Global existence in reaction–diffusion systems with mass control under relaxed assumptions merely referring to cross-absorptive effects

We introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.


Introduction
Reaction-diffusion equations arise in various applications in chemistry and biology (cf. [20,Ch. 2]) and form an important class of model problems in the study of systems of parabolic equations (see [24,Ch. 33]). Already at the stage of basic theories of solvability, a major challenge for the analysis of such systems consists in the presence of commonly superlinear source terms. While the possibility of blow up then is apparrent as long as suitably destabilizing reaction mechanisms are admitted (cf. e.g. [13]), even the requirement of dissipation of mass-which is sufficient to yield global existence and boundedness in the corresponding ODE systems-cannot preclude its occurrence, as impressively demonstrated by the counterexamples in [22]. Global classical solutions have, accordingly, been searched for and found under certain restrictive conditions.
In the context of boundary value problems for systems of the general form Another line of investigations pursues solutions in a weaker sense. Weak solutions can be constructed if L 1 -bounds for the reaction terms f i (u) are known [19], or if the reaction functions are at most quadratic [21]. For nonlinear diffusion of porous medium type, the existence of weak solutions to (1.4) with Dirichlet boundary data is shown in [15] under the assumptions that is such that f i (s 1  and that either a priori L 1 -bounds for the reaction terms are known, or f i (s 1 , .., s N ) ≥ −C N j=1 s β j j + 1 with β i < m i +1 for all i ∈ {1, . . . , N }. An analogous result has been achieved for the corresponding Neumann problem in [14] with a different proof and less restrictive conditions on the initial data, and in the cases when additionally all m i are sufficiently large compared to max i β i , higher regularity and convergence of solutions were shown in [8].
The concept of renormalized solutions, that is, the idea that not u itself, but a transformed quantity ρ(u) solves (a weak form) of the equation, makes it possible to bypass even further restrictions on the form of the system. This concept has been successfully introduced for the Boltzmann equation by DiPerna and Lions [7] and was employed for reaction diffusion equations with quadratic reaction functions and linear diffusion in [6]. The apparently most far-reaching application of this idea to reactiondiffusion systems (with linear diffusion) can be found in [10], where essentially no growth restriction on the f i is needed, but where the reaction function is supposed to obey a certain entropy condition. The term in the definition of solutions for whose treatment this entropy condition is essential arises from the choice of renormalization functions ξ : [0, ∞) N → R with compactly supported Dξ , which in particular depend on all solution components simultaneously.
Main results. In the present manuscript, we intend to introduce an approach by which it becomes possible to avoid any requirement of the latter type, and it turns out that this can in fact be achieved by resorting to separate renormalization functions for each component u i . Thereby, our main result, as stated in Theorem 1.1 below, partially answers the open problem [10, p.585] to find a similar notion of solution without requiring an entropy condition.
Specifically, we shall be concerned with the Neumann problem ⎧ ⎨ Here we recall that the quasipositivity condition in (1.2) is important in order to avoid negative concentrations, and that (1.3) is a slightly generalized mass dissipation condition, and includes some stoichiometric coefficients a. In addition to this, (1.5) signifies a growth condition for the negative parts of the reaction functions, where in the special case of linear diffusion, subquadratic growth is admissible. It is important to note, however, that this restriction only applies to the cross-absorptive effects: For ( f i ) − , the possible growth with respect to the i-th argument remains unrestricted. As for the initial data in (1.4), throughout this paper we shall suppose that (1.6) Postponing the precise description of the solution concept to be pursued here to Sect. 2, let us introduce our main result obtained in this framework, and give a few examples of its application.
Remark. (i) The required smoothness of the domain is not the focus of our investigation and could be weakened -in fact, already for the present construction, C 2+α regularity for some α ∈ (0, 1) -entering in the construction of classical approximate solutions -would be sufficient. (ii) Likewise, in order to avoid additional technicalities we do not investigate in detail here whether covering less regular sources, such as e.g. merely continuous f i , i ∈ {1, ..., N }, might be possible at the cost of an additional approximation argument in the context of the regularized versions (3.1) of (1.4) below.
(iii) An interesting question left open here is how far the regularity requirements in (1.6) could further be relaxed, so as to require integrability of (u i0 ) i∈{1,...,N } only, for instance. As will become clear in the proof of Lemma 3.2 below, our currently pursued strategy will crucially rely on (1.6) in order to appropriately control certain initial data appearing in the course of a duality-based reasoning.
Application #1. A first application of Theorem 1.1 addresses the system which describes a general reversible reaction of the form and for which we obtain the following.

Proposition 1.2.
Let N ≥ 2, and suppose that k 1 > 0 and k 2 > 0, and that for  For linear diffusion, weak solutions of (1.7) have been found in [23] if the reaction functions grow at most quadratically or if the diffusion coefficients are sufficiently close to each other. The same article also deals with their exponential convergence.
Proof of Proposition 1.3. Taking any nonnegative φ i ∈ C 1 ([0, ∞)) such that φ i > 0 and φ i ≥ g i on (0, ∞) for i ∈ {1, 2}, one can readily verify that for we have as well as and, similarly, The assumptions β i < m i + 1, i ∈ {1, 2}, therefore warrant applicability of Theorem 1.1 with the intended result.
Let us remark that since in Proposition 1.
Application #3. We shall next briefly address for which without imposing any smallness condition on q 2 nor p 1 we obtain the following. Corollary 1.4. Let k 1 , k 2 , d 1 , d 2 , m 1 and m 2 be positive, and let p 1 ≥ 1, p 2 ≥ 1, q 1 ≥ 1 and q 2 ≥ 1 be such that Then for all (u 01 , u 02 ) fulfilling (1.6), the problem (1.12) possesses a generalized solution in the sense of Definition 2.2.
Application #4. As final example, let us consider the generalized Lotka-Volterra system which does not obey the typical entropy condition (that is required for the renormalized solutions in [10] and for classical solvability e.g. in [26]). In [9], global classical solutions are shown to exist for the classical Lotka-Volterra system (β i j = 1 for all i, j) with linear diffusion. If m i = 1 for all i and β i j = β for all i, j, then the result of [9] covers 2β < 2 + ε (for sufficiently small ε ∈ (0, 1), see [9, Step 1, (7)]). Within the generalized solvability framework considered here, the following consequence of Theorem 1.1 shows that here actually the entire range β < 2 can be exhausted.
. . , N }, and suppose that for i, j ∈ {1, ..., N } the numbers a i j ∈ R and β i j > 0 are such that a i j + a ji ≤ 0, and that if i = j and a i j < 0, then β i j < m i + 1. (1.14) Then for all initial data u 01 , ..., u 0n as in (1.6), (1.13) has a generalized solution in the sense of Definition 2.2.
for any such s, so that, according to (1.14), (1.5) is fulfilled and Theorem 1.1 is applicable.

Solution concept
The first step toward the design of our solution concept is concerned with an appropriate supersolution feature required in each of the equations making up (1.4): Definition 2.1. Suppose that for i ∈ {1, ..., N }, u 0i : → R and u i : × (0, ∞) → R are measurable and nonnegative. Then (u 1 , ..., u N ) will be called a renormalized supersolution of (1.
and if moreover Remark. (i) In the above situation, both integrals on the left of (2.3) as well as the second integral on the right-hand side therein exist due to the readily verified fact that ρ and P i (u i )| 2 , which will be estimated by means of lower semicontinuity, cf. (3.28). (iv) The definition of P (1) i , crucial in making use of the comparatively weak limit information for gradients, c.f. iii), contains √ ρ , thereby causing the requirement ρ ≥ 0 -and, consequently, due to compact support of ρ also ρ ≤ 0 -, whereas comparable definitions in other systems only ask for the renormalization to be smooth with compactly supported derivative.
As discussed in several previous related approaches toward generalized solvability on the basis of supersolution features of the above flavor [28,29], supplementing Definition 2.1 by a mere requirement on mass control is already sufficient to create a notion of solvability which within classes of suitably smooth functions indeed reduces to classical ones (see, e.g, [17] and [27] for detailed reasonings in this regard): Definition 2.2. By a generalized solution of (1.4) we mean a vector (u 1 , ..., u N ) of nonnegative measurable functions on × (0, ∞) such that (u 1 , ..., u N ) is a renormalized supersolution of (1.4) in the sense of Definition 2.1, that with some a ∈ (0, ∞) N u 1 , ..., u N and 4) and that ..., u N ) for a.e. t > 0. (2.5)

Approximate systems
In order to construct such solutions through an essentially standard type of approximation, for ε ∈ (0, 1) we consider which, moreover, is nonnegative.
The following basic observation concerning L 1 -boundedness of these solutions is a fairly immediate consequence of (1.3).
Proof. By integrating in (3.1), we see that since The following estimate rests on a duality-based reasoning inspired by a corresponding argument from [15]. An important difference is given by the change of boundary conditions: Where [15] dealt with Dirichlet boundary data and thus could use the solution of Poisson's equation as test function, the non-invertibility of the Laplacian with homogeneous Neumann boundary data leads us to employ the solution of a Helmholtz equation instead. An alternative approach of working with the Neumann Laplacian after subtracting the mean value has been followed in [14], but it seems unclear how far strategies of this type can be applied so as to successfully cover the present setting. and for ε ∈ (0, 1) and t ≥ 0, we have and hence w ε (·, t) ≤ w ε (·, 0) + z ε (·, t) + 1 in for all t > 0 and ε ∈ (0, 1), (3.5) because t 0 K e −K s ds = 1 − e −K s ≤ 1 for all t > 0. Upon multiplication by ∂ t z ε ≥ 0 and integration over ×(0, T ) for T > 0, as in [15] we obtain that since z ε (·, 0) ≡ 0, where the right-hand side is now estimated in a way slightly deviating from that in [15] due to the different boundary conditions considered here. In fact, by nonnegativity of w ε the inequality in (3.5) implies that for each fixed T > 0 and arbitrary ε ∈ (0, 1), − z ε (·, T ) + z ε (·, T ) ≤ w ε (·, 0) + 1 + z ε (·, T ) in and ∂ ν z ε (·, T ) = 0 on ∂ , so that since the Helmholtz operator − + 1 admits a comparison principle under homogeneous Neumann boundary conditions, we obtain that where z ε denotes the solution of Now without loss of generality assuming that the number r ≥ 1 in (1.6) satisfies r < 2, we take r ∈ (2, ∞] such that 1 r + 1 r = 1, and employ a Sobolev embedding theorem and elliptic regularity theory [11] to find c 1 > 0 and c 2 > 0 such that for all ε ∈ (0, 1), because the restrictions on r in (1.6) warrant that 2 − n r ≥ − n r if n ≥ 3, and that 2 − n r > 0 if n ≤ 2. Here we note that according to Lemma 3.1 we know that if we let θ ∈ (0, 1) small enough such that θ m i ≤ 1 for all i ∈ {1, ..., N }, then we can find c 3 (T ) > 0 fulfilling for all ε ∈ (0, 1), whence using that then θ < 1 < r < 2 < r we may invoke Hölder's and Young's inequality to see that with λ = ( 1 θ − 1 r )/( 1 θ − 1 r ) ∈ (0, 1) and some c 4 > 0 we have In view of (3.7) and the nonnegativity of z ε , (3.8) thus implies that for all ε ∈ (0, 1), so that in (3.6) we can use the Hölder inequality to estimate for all ε ∈ (0, 1). As sup ε∈(0,1) w ε (·, 0) L r ( ) is finite according to the hypothesis (3.2), (3.6) thereby entails the existence of c 5 (T ) > 0 such that for all ε ∈ (0, 1), from which (3.4) readily follows.

upon further integration we find that
for all T > 0, (3.13) because i is nonpositive. Now on the right-hand side of (3.13) we use (1.5) to see that thanks to (3.12), for all T > 0, whence recalling Lemma 3.2 we obtain c 2i (T ) > 0 such that for all ε ∈ (0, 1), (3.14) As by (3.11), in view of Lemma 3.1 we thus conclude from (3.13) that there exists c 3i (T ) > 0 with the property that , for instance.
While the bound in Lemma 3.2 is sufficient for concluding relative compactness of {u iε | ε ∈ (0, 1)} in some weak topology, we are additionally interested in possible pointwise convergence of u iε j along some sequence (ε j ) j∈N 0. We thus strive to derive a suitable strong compactness property in L 2 ( × (0, T )), at least of a power of u iε which has been cut off at large values so as to ensure accessibility to the estimates of Lemma 3.3.

Lemma 3.4. Given
Then 1) is relatively compact in L 2 ( × (0, T )) for all T > 0. (3.16) Proof. Let us first make sure that for each T > 0, is bounded in L 2 ((0, T ); W 1,2 ( )). (3.17) To see this, we note that due to the compactness of supp ρ i it is clear that (ρ i (u iε )) ε∈(0,1) is bounded in L ∞ ( × (0, ∞)). Therefore, (3.17) results upon the observation that if we fix M > 0 such that ζ ≡ 0 on (M, ∞), then by Young's inequality and (3.9) we see that for all T > 0 there exists c 1 (T ) > 0 such that where we have used that 2κ i − m i + 1 ≥ 2κ i − m i − 1 ≥ 0 by hypothesis. We next fix any integer k ≥ 1 such that k > n, and claim that then for all T > 0, is bounded in L 1 (0, T ); (W k,2 ( )) . (3.18) To verify this in quite a straightforward manner, we pick ψ ∈ C ∞ ( ) and use (3.1) to see that for each t > 0 and any ε ∈ (0, 1), where finiteness of both ρ L ∞ ((0,∞)) and ρ i L ∞ ((0,∞)) is asserted by our restriction that κ i ≥ 2. Here we observe that in the case m i ≥ 1 we have whereas if m i < 1 then As furthermore W k,2 ( ) → W 1,∞ ( ) due to our restriction that k > n, from (3.19) we thus infer the existence of c 2 > 0 such that for all t > 0 and any ε ∈ (0, 1), which in light of (3.9) and (3.10) establishes (3.18) upon a time integration. We finally only need to combine (3.17) with (3.18) to conclude that (3.16) is a consequence of an Aubin-Lions type lemma ( [25,Cor. 4]).
Based on this compactness statement, we may conclude the existence of a limit object. as ε = ε j 0.
Our next goal is to show that the functions just constructed actually form a solution. We begin by confirming that they enjoy a renormalized supersolution property in the style of Definition 2.1. The most crucial ingredient in our verification of this -and actually the reason for dealing with supersolutions -becomes apparent in (3.28), which is enlisted to control the integral involving the gradient (of P (1) i (u)) from above by means of lower semicontinuity. Lemma 3.6. Let u 1 , ..., u N be as given by Lemma 3.5. Then u = (u 1 , ..., u N ) forms a renormalized supersolution of (1.4) in the sense of Definition 2.1.
In view of (3.21) applied to p := 1, a version of the dominated convergence theorem thus ensures that For any such t, this firstly implies that F(u 1 , . . . , u N ) belongs to L 1 ( × (0, t)), and secondly entails that (2.5) holds.
The previous two lemmata already demonstrate that u is a generalized solution in the sense of Definition 2.2: Proof of Theorem 1.1. We take u 1 , ..., u N as given by Lemma 3.5 and then only need to combine Lemma 3.6 with Lemma 3.7.