Z

Ωu^r+a−1v^p+1−a(s⁰)ϕ₁dxds⁰ _p+^p_1−a

J

# , where

J = J(t):= _{Z t}

0

(t−s⁰)^−N+21(_M¹⁻_K¹)^p+₁₋¹⁻_a^a _ds0

_p+¹⁻₁₋^a_a .

Notice that κ is equal to 1. Observe that I0 is finite on[_0,∞)_{, since} ^N₂⁺¹ _M¹ − _K^{1 p+}₁₋¹⁻_a^a < _1.

This follows from the definition (3.2) of functionK⁰ and our choice ofK. Sincek <K, we can use (2.12) to obtain

ku(t)k_K,δ ≤C

"

ku₀k_K,δ+C(T) sup

s⁰∈[0,T]

ku(s⁰)k_K,δ

# . This estimate implies the assertion (i) with γ ∈ ^N_N⁺₊³₁,∞

for T small, hence this assertion is actually true for everyT ≥0.

IfM ∈^(p+r)(₂^N+1), _p+^p+_r−^r₁

, then we can chooseK =_∞andk ∈(k⁰(M),∞).

The proof of the assertion (iii) forγ ∈ ^N_N⁺₊³_1,∞is similar to the proof of the assertion (i) forγ∈^N_N⁺₊³₁,∞

. One would use (3.1), (3.3) instead of (3.2), (3.4).

Lemma 3.3. Let p+r < ^N_N⁺₊³₁, p > 0 and conditions (2.4), (2.7) be true. Let (u,v) be a global nonnegative solution of problem(1.1).

(i) Assume r > 1. Then for γ ∈ 1,₂₋₍¹_p+r)i

and T ≥ 0, there exists C = C(p,q,r,s,Ω,T)such that

Z _T

0

ku(s⁰)k_γ,δ ds⁰ ≤ Cku(T)k_1,δ. (ii) Assume r ≤ 1, p+r > 1. Then for γ ∈ 1,₂₋₍¹_p+r)i

and T ≥ 0, there exists C = C(p,q,r,s,Ω,T)such that

Z _T

0

ku(s⁰)k_γ,δ ds⁰ ≤C(1+ku(T)k_1,δ). If p+r≤1, then this estimate is true forγ∈1, ^N_N⁺₋¹₁

. Proof. We define exponentγ = ¹

2−(p+r). The conditions 1 < p+r < ^N_N⁺₊³₁ imply p+r < γ <

N+1

N−1. For T≥0 andt∈ (0,T]we estimate ku(t)k_γ,δ ≤ C

t^−N+21(1−¹_γ)ku₀k_1,δ+

Z _t

0

(t−s⁰)^−N+21(1−¹_γ)ku^rv^p(s⁰)k_1,δ ds⁰

.

Integrating this estimate on interval [0,T] with respect to t and using Fubini’s theorem we obtain

Z _T

0

ku(t)k_γ,δ dt≤CT^1−N+21(1−¹_γ)ku0k_1,δ+

Z _T

0

ku^rv^p(s⁰)k_1,δ ds⁰

. (3.12)

Note that ^N+₂¹ 1− ¹

γ

<1, sinceγ< ^N_N⁺₋¹₁.

As in the proof of Lemma3.1we use (3.11) with M =1,k= p+r to obtain Z _T

0

ku(t)k_γ,δ dt≤C

"

ku0k_1,δ+C(T) Z _T

0

ku(s⁰)k^p_p⁺₊^r_r,δ ds⁰

_p+¹⁻₁₋^a_a#

. (3.13)

Notice that ^γ(p_γ⁺₋^r₁⁻¹⁾ =1. We use the interpolation inequality ku(s⁰)k^p_p⁺₊^r_r,δ ≤ ku(s⁰)k

γ−(p+r) γ−1

1,δ ku(s⁰)k

γ(p+r−1) γ−1

γ,δ , s⁰ ∈[0,T] and Young’s inequality to deduce

Z _T

0

ku(t)k_γ,δ dt≤C(T)



ku₀k_1,δ+ _sup

s⁰∈[0,T]

ku(s⁰)k_1,δ

!β

,

whereβ= ^γ−(_γ−^p+₁^r)1−_p^a. Using this estimate we are ready to prove the assertions of the Lemma.

First we prove the assertion (i). Ifr >1, then we choosea= _p+^r−_r−¹₁ in the definition ofβ, hence β=1. Finally, we use (2.3) to obtain the assertion (i).

To prove the assertion (ii) for p+_r > 1, we choose arbitrarya ∈ Ain the definition of β, henceβ<1. One can use Young’s inequality to obtain the assertion.

If p+r ≤ 1, then for γ ∈ 1, ^N_N⁺₋¹₁

we obtain estimate similar to (3.13) a then we use Young’s inequality. The proof of Lemma3.3 is complete.

In Lemma3.4 we will use the following notation. Forr>_{1 denote} K⁰₀:

1, p+r p+r−₁

−→_R∪ {_∞}, K₀⁰(M) =







M(N+1)

(N+1)−2M, M∈ 1,^N+₂¹ ,

∞, M∈ ^hN+₂¹,_p+^p+_r−^r₁ .

(3.14)

Lemma 3.4. Let p+r < ^N_N⁺₊³₁, p > 0 and conditions (2.4), (2.7) be true. Let (u,v) be a global nonnegative solution of problem(1.1).

(i) Assume r >1. Then for T≥0, there exists C=C(p,q,r,s,Ω,T)such that Z _T

0

ku(s⁰)k_K,δ ds⁰ ≤Cku₀k_k,δ for K⁰₀(M)>K> k=k⁰(M), M ∈1,^N+₂¹

. If M∈ ^N+₂¹,_p+^p+_r−^r₁

, then we can take K=_∞.

(ii) Assume r≤1,_N²₊₁ > p. Then for T≥0, there exists C=C(p,q,r,s,Ω,T)such that Z _T

0

ku(s⁰)k_K,δ ds⁰ ≤C(1+ku₀k_max{M,k},δ) for K₀(M) > K > k > bk(M), k ≥ 1, M ∈ 1,^N+₂¹

. If M ∈ ^N+₂¹, ^p+_p¹

, then we can take K= _∞.

Proof. We chooseaas follows a= ^r−1

p+r−1 for part (i), (3.15)

a>0 sufficiently close to 0 for part (ii). (3.16) We only prove (i), since the proof of (ii) is similar. Observe that ^N₂⁺¹ < _p+^p+_r−^r₁ and K⁰₀(M) >

K⁰(M)for everyM ∈1, _p+^p+_r−^r₁

due to conditions 1< p+r< ^N_N⁺₊³₁ (see the definition (3.2) of functionsK⁰, k⁰ and the definition (3.14) ofK₀⁰). Hence (3.4) implies

K₀⁰(M)>k⁰(M)> M for all M∈

1, p+r p+r−1

. (3.17)

Let K⁰₀(M) > K > k = k⁰(M), M ∈ 1, ^N₂⁺¹

, T ≥ 0 and t ∈ (0,T]. Then, there holds

N+1

2 1

M − _K¹<1. As in the proof of Lemma 3.3we obtain Z _T

0

ku(t)k_K,δ dt≤CT^1−N+21(_M¹⁻¹_K)ku₀k_M,δ+

Z _T

0

ku^rv^p(s⁰)k_M,δ ds⁰

.

Using Lemma 3.1 (i), (3.11) and similar arguments as in the proof of Lemma 3.1 (with a is defined by (3.15)) we have

Z _T

0

ku(t)k_K,δ dt ≤C(T) (ku₀k_M,δ+ku₀k_k,δ)≤C(T)ku₀k_k,δ, (3.18) since k> M.

If M ∈ ^N+₂¹,_p+^p+_r−^r₁

, then in previous estimates, we can chooseK = _∞ andk⁰(M)< k <

∞. Hence we proved (i).

Lemma 3.5. Let p+r < ^N_N⁺₊³₁, p > 0 and conditions (2.4), (2.7) be true. Let (u,v) be a global nonnegative solution of problem(1.1).

(i) Assume r>1. Then for everyτ>0, there exists C=C(p,q,r,s,Ω,τ)such that ku(t)k_∞ ≤Cku(t)k_1,δ

for every t≥τ.

(ii) Assume r≤1,p< _N²₊₁. Then for everyτ>0, there exists C=C(p,q,r,s,Ω,τ)such that ku(t)k_∞ ≤C(1+ku(t)k_1,δ)

for every t≥τ.

Remark 3.6. The constantCfrom both assertions of Lemma3.5may explode if τ→0⁺. Proof. We prove only (i). Letγ= ¹

2−(p+r). Conditions 1< p+r< ^N_N⁺₊³_{1 imply}p+r< γ< ^N_N⁺₋¹_1. Fix 1> τ₀ > 0 and lett > 0 be arbitrary. Note that there existsτ⁰ ∈ [τ₀+t, 2τ₀+t]such that ku(τ⁰)k_γ,δ = τ₀⁻¹R2τ0+t

τ0+t ku(s⁰)k_γ,δ ds⁰. Obviously, this τ⁰ may depend on t and u. Note that 2τ₀+t∈ [τ⁰,τ⁰+τ₀]. We use Lemma3.3(i) and Lemma3.1(i) to obtain

sup

s⁰∈[τ⁰,τ⁰+τ0]

ku(s⁰)k_γ,δ ≤Cku(τ⁰)k_γ,δ ≤C Z _2τ0+t

τ₀+t

ku(s⁰)k_γ,δ ds⁰ ≤Cku(2τ₀+t)k_1,δ