A Priori Estimates of Global Solutions of Superlinear Parabolic Systems

We consider the parabolic system u t − ∆u = u r v p , v t − ∆v = u q v s in Ω × (0, ∞), complemented by the homogeneous Dirichlet boundary conditions and the initial conditions (u, v)(·, 0) = (u 0 , v 0) in Ω, where Ω is a smooth bounded domain in R N and u 0 , v 0 ∈ L ∞ (Ω) are nonnegative functions. We find conditions on p, q, r, s guaranteeing a priori estimates of nonnegative classical global solutions. More precisely 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.


Introduction
Superlinear parabolic problems represent important mathematical models for various phenomena occurring in physics, chemistry or biology.Therefore such problems have been intensively 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.

J. Pačuta
In this paper we study global classical positive solutions of the model problem u t − ∆u = u r v p , (x, t) ∈ Ω × (0, ∞), v t − ∆v = u q v s , (x, t) ∈ Ω × (0, ∞), u(x, t) = v(x, t) = 0, (x, t) ∈ ∂Ω × (0, ∞), where p, q, r, s ≥ 0 and Ω ⊂ R N is smooth and bounded, u 0 , v 0 ∈ L ∞ (Ω) are nonnegative. (1.2) In this case, sufficient conditions on the exponents p, q, r, s guaranteeing a priori estimates and existence of positive stationary solutions have been obtained in [3,13,[16][17][18][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 form u a v 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 particular, 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 superlinear 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 either r > 1 or s ≤ 1, ( and we denote by • 1,δ the norm in the weighted Lebesgue space L 1 (Ω; dist(x, ∂Ω) dx).
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 and 1.2 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 where Ω, p, q, r, s and u 0 , v 0 are as above, λ > 0 and ν is 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+3 N+1 becomes p + r < N+2 N in this case).As other particular application of our method, we present the following theorem.
where Ω is a bounded domain with smooth boundary, N ≤ for every global nonnegative solution (u, v) of problem (1.6).
More detailed proofs of Theorems 1.1-1.3can be found in [9].If r = s = 0 and p, q > 1, then a very easy argument in [6] yields a universal estimate of u(τ) 1,δ , v(τ) 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+3 N+1 which is different from that in Theorem 1.2 (we do not require q < N+3 N+1 , for example).Of course, if r = 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 assumption r = s = 0.

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 We will use the notation • p for the norm in L p (Ω) for p ∈ [1, ∞), as well.
Let λ 1 be the first eigenvalue of the problem and ϕ 1 to be the corresponding positive eigenfunction satisfying ϕ 1 2 = 1.There holds Therefore the norm u p,ϕ 1 = Ω |u(x)| p ϕ 1 (x) dx 1/p is equivalent to the norm u p,δ in Let (u, v) be a solution of system (1.1).Then (u, v) solves the system of integral equations 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].
Assertions (iii) and (iv) from Lemma 2.1 for 1 ≤ p < q ≤ ∞, t > 0 and ε ∈ (0, 1) imply If we multiply the equations in (2.2) by ϕ 1 and integrate on Ω, then assertions (i) and (ii) from Lemma 2.1 imply We will also use the following estimate of the semigroup e t∆ t≥0 ; see e.g.[11].Lemma 2.2.Let Ω be smooth bounded domain.For every f ∈ L 1 δ (Ω), f ≥ 0, there holds 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 equations similar to (2.2) with e tL instead of e t∆ , where e tL := e −λt e t∆ N , t ≥ 0 is the semigroup corresponding to operator L := ∆ − λ with homogeneous Neumann boundary condition and e t∆ N t≥0 is the Neumann heat semigroup in Ω.For the Neumann semigroup, estimates similar to those from Lemma 2.1 are true; see [4,8].One can also obtain inequalities similar to (2.3) with ϕ 1 replaced by 1 and λ 1 replaced by λ.
In the following we will use the notation from [10].We set where Note that the set A is nonempty provided there holds if p = 0, then either s > 1 or r ≤ 1, if q = 0, then either r > 1 or s ≤ 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 = (1 − r)(1 − s) and (2.4).For given a ∈ A, there exists κ ≥ 0 and C = C(p, q, r, s, a) such that any global nonnegative solution of (1.1) satisfies where Similarly, for any global nonnegative solution of (1.5), there holds then κ > 1.
Let (u, v) be a global nonnegative solution of system (1.1).Denote w = w(t) := Ω u a v 1−a (t)ϕ 1 dx.The following estimates are based on ideas from [7].Let a ∈ A and condition (2.7) be true (then κ > 1).Then due to Lemma 2.3 and due to Jensen's inequality, it holds where C = C(Ω, p, q, r, s, a) is independent of w.Since w is global and satisfies the inequality (2.8) for all t > 0, it holds for all t ≥ 0 and a ∈ A. (2.9) Lemma 2.3 also implies Multiplying inequality (2.10) by e λ 1 s , integrating on interval [0, t] with respect to s and using 0 ≤ w ≤ C, we deduce that Since there holds where C = C (Ω, p, q, r, s, a, t).Let (u, v) be a global nonnegative solution of system (1.5).Since (u, v) satisfies homogeneous Neumann boundary conditions, so does u a v 1−a and hence Green's formula implies Ω ∆(u a v 1−a (t)) dx = 0 for t ≥ 0 and a ∈ A. We obtain estimates similar to (2.9), (2.11), (2.12) with ϕ 1 , λ 1 replaced by 1, λ, respectively, in (2.9), (2.11), (2.12) if (2.7) is true.

Proofs of Theorems 1.1-1.3
In the following proofs, every constant may depend on Ω, p, q, r, s, however we do not denote this dependence.The constants may vary from step to step.
In fact, in the following proof we will choose a > 0 sufficiently small in case (iii) or (ii) for r ≤ 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−a 1−a is close to p + r and condition p + r < N+3 N+1 implies the inequality (3.6).If a is defined by (3.9), then p+1−a 1−a is close to p + 1 and condition p < 2 N+1 implies the inequality (3.6).Note that the function I is bounded by a constant independent of T.
First we prove (ii).In the estimate (3.5) we choose a defined by (3.8), if r > 1, or by (3.9), if r ≤ 1.In both cases we have κ < 1, hence the assertion (ii) follows from Young's inequality.

J. Pačuta
As in the proof of Lemma and Young's inequality to deduce where Using this estimate we are ready to prove the assertions of the Lemma.First we prove the assertion (i).If r > 1, then we choose a = r−1 p+r−1 in the definition of β, hence β = 1.Finally, we use (2.3) to obtain the assertion (i).
To prove the assertion (ii) for p + r > 1, we choose arbitrary a ∈ A in the definition of β, hence β < 1.One can use Young's inequality to obtain the assertion.
If p + r ≤ 1, then for γ ∈ 1, N+1 N−1 we obtain estimate similar to (3.13) a then we use Young's inequality.The proof of Lemma 3.3 is complete.
In Lemma 3.4 we will use the following notation.For r > 1 denote (i) Assume r > 1.Then for T ≥ 0, there exists C = C(p, q, r, s, Ω, T) such that p+r p+r−1 , then we can take K = ∞.
Proof.We choose a as follows a > 0 sufficiently close to 0 for part (ii). (3.16) We only prove (i), since the proof of (ii) is similar.Observe that N+1

2
< p+r p+r−1 and K 0 (M) > K (M) for every M ∈ 1, p+r p+r−1 due to conditions 1 < p + r < N+3 N+1 (see the definition (3.2) of functions K , k and the definition (3.14) of K 0 ).Hence (3.4) implies , T ≥ 0 and t ∈ (0, T].Then, there holds As in the proof of Lemma 3.3 we obtain 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 since k > M.
Proof.We use Lemmas 3.9 and 3.10 and arguments as in the proof of Lemma 3.5 to obtain

J. Pačuta
Hence we obtain To finish the proof we use bootstrap argument.