Damian Borbolla

Grégoire Ficheur

Clinical Decision Support Systems and Computerized Provider Order Entry: Contributions from 2020

Keywords: Health informatics, International Medical Informatics Association, Review, clinical decision support systems

Objectives: To summarize research contributions published in 2020 in the field of clinical decision support systems (CDSS) and computerized provider order entry (CPOE), and select the best papers for the Decision Support section of the International Medical Informatics Association (IMIA) Yearbook 2021. Methods: Two bibliographic databases were searched for papers referring to clinical decision support systems. From search results, section editors established a list of candidate best papers, which were then peer-reviewed by seven external reviewers. The IMIA Yearbook editorial committee finally selected the best papers on the basis of all reviews including the section editors' evaluation. Results: A total of 1,919 articles were retrieved. 15 best paper candidates were selected, the reviews of which resulted in the selection of two best papers. One paper reports on the use of electronic health records to support a public health response to the COVID-19 pandemic in the United States. The second paper proposes a combination of CDSS and telemedicine as a technology-based intervention to improve the outcomes of depression as part of a cluster trial. Conclusions: As shown by the number and the variety of works related to clinical decision support, research in the field is very active. This year's selection highlighted the application of CDSS to fight COVID-19 and a combined technology-based strategy to improve the treatment of depression.

Introduction

This paper serves as the synopsis of the decision support section of the International Medical Informatics Association (IMIA) Yearbook 2021. It complements the survey paper authored by Taber et al. where the authors sought to identify the current state of research of clinical decision support system (CDSS) using interoperability standards [START_REF] Taber | New Standards for Clinical Decision Support: A Survey of The State of Implementation[END_REF]. Our aim is to summarize recent research in the domain of decision support and to select the best papers published in this field in 2020. This literature review focused on research studies related to CDSS and computerized provider order entry (CPOE) systems. The synopsis is organized as follows: the next section summarizes the process for selecting the best papers on the decision support topic; then we present the results of this year's selection process, and the last section comments the contributions of the two best papers, as well as noticeable research works in the field identified during the whole process.

Methods

A comprehensive literature search on topics related to CDSSs and CPOE systems was performed to identify candidate best papers in two bibliographic databases, the PubMed/Medline database (from the US National Center for Biotechnology Information) and the Web of Science® (WoS, from Clarivate Analytics). Both databases were searched with similar queries, tailored to the specificities of each one, targeting journal articles published in 2020, written in English, and related to the topics of interest. The adopted strategy was the one described in the paper of Lamy et al. [START_REF] Lamy | Toward a formalization of the process to selec IMIA Yearbook best papers[END_REF].

A first review of the retrieved citations was performed by the two section editors to select 15 candidate best papers. Following the IMIA Yearbook protocol, these candidate best papers were then individually reviewed and rated by both section editors, the chief editor of the Decision Support section, and external reviewers from the international Health Informatics community. Based on the reviewers' ratings and comments, the Yearbook editorial committee then selected the best papers of the decision support domain.

Results

A total of 1,919 unique references were obtained. After a first individual screening independently performed by both section editors based on the title of papers, 1,816 were rejected by both of them. 103 studies were then discussed by the two editors to achieve a final selection of 15 candidate best papers. After the external review of these 15 articles, the editorial committee finally selected two of them as best papers for 2020. Table 1 summarizes 15 papers initially selected by the section editors. The two best papers of 2020 are discussed in the next section.

Discussion and Outlook

The first paper, by Wu et al. [START_REF] Wu | Development of a clinical decision support system for severity risk prediction and triage of COVID-19 patients at hospital admission: an international multicentre study[END_REF], proposed a clinical decision support system for severity risk prediction of COVID-19 patients at hospital admission. The model was built in a supervised manner from 725 patients in order to predict the onset of severe or critical illness during hospitalization. It was an international multicenter study including retrospective and prospective cohorts from China, Italy and Belgium. The final list of features included in the model were age, lymphocytes (proportion), CRP, LDH, creatine kinase, urea and calcium. This modelisation is proposed as an online calculator tool, as well as transcribed through a nomogram and trial tool for clinicians. The very early conduct of this large-scale multicenter study during the epidemic, as well as the pragmatism of the dissemination of the model through these tools, seems to the editors to be highlighted.

In the second paper, Balestrieri et al. [START_REF] Balestrieri | Effectiveness of clinical decision support systems and telemedicine on outcomes of depression: a cluster randomized trial in general practice[END_REF] aimed to test the effectiveness of a composite CDSS and telemedicine combined approach designed to treat depression in primary care. This intervention was assessed through a cluster randomized trial in four clinics in Italy comparing the technology-based intervention versus usual treatment. The algorithm used in this project was a computerized treatment system developed by expert consensus; it provided all prompts necessary to guide the GPs in the choice of the best treatment and provided guidance about the need of a referral to a specialist service. The percentage of remitters at 6 months was significantly lower in the intervention group (24.1% vs 6.6%). This work seems to us to be a reference article for the global evaluation of a decision support tool including a content and an expert decision tree. The study design, always a challenge when implementing CDSS interventions, used in this article is an excellent example of how CDSS can be disseminated.

Besides the two best papers selected for the Decision Support section of the 2021 edition of the IMIA Yearbook, several other works deserve to be cited. One paper describes the use of electronic health records to support a public health response to the COVID-19 pandemic with a focus on academic centers in the United States [START_REF] Madhavan | Use of electronic health records to support a public health response to the COVID-19 pandemic in the United States: a perspective from 15 academic medical centers[END_REF]. Other papers focus on the integration of Artificial Intelligence (AI) and machine learning-based with clinical decision support system to help in different clinical situations [START_REF] Corny | A machine learning-based clinical decision support system to identify prescriptions with a high risk of medication error[END_REF].

As showed in the amount and the variety of studies found in this synopsis, research in the field of CDSS is very active. As expected, this year's selection highlighted the impact that COVID-19 and the global crisis had on public health and how information system and especially CDSS can help to implement technology-based solutions. Telehealth and machine learning tools associated to CDSS were also emerging in the scientific literature.

Table 2

 2 Article published online: 2021-09-03 Best paper selection of articles for the IMIA Yearbook of Medical Informatics 2021 in the special section 'Decision Support'. The articles are listed in alphabetical order of the first author's surname. Section Decision Support  Balestrieri M, Sisti D, Rocchi M, Rucci P, Simon G, Araya R, de Girolamo G. Effectiveness of clinical decision support systems and telemedicine on outcomes of depression: a cluster randomized trial in general practice. Fam Pract 2020 Nov 28;37(6):731-7.  Wu G, Yang P, Xie Y, Woodruff HC, Rao X, Guiot J, Frix AN, Louis R, Moutschen M, Li J, Li J, Yan C, Du D, Zhao S, Ding Y, Liu B, Sun W, Albarello F, D'Abramo A, Schininà V, Nicastri E, Occhipinti M, Barisione G, Barisione E, Halilaj I, Lovinfosse P, Wang X, Wu J, Lambin P. Development of a clinical decision support system for severity risk prediction and triage of COVID-19 patients at hospital admission: an international multicentre study. Eur Respir J 2020 Aug 20;56(2):2001104.

Table 1

 1 The 15 papers initially selected Guangyao Wu et al. Development of a clinical decision support system for severity risk prediction and triage of COVID-19 patients at hospital admission: an international multicentre study. Eur Respir J 2020 Aug 20;56(2):2001104. doi: 10.1183/13993003.01104-2020. PMID: 32616597; PMCID: PMC7331655.  Mengchun Gong et al. Cloud-Based System for Effective Surveillance and Control of COVID-19: Useful Experiences From Hubei, China. J Med Internet Res 2020 Apr 22;22(4):e18948. doi: 10.2196/18948. PMID: 32287040; PMCID: PMC7179239.  Paul M McKie et al. Computerized Advisory Decision Support for Cardiovascular Diseases in Primary Care: A Cluster Randomized Trial. Am J Med 2020 Jun;133(6):750-756.e2. doi: 10.1016/j.amjmed.2019.10.039. Epub 2019 Dec 18. PMID: 31862329.  Jennifer Corny et al. A machine learning-based clinical decision support system to identify prescriptions with a high risk of medication error. J Am Med Inform Assoc 2020 Nov 1;27(11):1688-1694. doi: 10.1093/jamia/ocaa154. PMID: 32984901; PMCID: PMC7671619.  Subha Madhavan et al. Use of electronic health records to support a public health response to the COVID-19 pandemic in the United States: a perspective from 15 academic medical centers. J Am Med Inform Assoc 2021 Feb 15;28(2):393-401. doi: 10.1093/jamia/ocaa287. PMID: 33260207; PMCID: PMC7665546.  David Gallagher et al. Implementation and Continuous Monitoring of an Electronic Health Record Embedded Readmissions Clinical Decision Support Tool. J Pers Med 2020 Aug 26;10(3):103. doi: 10.3390/jpm10030103. PMID: 32858890; PMCID: PMC7565687.  Nicolas Delvaux et al. Clinical decision support improves the appropriateness of laboratory test ordering in primary care without increasing diagnostic error: the ELMO cluster randomized trial. Implement Sci 2020 Nov 4;15(1):100. doi: 10.1186/s13012-020-01059-y. PMID: 33148311; PMCID: PMC7640389.  Peter G Szilagyi et al. Effect of Patient Portal Reminders Sent by a Health Care System on Influenza Vaccination Rates: A Greer et al. Randomized Trial of a Smartphone Mobile App to Improve Symptoms and Adherence to Oral Therapy for Cancer. J Natl Compr Canc Netw 2020 Feb;18(2):133-141. doi: 10.6004/jnccn.2019.7354. PMID: 32023526.  Jasmeet S Dhaliwal et al. Reduced admission rates and resource utilization for chest pain patients using an electronic health record-embedded clinical pathway in the emergency department. J Am Coll Emerg Physicians Open 2020 Nov 17;1(6):1602-1613. doi: 10.1002/emp2.12308. PMID: 33392569; PMCID: PMC7771814.  Jessica P Ridgway et al. A Randomized Controlled Trial of an Electronic Clinical Decision Support Tool for Inpatient Antimicrobial Stewardship. Clin Infect Dis 2021 May 4;72(9):e265-e271. doi: 10.1093/cid/ciaa1048. PMID: 32712674.  Matteo Balestrieri et al. Effectiveness of clinical decision support systems and telemedicine on outcomes of depression: a cluster randomized trial in general practice. Fam Pract 2020 Nov 28;37(6):731-737. doi: 10.1093/fampra/cmaa077. PMID: 32766705.

	 J Jeffery Reeves et al. Rapid response to COVID-19: health informatics support for outbreak management in an academic
	health system. J Am Med Inform Assoc 2020 Jun 1;27(6):853-859. doi: 10.1093/jamia/ocaa037. PMID: 32208481; PMCID:
	PMC7184393.
	 Elisha S Grange et al. Responding to COVID-19: The UW Medicine Information Technology Services Experience. Appl Clin Inform
	2020 Mar;11(2):265-275. doi: 10.1055/s-0040-1709715. Epub 2020 Apr 8. PMID: 32268390; PMCID: PMC7141898.
	

IMIA Yearbook of Medical Informatics 2021Borbolla et al.

In this paper the authors aimed to develop a machine learning algorithm to assess and triage patients with COVID-19. They used 725 patients to train the model, including cohorts from hospitals in China, Italy and Belgium. The main outcome was the onset of severe or critical illness during hospitalization. Model performances were quantified using the area under the receiver operating characteristic curve (AUC) and other metrics. The model was validated yielding AUCs ranging from 0.84 to 0.93, with accuracies ranging from 74.4% to 87.5%, sensitivities ranging from 75.0% to 96.9%, and specificities ranging from 55.0% to 88.0%, most of which performed better than the pneumonia severity index. The authors concluded that the machine-learning algorithm is useful to triage the severity and critical illness among COVID-19 patients at hospital admission. The online calculators can be found at www.covid19risk.ai.

Balestrieri M, Sisti D, Rocchi M, Rucci P, Simon G, Araya R, de Girolamo G

Effectiveness of clinical decision support systems and telemedicine on outcomes of depression: a cluster randomized trial in general practice

Fam Pract 2020 Nov 28;37(6):731-7

In this paper, the authors combined a Computerized Clinical Decision Support Systems (CCDSS) tool with a Telehealth Intervention and tested the effectiveness of this combined approach to trat depression in primary care. The design was a randomized trial involving General Practice clinics in Italy. The study compared the combined Intervention (TG -Telehealth visit + CCDSS) with the control group (CG) in which GPs provided usual treatment. 2810 patients were included. 66 patients were included in the TG group and 32 in the CG. The percentage of remitters at 6 months was significantly higher in the TG than in the CG group (24.1% versus 3.1%, χ 2 = 6.6, P = 0.01). The combined intervention was more effective than the usual care offered by GPs to patients with depression in the study.


Symposium on Ammonia Energy

Introduction

To face the Green-House Gas emission impact and the climate global change, wing or solar energies are envisaged by most of the countries [START_REF] Valera-Medina | Review on Ammonia as a Potential Fuel: From Synthesis to Economics[END_REF]. To circumvent the problem of the intermittency supply, energy can be stored as efuels like hydrogen or ammonia. Ammonia can be envisaged either as H2-carrier or as fuel supply to combustion devices. The use of NH3 as fuel is quite challenging because of its low laminar burning velocity and high ignition temperature, and requires to be blended with another fuel, H2 being carbon-free is preferable. One of the main issue comes from the expected high levels of NOx and N2O emissions. In order to better understand the conditions in which these emissions could be limited, it is necessary to develop reliable detailed kinetic models. They are already many of them, mostly validated against global kinetic parameters; laminar burning velocities [START_REF] Lhuillier | Experimental investigation on laminar burning velocities of ammonia/hydrogen/air mixtures at elevated temperatures[END_REF][START_REF] Okafor | Experimental and numerical study of the laminar burning velocity of CH4-NH3-air premixed flames[END_REF][START_REF] Mei | Experimental and kinetic modeling investigation on the laminar flame propagation of ammonia under oxygen enrichment and elevated pressure conditions[END_REF], and ignition delay times [START_REF] Dai | Experimental and numerical analysis of the autoignition behavior of NH3 and NH3/H2 mixtures at high pressure[END_REF][START_REF] Pochet | Experimental and numerical study, under LTC conditions, of ammonia ignition delay with and without hydrogen addition[END_REF][START_REF] Mathieu | Experimental and modeling study on the high-temperature oxidation of Ammonia and related NOx chemistry[END_REF]. Recently few models have been validated thanks to experimental data obtained in Jet-Stirred Reactors in a temperature ranging 700-1300K [START_REF] Sabia | Ammonia oxidation features in a Jet Stirred Flow Reactor. The role of NH2 chemistry[END_REF][START_REF] Manna | New insight into NH-H2 mutual inhibiting effects and dynamic regimes at low-intermediate temperatures[END_REF][START_REF] Zhang | Combustion chemistry of ammonia/hydrogen mixtures: Jet-stirred reactor measurements and comprehensive kinetic modeling[END_REF][START_REF] Osipova | Ammonia and ammonia/hydrogen blends oxidation in a jet-stirred reactor: Experimental and numerical study[END_REF]. To improve the models, it is important to broaden the range of conditions in which species measurements are performed. Only few chemical flame structure analysis have been performed in low-pressure [START_REF] Bian | Experimental study of the structure of an ammonia-oxygen flame[END_REF][START_REF] Duynslaegher | Flame structure studies of premixed ammonia/hydrogen/oxygen/argon flames: Experimental and numerical investigation[END_REF] or atmospheric pressure [START_REF] Brackmann | Structure of premixed ammonia + air flames at atmospheric pressure: Laser diagnostics and kinetic modeling[END_REF][START_REF] Osipova | Chemical structure and laminar burning velocity of atmospheric pressure premixed ammonia/hydrogen flames[END_REF] flames. Using laser spectroscopic diagnostics, Brackmann et al. [START_REF] Brackmann | Structure of premixed ammonia + air flames at atmospheric pressure: Laser diagnostics and kinetic modeling[END_REF] obtained NH, OH and NO species profiles in NH3/air premixed flames. To help the flame stabilization, a disk at 16.2 mm above the burner surface serves as stagnation plate. Species concentration profiles were measured using Molecular Beam Mass Spectrometry in premixed flames either in pure NH3/O2 [START_REF] Bian | Experimental study of the structure of an ammonia-oxygen flame[END_REF] or in Ar diluted [START_REF] Duynslaegher | Flame structure studies of premixed ammonia/hydrogen/oxygen/argon flames: Experimental and numerical investigation[END_REF][START_REF] Osipova | Chemical structure and laminar burning velocity of atmospheric pressure premixed ammonia/hydrogen flames[END_REF] along the flame axis. With the addition of H2 in the premixed gases, a synergic effect with a regeneration/recovery mechanism is described in [START_REF] Zhang | Combustion chemistry of ammonia/hydrogen mixtures: Jet-stirred reactor measurements and comprehensive kinetic modeling[END_REF][START_REF] Otomo | Chemical kinetic modeling of ammonia oxidation with improved reaction mechanism for ammonia/air and ammonia/hydrogen/air combustion[END_REF][START_REF] Gotama | Measurement of the laminar burning velocity and kinetics study of the importance of the hydrogen recovery mechanism of ammonia/hydrogen/air premixed flames[END_REF]. In laminar flames studies [START_REF] Duynslaegher | Flame structure studies of premixed ammonia/hydrogen/oxygen/argon flames: Experimental and numerical investigation[END_REF][START_REF] Osipova | Chemical structure and laminar burning velocity of atmospheric pressure premixed ammonia/hydrogen flames[END_REF], it seems that the addition of H2 does not impact the NO formation in stoichiometric condition. The present work aims to complete these scarce experimental data by measuring OH and NO species using spectroscopic laser diagnostics in four premixed flames stabilized at 10 kPa. The experimental results are compared to four kinetic models [START_REF] Mei | Experimental and kinetic modeling investigation on the laminar flame propagation of ammonia under oxygen enrichment and elevated pressure conditions[END_REF][START_REF] Zhang | Combustion chemistry of ammonia/hydrogen mixtures: Jet-stirred reactor measurements and comprehensive kinetic modeling[END_REF][START_REF] Otomo | Chemical kinetic modeling of ammonia oxidation with improved reaction mechanism for ammonia/air and ammonia/hydrogen/air combustion[END_REF][START_REF] Stagni | An experimental, theoretical and kinetic-modeling study of the gas-phase oxidation of ammonia[END_REF].

Materials and Methods

Premixed flames were stabilized above a stainless-steel porous burner (Holthuis). The burner is enclosed in a stainless-steel enclosure maintained at a constant pressure (p=10kPa) thanks to a vacuum pump and a regulating valve. The porous burner is water cooled at 323 K. It is vertically mobile in order to describe the species and temperature profiles as function of the height above the burner (HAB). The premixed gases mixtures are supplied through mass flow controllers, while the flow of the shroud N2 gas is maintained constant through a rotameter. The equivalence ratio of 1 is calculated according to a full conversion into N2 and H2O as products. Three flames are NH3/O2 diluted with 24% N2 (with an equivalence ratio () equal to 0.87, 1.10 and 1.33), and the fourth one is NH3/H2 blend (90%/10%) (=0.87) diluted with 40%N2. The flame conditions are given in Table 1. Four optical access allowed for the laser diagnostics arrangement. Laser Induced Fluorescence and absorption measurements were performed using a Nd/YAG laser pumping a tunable dye laser. The laser beam diameter was restricted through a diaphragm of 2 mm and was strictly parallel to the burner surface. For both species, measurements were obtained in the linear regime of LIF. LIF signals were collected at right angle through a spectrometer and a photomultiplier tube (PMT). Fluorescence was collected through a two-lenses arrangement, and the entrance slit of the spectrometer was limited to obtain a spatial vertical resolution of 250 µm in the flame.

The mole fraction profiles of OH and NO were measured by LIF calibrated respectively by single pass absorption or by the method of the added values as previously performed in methane flames [START_REF] Lamoureux | In Situ Laser-Induced Fluorescence and Ex Situ Cavity Ring-Down Spectroscopy Applied to NO Measurement in Flames: Microprobe Perturbation and Absolute Quantification[END_REF]. The temperature was measured by single-pass absorption across the lean NH3(087) flame: by scanning the laser wavelength from 306.8 to 307.1 nm, 7 isolated absorption lines of OH could be selected from which the temperature was determined at different HAB. From the spectral integrated absorption measurement along the R2(7) line, the OH mole fraction was measured at 30 mm in the lean NH3(087) flame and used to calibrate the relative LIF profiles measured in the 4 flames.

The LIF profiles of OH and NO were measured by probing the transition R21(7) and Q2(27) lines, at 307.0 and 205.6 nm respectively. The LIF signal is proportional to the absolute rotational population of the probed species. Knowing the temperature, and the absolute concentration species at a given location in NH3(087) flame, the rotational population is converted into absolute species concentration, and mole fraction. Probing NO around 225 nm, a broadband absorption due to NH3 [START_REF] Chou | Laser induced fluorescence and absorption measurements of NO in NH3/O2 and CH4/air flames[END_REF][START_REF] Weng | Ultraviolet Absorption Cross-Sections of Ammonia at Elevated Temperatures for Nonintrusive Quantitative Detection in Combustion Environments[END_REF] along the laser axis in the flame (mostly close to the burner surface) was observed. The LIF signal was corrected for the laser energy (including the absorption) to get the rotational population along the flame axis.

Results and Discussions

The temperature profile along the flame axis was measured from absorption OH spectra. The determination of T relies on the Boltzmann distribution of the rotational OH population and is derived by plotting the Bolzmann plot.

T profile close to burner surface was completed by measurements obtained directly from commercial thermocouple. During this on-going work, the full profile was obtained in the lean NH3 flame only, and was assumed identical for all flames. From the experimental results, OH mole fraction profiles present similar shape in all flames with a peak value around 6-8 mm followed by a decrease in the burned gases. It is shown that both OH peak mole fractions and NO mole fractions in the burned gases region decrease with the increase of . In the two lean flames (with and w/o H2), the NO mole fraction profiles show an increase at HAB>17 mm. The addition of H2 in the lean flames affects moderately the OH and NO with an increase of 15% of the mole fractions (within the experimental uncertainties). Kinetic modeling of the four flames was performed using four different kinetic models from the literature. The simulated OH mole fraction profiles agree well with the experimental ones (shape and quantitatively) as shown in Fig. 1. Small deviations are noticed between the kinetic models. The predicted NO mole fraction profiles show important discrepancies between the models with a mole fraction of NO equal to 1.6% (±20%) in the burned gases of the lean NH3(087) flame. Moreover, the simulated NO mole fraction profiles are at least 5 times larger than the experimental data whatever the equivalence ratio. In the lean flames, the simulated NO mole fraction profiles remain flat in the burned gases contrary the experimental increase.

Fig. 1 .

 1 Fig. 1. Experimental and simulated NO and OH mole fraction profiles in NH3/O2/N2 flames at p=10 kPa

  NO mole fraction (simulation)

Table 1 .

 1 Flame composition (in slpm).

	Flame #	NH3	H2	O2	N2
	NH3(087)	1.07		0.92	0.64
	NH3(110)	1.23		0.84	0.64
	NH3(133)	1.29		0.73	0.64
	NH3-10H2(087)	1.16	0.13	1.07	1.58


Michel Pierre
email: michel.pierre@ens-rennes.fr

Didier Schmitt
email: didier.schmitt@univ-lorraine.fr

EXAMPLES OF FINITE TIME BLOW UP IN MASS DISSIPATIVE REACTION-DIFFUSION SYSTEMS WITH SUPERQUADRATIC GROWTH

Keywords: 2020 Mathematics Subject Classification. Primary: 35B44, 35K57, 35K40;Secondary: 35K91 Reaction-diffusion, blow up, global existence, dissipative evolution, quasipositivity

We provide explicit examples of finite time L ∞ -blow up for the solutions of 2 × 2 reaction-diffusion systems for which three main properties hold: positivity is preserved for all time, the total mass is uniformly controlled and the growth of the nonlinear reaction terms is superquadratic. They are obtained by choosing the space dimension large enough. This is to be compared with recent global existence results of uniformly bounded solutions for the same kind of systems with quadratic or even slightly superquadratic growth depending on the dimension. Such blow up may occur even with homogeneous Neumann boundary conditions. All these L ∞ -blowing up solutions may be extended as weak global solutions. Blow up examples are also provided in space dimensions one, two and three with various growths.

1. Introduction. The main goal of this paper is to provide examples of blow up in finite time for 2 × 2 reaction-diffusion systems of the form

       ∂ t u 1 -d 1 ∆u 1 = f 1 (u 1 , u 2 ) on (0, T ) × B N , ∂ t u 2 -d 2 ∆u 2 = f 2 (u 1 , u 2 ) on (0, T ) × B N , "good " boundary conditions on (0, T ) × ∂B N , u 1 (0, •) = u 0 1 ≥ 0, u 2 (0, •) = u 0 2 ≥ 0, (1)
where B N is the open unit ball in R N , d 1 , d 2 ∈ (0, +∞), f 1 , f 2 : [0, +∞) 2 → R are regular nonlinearities such that the nonnegativity of the solutions to (1) is preserved for all time which, as it is well-known, is equivalent to

(P) quasipositivity: f 1 (0, s 2 ) ≥ 0, f 2 (s 1 , 0) ≥ 0, ∀s 1 , s 2 ∈ [0, +∞), (2)
and such that the following 'mass dissipativity' condition holds (M) dissipativity: f 1 (s 1 , s 2 ) + f 2 (s 1 , s 2 ) ≤ 0, ∀s 1 , s 2 ∈ [0, +∞).

(

) 3
This last property implies that the L 1 (B N )-norm of the solutions u 1 (t), u 2 (t) of [START_REF] Cañizo | Improved duality estimates and applications to reaction-diffusion equations[END_REF] does not blow up in finite time (= control of the total mass for all time).

It is well-known that, for u 0 1 , u 0 2 ∈ L ∞ (B N ) + , there exist T ∈ (0, +∞] and a classical solution to (1) on [0, T ). If all the d i are equal, then T = +∞ so that global classical existence holds. However, if the d i are different from each other, it is known that the two conditions (P), (M) are in general not sufficient to provide global classical solutions on [0, +∞) as proved in [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF], [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF] where examples of solutions blowing up in L ∞ (B N ) in finite time are described for this class of systems.

A main new result proved here (see Theorem 2.1) is that actually, for all d = 2 + η, η > 0, there exists a choice of d 1 , d 2 ∈ (0, +∞), of the space dimension N and of the nonlinearities f 1 , f 2 satisfying (P), (M) and the growth condition (G) below, for which the solution to [START_REF] Cañizo | Improved duality estimates and applications to reaction-diffusion equations[END_REF] blows up in L ∞ (B N ) in finite time T < +∞ :

(G) d-growth: |fi(s1, s2)| ≤ C0 + C1(s1 + s2) d , ∀s1, s2 ∈ [0, +∞), i = 1, 2, (4)
where C 0 , C 1 ∈ [0, +∞). The point is that finite time blow up may occur for any superquadratic growth.

This result has to be analyzed in parallel with the recent results in [START_REF] Fellner | Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions[END_REF], [START_REF] Fellner | Global classical solutions to quadratic systems with mass control in arbitrary dimensions[END_REF] saying that, for any dimension N , there exists η > 0, depending on the dimension N , such that, if the growth of f 1 , f 2 at infinity is at most d = 2 + η, then all solutions of [START_REF] Cañizo | Improved duality estimates and applications to reaction-diffusion equations[END_REF] are global in time: in particular, the L ∞ (B N )-norm of u 1 (t), u 2 (t) is bounded on all intervals [0, T ]. Note that these global existence results are proved in [START_REF] Fellner | Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions[END_REF], [START_REF] Fellner | Global classical solutions to quadratic systems with mass control in arbitrary dimensions[END_REF] even for all m×m systems, m ≥ 2, with the properties (P), (M), (G). Theorem 2.1 shows that these results are in some sense optimal. Besides [START_REF] Fellner | Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions[END_REF], [START_REF] Fellner | Global classical solutions to quadratic systems with mass control in arbitrary dimensions[END_REF], several papers had provided global existence results of classical solutions for close to quadratic systems like [START_REF] Ya | Solvability in the large of a system of reaction-diffusion equations with the balance condition[END_REF], [START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF], [START_REF] Caputo | Solutions of the 4-species quadratic reactiondiffusion systems are bounded and C ∞ -smooth, in any space dimension[END_REF], [START_REF] Ph | Global existence for reaction-diffusion systems with dissipation of mass and quadratic growth[END_REF]. More results may also be found in [START_REF] Cupps | Uniform boundedness for reaction-diffusion systems with mass dissipation[END_REF], [START_REF] Tang | Global classical solutions to reaction-diffusion systems in one and two dimensions[END_REF], [START_REF] Laamri | Global existence of classical solutions for a class of reaction-diffusion systems[END_REF], [START_REF] Cañizo | Improved duality estimates and applications to reaction-diffusion equations[END_REF], [START_REF] Pierre | Dissipative reaction-diffusion systems with quadratic growth[END_REF]. We refer to the introduction in [START_REF] Fellner | Uniform-in-time bounds for quadratic reaction-diffusion systems with mass dissipation in higher dimensions[END_REF] where a nice state of art on these questions is described.

Obviously, given η > 0 and f 1 , f 2 with a 2+η growth, to obtain blow up examples, it is necessary to work in space dimensions N which are higher and higher when η is smaller and smaller.

Four more facts are also interesting about these blow up examples.

1) First it turns out that, not only the L ∞ -norm blows up at time t = T , but also the L m -norm for all m ≥ N (1 + η)/2, N large.

2) Next these T -blowing up solutions can nevertheless be extended as global weak solutions on [0, +∞) of the corresponding system (1) (see Section 7).

3) We also prove (see Theorem 2.2) ) that the same kind of solutions provide blow up examples for the following kind of systems:

       ∂ t u 1 -d 1 ∆u 1 = c 1 (t, x)u α 1 u β 2 on (0, T ) × B N , ∂ t u 2 -d 2 ∆u 2 = c 2 (t, x)u α 1 u β 2 on (0, T ) × B N , "good " boundary conditions on (0, T ) × ∂B N , u 1 (0, •) = u 0 1 ≥ 0, u 2 (0, •) = u 0 2 ≥ 0, (5)
where

c 1 + c 2 ≤ 0, c 1 , c 2 ∈ L ∞ (Q T )
g i = g i (t, x, s 1 , s 2 ). This is indicated in Theorem 2.3.
An optimal dependence of N in terms of η or d is not explicitly known in the main result of Theorem 2.1. What comes out in the proof of this theorem [see (28)], is that one should choose N large enough so that

2N -θ N -θ ≤ d = 2 + η ⇔ N ≥ θ(1 + η -1 ),
where θ ∈ (4/3, +∞). This means that we should have N > 4(1 + η -1 )/3.

Another natural question is, given the dimension N , to find the optimal growth which provides global existence (or blow up in finite time) for nonlinearities satisfying the main properties (P), (M). As proved in [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF], [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF], examples of blow up may occur even in dimension N = 1 by choosing a large enough growth for the nonlinearity. It is known that in dimension N = 1, global existence is proved for at most cubic growth [see [START_REF] Goudon | Regularity analysis for systems of reaction-diffusion equations[END_REF], [START_REF] Tang | Global classical solutions to reaction-diffusion systems in one and two dimensions[END_REF]], but the optimal growth is not known.

To progress in this understanding, we give in Section 8 explicit blow up examples in the following cases :

N = 1, d = 6; N = 2, d = 7/2; N = 3, d = 3.
2. Statement of the main results. Some notation. Here B N denotes the open euclidian unit ball in R N , Q T = (0, T )×B N , Σ T = (0, T )×∂B N . We also denote by

C ∞ (A 1 , A 2 ) the family of C ∞ -mappings from A 1 to A 2 where A i ⊂ R Ni , i = 1, 2,
and N 1 , N 2 are positive integers. If A 2 = R, we simply denote C ∞ (A 1 ).

Theorem 2.1. Let d := 2 + η, η ∈ (0, +∞) and T ∈ (0, +∞). Then there exist

f 1 , f 2 ∈ C ∞ [0, +∞) 2 satisfying (P) + (M) + (G), d 1 , d 2 ∈ (0, +∞), a dimension N large enough, u 0 1 , u 0 2 ∈ C ∞ (B N ) + , α 1 , α 2 ∈ C ∞ ([0, T ]) + and u 1 , u 2 nonnegative C ∞ -solutions of (1) with u 1 (t, x) = α 1 (t), u 2 (t, x) = α 2 (t) on Σ T ,
and such that

lim t→T - u 1 (t) L m (B N ) = lim t→T - u 2 (t) L m (B N ) = +∞, for m ≥ N (d -1)/2. Moreover, for λ close to 1 (M λ ) f 1 (s 1 , s 2 ) + λf 2 (s 1 , s 2 ) ≤ 0, ∀s 1 , s 2 ∈ [0, +∞). ( 6
)
Remark 2.1. Despite the blow up of u 1 (t), u 2 (t) at time t = T , it turns out that the solutions u i provided by Theorem 2.1 can be extended to the whole interval [0, +∞)

as weak global solutions of the system (1) on the whole interval [0, +∞). By weak solution, we mean that the nonlinear terms

f i (u 1 , u 2 ), i = 1, 2 are in L 1 ((0, τ ) × B N )
for all τ ∈ (0, +∞) and u 1 , u 2 are solutions in the sense of distributions or in the sense of the 'variation of constants' formula. This is a consequence of the fact that, in the blow up examples, not only f 1 + f 2 ≤ 0 holds, but also f 1 + λf 2 ≤ 0 for some λ = 1. This provides the L 1 -bound on the nonlinear terms and this is sufficient for the existence of weak global solutions. All this is made more precise in Section 7.

Theorem 2.2. Let d := 2 + η, η ∈ (0, +∞), α, β ∈ (1, +∞) with α + β = d and T ∈ (0, +∞). Then there exist a dimension N large enough, d 1 , d 2 ∈ (0, +∞) and

c 1 , c 2 ∈ C ∞ [0, T ) × B N ∩ L ∞ (Q T ), u 0 1 , u 0 2 ∈ C ∞ (B N ) + , α 1 , α 2 ∈ C ∞ ([0, T ]) + ,
and u 1 , u 2 nonnegative C ∞ -solutions of the system (5) with u i = α i , i = 1, 2 on Σ T and such that It turns out that we can also use the same kind of solutions to construct new blowing up solutions for similar systems, but with homogeneous Neumann boundary conditions. Here the nonlinearities depend also on (t, x). This is the purpose of the next theorem.

c 1 (t, x) + c 2 (t, x) ≤ 0, ∀(t, x) ∈ Q T , lim t→T - u 1 (t) L m (B N ) = lim t→T - u 2 (t) L m (B N ) = +∞, for m ≥ N (d -1)/2.
Theorem 2.3. Let d := 2 + η, η ∈ (0, +∞) and T ∈ (0, +∞). Then there exist τ 0 ∈ [0, T ), g 1 , g 2 ∈ C ∞ [τ 0 , T ) × B N × [0, +∞) 2 satisfying (P ) + (M λ ) + (G ) below, d 1 , d 2 ∈ (0, +∞), a dimension N large, v 0 1 , v 0 2 ∈ C ∞ (B N ) + and v 1 , v 2 nonnegative C ∞ -solutions of        ∂ t v 1 -d 1 ∆v 1 = g 1 (t, x, v 1 , v 2 ) in (τ 0 , T ) × B N , ∂ t v 2 -d 2 ∆v 2 = g 2 (t, x, v 1 , v 2 ) in (τ 0 , T ) × B N , ∂ ν v 1 = 0 = ∂ ν v 2 on (τ 0 , T ) × B N , v i (τ 0 , •) = v 0 i ≥ 0, (7)
and such that

lim t→T - v 1 (t) L m (B N ) = lim t→T - v 2 (t) L m (B N ) = +∞, for m ≥ N (d -1)/2.
Here the conditions (P ) + (M λ ) + (G ) are the same as (P) + (M λ ) + (G), but with a (t, x)-dependence, namely,

∀(t, x) ∈ [τ0, T ] × BN , ∀(s1, s2) ∈ [0, +∞) 2 , (P ) g1(t, x, 0, s2) ≥ 0, g2(t, x, s1, 0) ≥ 0, (M λ ) g1(t, x, s1, s2) + λg2(t, x, s1, s2) ≤ 0, ∀λ close to 1, (G ) |g1(t, x, s1, s2)| + |g2(t, x, s1, s2)| ≤ C0 + C1(s1 + s2) d . (8)
3.

Steps of the proof of Theorem 2.1. The main idea in the proof of Theorem 2.1 is similar to the approach in [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF], [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF]. It consists in working with functions u1, u2 of the form

ui(t, x) = ai(T -t) + bir 2 [(T -t) + r 2 ] γ , r = |x|, i = 1, 2, (9)
where a1, a2, b1, b2 will be well-chosen in (0, +∞), as well as γ ∈ (1, 2) and the dimension N (large enough), in such a way that there exist d1, d2 ∈ (0, +∞) and two functions f1, f2

as described in Theorem 2.1 for which

∂tui -di∆ui = fi(u1, u2), in (0, T ) × BN , i = 1, 2. ( 10
)
The values of u1, u2 on ∂BN are obviously C ∞ -functions. Moreover the L ∞ (BN )-norm of u1(t), u2(t) blows up as t → T -since γ > 1. More precisely, we directly check (see Lemma

3.1) that lim t→T - ui(t) L m (B N ) = +∞, ∀m ≥ N 2(γ -1) , i = 1, 2. (11)
Thus Theorem 2.1 will be proved if we can achieve [START_REF] Pierre | Global Existence in Reaction-Diffusion Systems with Dissipation of Mass : a Survey[END_REF] with regular functions f1, f2 satisfying (P), (M), (G) with d = 2 + η ≥ γ/(γ -1). It will be a consequence of the three following lemmas.

First note that ui, i = 1, 2, is of the following form:

ui(t, x) = (T -t) 1-γ ρi r 2 T -t , ρi(σ) = ai + biσ (1 + σ) γ , i = 1, 2. ( 12
)
We have the following technical lemma.

Lemma 3.1. Let u1, u2 be given by [START_REF] Pierre | Weak solutions and supersolutions in L 1 for reaction-diffusion systems[END_REF].

For i = 1, 2, we have in (0, T ) × BN ∂tui -di∆ui = (T -t) -γ (1 + σ) -(γ+2) [Ai + Biσ + Ciσ 2 ], σ = r 2 /(T -t), (13)
where Ai, Bi, Ci are real numbers given by

   Ai = (γ -1)ai -2diN (bi -γai), Bi = (γ -1)[2ai + bi(1 + 2diN + 4di)] + (bi -γai)[4di(γ + 1) -2diN + 1], Ci = (γ -1)[ai + bi(1 + 2diN -4γdi)] + bi -γai. ( 14
)
As a consequence

∂tu1 -d1∆u1 + ∂tu2 -d2∆u2 = (T -t) -γ (1 + σ) -(γ+2) [A + Bσ + Cσ 2 ], (15)
   A = (γ -1)a -2N E + 2N γG, B = (γ -2)a + γb + [2N (γ -2) + 8γ]E -2γ[2(γ + 1) -N ]G, C = -a + γb + 2(γ -1)(N -2γ)E, ( 16
)
where only the four following quantities are involved

a := a1 + a2, b := b1 + b2, E := b1d1 + b2d2, G := d1a1 + d2a2. (17)
Moreover, lim t→T -ui(t) L m (B N ) = +∞ if m ≥ N/2(γ -1).
The proof of this lemma is elementary (see below). Next we check that ∂tui -di∆ui may be written as a function of u1, u2. This is the purpose of the following lemma. Here Q denotes the set of rational numbers.

Lemma 3.2. Let u1, u2 be given by [START_REF] Pierre | Weak solutions and supersolutions in L 1 for reaction-diffusion systems[END_REF]. Assume

γ ∈ (1, 2) ∩ Q, a1b2 -a2b1 = 0. ( 18
)
Then for i = 1, 2, there exists a C ∞ -function Pi : (0, +∞) 2 → R 2 which is homogeneous of degree γ = γ/(γ -1) and such that

∂tui -di∆ui = Pi(u1, u2) in (0, T ) × BN , (19) |Pi
(s1, s2)| ≤ K(s1 + s2) γ , ∀(s1, s2) ∈ (0, +∞) 2 , for some K ∈ (0, +∞), (20)
Pi(a1

+ b1σ, a2 + b2σ) = (1 + σ) 2-γ γ-1 [Ai + Biσ + Ciσ 2 ], ∀σ ∈ (0, +∞). ( 21
)
Remark 3.1. By homogeneous of degree γ , we mean that

Pi(λs1, λs2) = λ γ Pi(s1, s2), ∀λ ∈ (0, +∞), (s1, s2) ∈ (0, +∞) 2 . ( 22
)
This lemma will be proved below. But let us continue describing the scheme of the proof of Theorem 2.1. A main step is now to prove that we can choose the various parameters ai, bi, di, i = 1, 2, d, γ and the dimension N such that the functions P1, P2 appearing in Lemma 3.2 satisfy

P1(s1, s2) + P2(s1, s2) ≤ 0, ∀(s1, s2) ∈ (0, +∞) 2 . ( 23
)
By (21), this inequality will be satisfied for (s1, s2) = (a1 + b1σ, a2 + b2σ), σ ∈ (0, +∞) if and only if A + Bσ + Cσ 2 ≤ 0, ∀σ ∈ (0, +∞), where A, B, C are defined in [START_REF] Tang | Global classical solutions to reaction-diffusion systems in one and two dimensions[END_REF] and ( 16) of Lemma 3.1. It will be the case if

A < 0, C < 0, B 2 < 4AC. ( 24
)
And we then construct the Pi so that this inequality extends to all (s1, s2) ∈ (0, +∞) 2 .

The main point in the proof of Theorem 2.1 is that we can choose the various parameters in the definition (9) so that (24) holds for A, B, C as defined in Lemma 3.1. This is the purpose of the following main lemma where we use the following notation.

Notation. Writing a function D : R → R as

D(N ) = O(N -α ), α ∈ R, means that lim sup N →+∞ |D(N )|N α < +∞.
Lemma 3.3. In the definition (9) of u1, u2, let us choose

a1 = 1/N 2 , a2 = 2N -N -2 , b1 = b2 = 1/ √ N , d1 = √ N , d2 = 1/N 3 , γ = 2 -θ/N, θ ∈ (4/3, +∞) ∩ Q. ( 25
)
Then as N → +∞, the values A, B, C, as defined in Lemma 3.1, satisfy

     A = -2θ + O(N -1 2 ) < 0, C = -(8 + 2θ) + O(N -1 2 ) < 0, B 2 -4AC = 64(4 -3θ) + O(N -1 2 ) < 0, A + Bσ + Cσ 2 ≤ A -B 2 /4C = 8(4-3θ) 4+θ + O(N -1 2 ) < 0, ∀σ ∈ [0, +∞). ( 26
)
Moreover ( 18) is satisfied and the functions P1, P2 can be chosen in Lemma 3.2 so that they satisfy (23) and even

P1(s1, s2) + λP2(s1, s2) ≤ 0, ∀(s1, s2) ∈ (0, +∞) 2 , (27)
for λ close to 1. We consider the functions u1, u2 as defined in [START_REF] Pierre | Weak solutions and supersolutions in L 1 for reaction-diffusion systems[END_REF] with the choice of parameters a1, a2, b1, b2, d1, d2, γ given in Lemma 3.3 with θ ∈ (4/3, +∞) ∩ Q and N large enough so that

γ = 2N -θ N -θ ≤ d = 2 + η. ( 28
)
Since by assumption θ ∈ Q, we have γ ∈ Q. Moreover γ ∈ (1, 2) for N > θ. For this choice of the parameters, a2b1 -a1b2 = 2N 1/2 + O(N -5/2 ) = 0 for N large.

Thus by Lemma 3.2, there exist two C ∞ -functions P1, P2 : (0, +∞)

2 → R such that ∂tui -di∆ui = Pi(u1, u2), in (0, T ) × BN , |Pi(s1, s2)| ≤ K(s1 + s2) γ , ∀(s1, s2) ∈ (0, +∞) 2 , i = 1, 2, for some K ∈ (0 + ∞).
By Lemma 3.3, we can even choose P1, P2 so that (27) holds for λ close to 1.

Note that there exists mN ∈ (0, +∞) such that

ui(t, x) ≥ mN , ∀(t, x) ∈ [0, T ] × BN , i = 1, 2.
Indeed, we may choose mN = (T + 1) 1-γ min{a1, a2, b1, b2} as it can be seen by writing

ui(t, x) = (T -t + r 2 ) 1-γ ai + biσ 1 + σ . Let us introduce ϕ ∈ C ∞ ([0, +∞), [0, +∞)) such that 0 ≤ ϕ ≤ 1, ϕ ≡ 0 on [0, mN /2], ϕ ≡ 1 on [mN , +∞).
Let us denote fi(s1, s2) := ϕ(s1)ϕ(s2)Pi(s1, s2), i = 1, 2 on [0, +∞) 2 . Then ∂tui -di∆ui = fi(u1, u2), in (0, T ) × BN , and f1, f2 are exactly as described in Theorem 2.1. More precisely, and for future reference, we even have

fi(s1, s2) = 0, ∀(s1, s2) ∈ [0, +∞) 2 \ [mN /2, +∞) 2 . ( 29
) Next we have u 0 i ∈ C ∞ (BN ) and αi ∈ C ∞ ([0, T ]) for i = 1, 2 since u 0 i (x) = (T + r 2 ) -γ (aiT + bir 2 ), αi(t) = (T -t + 1) -γ (ai(T -t) + bi), i = 1, 2.
Now, we know by Lemma 3.1, that the L m (BN )-norm of ui(t), i = 1, 2 blows up if Let us now prove these three lemmas.

m ≥ N [2(γ -1)] -1 . Here γ is chosen [see (28)] so that γ ≤ d, that is (γ -1) -1 ≤ d -1. If m ≥ N (d -1)/2, then m ≥ N [2(γ -1)] -
Proof of Lemma 3.1. This result is derived by technical but easy computations. We may use (see [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF]

) ui = (T -t) 1-γ ρi(σ), σ = r 2 /(T -t) so that ∂tui = (γ -1)(T -t) -γ ρi(σ) + r 2 (T -t) 2 (T -t) 1-γ ρ i (σ), ∂rui = (T -t) 1-γ 2r T -t ρ i r 2 T -t , ∂rrui = (T -t) 1-γ 4r 2 (T -t) 2 ρ i r 2 T -t + 2 T -t ρ i r 2 T -t , N -1 r ∂rui + ∂rrui = (T -t) -γ 2N ρ i (σ) + 4σρ i (σ) . ∂tui -di∆ui = (T -t) -γ (γ -1)ρi(σ) + (σ -2diN )ρ i (σ) -4diσρ i (σ) .
With the choice of ρi(σ) = (ai + biσ)/(1 + σ) γ , we have

ρ i (σ) = bi -γai + bi(1 -γ)σ (1 + σ) γ+1 , ρ i (σ) = bi(1 -γ) (1 + σ) γ+1 - (γ + 1)[bi -γai + bi(1 -γ)σ] (1 + σ) γ+2 .
This leads to

∂tui -di∆ui = (T -t) -γ (1 + σ) -(γ+2) Ai + Biσ + Ciσ 2 , (30)
where Ai, Bi, Ci are given as stated in Lemma 3.1.

By summing for i = 1, 2 these two expressions, we obtain the claims ( 15)-( 16)-(17) of Lemma 3.1. It is important to notice that this sum can be written only in terms of the four parameters a, b, E, G.

Finally we compute the L m (BN )-norm of ui(t). For some CN ∈ (0, +∞)

B N u m i (t, x)dx = CN (T -t) m(1-γ) 1 0 r N -1 ρ m i r 2 /(T -t) dr. By setting σ = r 2 /(T -t) ⇔ r = √ T -t √ σ, we have B N u m i (t, x)dx = CN 2 (T -t) N 2 +m(1-γ) (T -t) -1 0 σ N 2 -1 ρ m i (σ)dσ. As σ N 2 -1 ρ m i (σ) is equivalent to σ N 2 -1+m(1-γ) , up to a positive constant, as σ → +∞, we have N 2 < m(γ -1) ⇒ +∞ 0 σ N 2 -1 ρ m i (σ)dσ < +∞.
As a consequence, in this case B N u m i (t, x)dx behaves like (T -t) Proof of Lemma 3.2. We are looking for two functions Pi(•, •), i = 1, 2 such that

∂tui -di∆ui = Pi(u1, u2) = Pi (T -t) 1-γ ρ1(σ), (T -t) 1-γ ρ2(σ) , σ = r 2 /(T -t). ( 31
)
According to [START_REF] Pierre | Dissipative reaction-diffusion systems with quadratic growth[END_REF] in Lemma 3.1, and by the expected γ -homogeneity of Pi, this is equivalent to

(1 + σ) -(γ+2) [Ai + Biσ + Ciσ 2 ] = (1 + σ) γ 2 /(1-γ) Pi(a1 + b1σ, a2 + b2σ), ⇔ (1 + σ) 2-γ γ-1 [Ai + Biσ + Ciσ 2 ] = Pi(a1 + b1σ, a2 + b2σ), ∀σ ∈ [0, +∞). ( 32
)
Let us write γ = p/q ∈ (2, +∞) ∩ Q with p, q two coprime positive integers. Then

γ = p/(p -q), (2 -γ)/(γ -1) = (p -2q)/q > 0.
Thus (32) may be rewritten

(1 + σ) p-2q q [Ai + Biσ + Ciσ 2 ] = Pi(a1 + b1σ, a2 + b2σ), (33)
which also implies

(1 + σ) p-2q [Ai + Biσ + Ciσ 2 ] q = {Pi(a1 + b1σ, a2 + b2σ)} q . ( 34
)
Here p > 2q and the left-hand side is a polynomial of degree at most p in σ. It is easy to see that, since a1b2 -a2b1 = 0, the family of p + 1 polynomials

{(a1 + b1σ) k (a2 + b2σ) p-k , k = 0, ..., p},
is a linear basis of the linear space of polynomials of degree at most p. Indeed, let us analyze the relation

p k=0 µ k (a1 + b1σ) k (a2 + b2σ) p-k ≡ 0, µ k ∈ R, ∀k = 0, ..., p.
Assume by contradiction that there exists r ∈ {0, ..., p} such that µ k = 0, ∀k = 0, ..., r -1, µr = 0.

Then after dividing by (a1 + b1σ) r , the above relation may be rewritten

p k=r µ k (a1 + b1σ) k-r (a2 + b2σ) p-k = 0.
If r = p, this obviously implies µr = 0 whence a contradiction. If r < p, choosing σ = -a1/b1 leads to µr(a2 -b2a1/b1) p-r = 0, whence again µr = 0 thanks to the assumption (18) on ai, bi, i = 1, 2, and this is a contradiction.

We are now going to define the convenient function Pi. Going back to the polynomial of degree at most p involved in (34), we can claim that there exist

λ i k ∈ R, k = 0, ..., p such that for all σ ∈ R (1 + σ) p-2q [Ai + Biσ + Ciσ 2 ] q = p k=0 λ i k (a1 + b1σ) k (a2 + b2σ) p-k . ( 35
)
Assume first that q is odd. Let us define

Pi(s1, s2) := p k=0 λ k s k 1 s p-k 2 1/q , (s1, s2) ∈ [0, +∞) 2 . ( 36
)
Note that Pi is homogeneous of degree p/q and

(1 + σ)

p-2q q [Ai + Biσ + Ciσ 2 ] = Pi(a1 + b1σ, a2 + b2σ). ( 37
)
According to (30), this implies [see also (31), (32)]

∂tui -di∆ui = Pi(u1, u2) in QT . ( 38
)
Moreover, we directly see on the definition (36) of Pi that for (s1, s2) ∈ [0, +∞)

| Pi(s1, s2)| ≤ p k=0 |λ k | 1/q (s1 + s2) p/q = C(s1 + s2) γ .
Now, for the regularity of Pi, it is obvious from its definition (36) that Pi is C ∞ around each point (s1, s2) for which Pi(s1, s2) = 0. We are going to modify it into a new function Pi which will be in C ∞ (0, +∞) 2 with the same main properties. For this, let us assume without loss of generality in (18) that a1b2 -a2b1 > 0 or b2/b1 > a2/a1 and let us consider the function

s ∈ (0, +∞) → X := a2 + b2s a1 + b1s ∈ I := a2 a1 , b2 b1 .
It is C ∞ with derivative (a1b2 -a2b1)/(a1 + b1s) 2 > 0 on (0, +∞). Thus its inverse

ψ : I → (0, +∞) is also C ∞ .
From (36), we may write Pi(s1, s2) = s p/q 1 Pi(1, s2/s1), and thanks to the relation (37) and the p/q homogeneity of Pi, we have

Pi(1, X) = [a1 + b1ψ(X)] -p/q [1 + ψ(X)] (p-2q)/q [Ai + Biψ(X) + Ciψ(X) 2 ]. ( 39
)
Whence the C ∞ -property of X ∈ I → Pi(1, X) and therefore of Pi(s1, s2) on I where we set

I := {(s1, s2) ∈ (0, +∞) 2 ; s2/s1 ∈ I}.
Actually, X ∈ R → Pi(1, X) is continuous and does not vanish in a neighborhood of the extremities of I, namely

X = a2/a1 ⇔ ψ(X) = 0, Pi(1, X) = Aia -p/q 1 , X = b2/b1 ⇔ ψ(X) = +∞, Pi(1, X) = Cib -p/q 1 .
Thus, as noticed before thanks to the definition (36), X → Pi(1, X) is C ∞ even on some open interval J containing I. We now introduce a function χ ∈ C ∞ ([0, +∞)) such that:

0 ≤ χ ≤ 1, χ ≡ 1 on I, χ ≡ 0 on [0, +∞) \ J. ( 40
)
And we set Pi(s1, s2) := s p/q 1 χ(s2/s1) Pi(1, s2/s1) = χ(s2/s1) Pi(s1, s2). Thus Pi is C ∞ on all of (0, +∞) 2 and coincides with Pi on I. It follows that we may replace Pi by Pi in the relations (37) and (38). This ends the proof of Lemma 3.2 when q is odd.

Assume now that q is even. We start again from the relation (35) from which we deduce

(1 + σ) (p-2q)/q [Ai + Biσ + Ciσ 2 ] = Pi(a1 + b1σ, a2 + b2σ)sign{Ai + Biσ + Ciσ 2 }, (41)
where we again set

Pi(s1, s2) := p k=0 λ i k s k 1 s p-k 2 1/q = s p/q 1 Pi(1, s2/s1), ∀(s1, s2) ∈ (0, +∞) 2 , (42)
and where sign(r) = 1, ∀r ∈ (0, +∞), sign(0) = 0, sign(r) = -1, ∀r ∈ (-∞, 0).

We deduce from (41) and the p/q homogeneity of Pi that, for all X ∈ I,

Pi(1, X)sign Ai + Biψ(X) + Ciψ(X) 2 = (a1 + b1ψ(X)) -p/q (1 + ψ(X)) (p-2q)/q [Ai + Biψ(X) + Ciψ(X) 2 ] (43)
and this function of X is C ∞ on I. Since Pi(1, X) does not vanish near the extremities of the interval I, it is also locally C ∞ there. Moreover sign Ai + Biψ(X) + Ciψ(X) 2 is constant as X tends to these extremities from inside I. It follows that the function of X appearing in (43) may be extended in a C ∞ -way to an open interval J containing I.

Using the same function χ as above (see (40)), by setting Pi(s1, s2) := s p/q 1 χ(s2/s1) Pi(1, s2/s1)sign Ai + Biψ(s2/s1) + Ciψ 2 (s2/s1) we define a C ∞ -function on (0, +∞) 2 which is p/q-homogeneous and satisfies

Pi(a1 + b1σ, a2 + b2σ) = (1 + σ) (p-2q)/q [Ai + Biσ + Ciσ 2 ], ∀σ ∈ [0, +∞)
and therefore (see (31), (32), (33) )

∂tui -di∆ui = Pi(u1, u2) on QT .
This ends the proof of Lemma 3.2.

Proof of Lemma 3.3. As a consequence of the choice of the parameters in Lemma 3.3, we obtain

a = a1 + a2 = 2N, b = b1 + b2 = 2/ √ N , E = d1b1 + d2b2 = √ N (1/ √ N ) + N -3 (1/ √ N ) = 1 + O(N -7/2 ), G = d1a1 + d2a2 = √ N N -2 + N -3 [2N -(1/N 2 )] = O(N -3/2
). It follows from the formulas (16) in Lemma 3.1 that for N large

A = [1 -θ/N ]2N -2N [1 + O(N -7/2 )] + 2N [2 -θ/N ]O(N -3/2 ) = -2θ + O(N -1/2 ), C = -2N +[2-θ/N ]2/ √ N +2[1-θ/N ][N -4+2θ/N ][1+O(N -7/2 )] = -(8+2θ)+O(N -1/2 ), B = [-θ/N ]2N + [2 -θ/N ][2/ √ N ] + [2N (-θ/N ) + 8(2 -θ/N )][1 + O(N -7/2 )] -2[2 -θ/N ][2(3 -θ/N ) -N ][O(N -3/2 )] = 16 -4θ + O(N -1/2 ). B 2 -4AC = 16[4 -θ] 2 -8θ(8 + 2θ) + O(N -1/2 ) = 64[4 -3θ] + O(N -1/2 ).
With the choice of θ > 4/3, we obtain A < 0, C < 0, B 2 -4AC < 0 for N large as claimed in Lemma 3.3.

Since the maximum of σ ∈ R → A + Bσ + Cσ 2 is reached for σ = -B/2C, we deduce

A + Bσ + Cσ 2 ≤ A -B 2 /4C = -K θ + O(N -1 2 ), ∀σ ∈ [0, +∞) (44)
with K θ := 8(3θ -4)/(4 + θ) > 0.

On the other hand, a2b1 -a1b2 = 2N 1/2 + O(N -5/2 ) = 0 and γ ∈ (1, 2) ∩ Q so that (18) is satisfied and we may apply Lemma 3.2. Using now the relation (21) in this Lemma 3.2 together with (44), we obtain that

(P1 + P2)(a1 + b1σ, a2 + b2σ) ≤ -K θ + O(N -1 2 ), ∀σ ∈ [0, +∞).
By the p/q-homogeneity of P1 + P2, we deduce that

(P1 + P2)(1, X) ≤ -K θ a -p/q 1 + O(N -1 2 ), ∀X ∈ I,
where I is defined as in the previous proof of Lemma 3.2. Since I is bounded, so is P2(1, X) for X ∈ I. It follows that for λ close to 1 and N large enough, we may claim that

(P1 + λP2)(1, X) ≤ - K θ a -p/q 1 2 , ∀X ∈ I.
Up to reducing J and the support of the function χ in (40) of the proof of Lemma 3.2, we may claim that

(P1 + λP2)(1, X) ≤ - K θ a -p/q 1 4 , ∀X ∈ J.
By homogeneity, this implies that

(P1 + λP2)(s1, s2) ≤ 0, ∀(s1, s2) ∈ (0, +∞) 2 .
This ends the proof of Lemma 3.3.

5.

Proof of Theorem 2.2. We use the same functions u1, u2 as those introduced in the proof of Theorem 2.1 and defined in [START_REF] Pierre | Weak solutions and supersolutions in L 1 for reaction-diffusion systems[END_REF], and we choose γ as in (28), namely such that

γ ≤ d ⇔ γ ≥ d/(d -1). ( 45
)
We deduce from (30) that

∂tui -di∆ui = Ai(T -t) 2 + Bi(T -t)r 2 + Cir 4 (T -t + r 2 ) γ+2 , i = 1, 2.
Given α, β as indicated in the statement of Theorem 2.2, we may understand this expression as being of the form ∂tui -di∆ui = ci(t, x)u α 1 u β 2 , where we define

ci(t, x) := Ai(T -t) 2 + Bi(T -t)r 2 + Cir 4 (T -t + r 2 ) γ+2 u -α 1 u -β 2 , ⇒ ci(t, x) = (T -t + r 2 ) γ(d-1)-2 [Ai(T -t) 2 + Bi(T -t)r 2 + Cir 4 ] [a1(T -t) + b1r 2 ] α [a2(T -t) + b2r 2 ] β . Obviously ci is C ∞ on [0, T ) × BN . Let us check that it is in L ∞ (QT ) as follows. Let H := T -t + r 2 . We use T -t ≤ H, r 2 ≤ H, ai(T -t) + bir 2 ≥ min{ai, bi}H.
Thus for some K1, K2 ∈ (0, +∞), and using that γ(d -1)

-2 ≥ γ(d -1) -d ≥ 0, we have |ci(t, x)| ≤ H γ(d-1)-2 [[Ai| + |Bi| + |Ci|]H 2 K1H α+β ≤ K2H γ(d-1)-d ≤ K2(T + 1) γ(d-1)-d .
Finally, we see that the sign of c1 +c2 is the same as the sign of A(T -t) 2 +B(T -t)r 2 +Cr 4 , that is c1 + c2 ≤ 0 since A < 0, C < 0, B 2 -4AC < 0 by Lemma 3.3. This ends the proof of Theorem 2.2.

6. Proof of Theorem 2.3. Let u1, u2 be the solutions obtained in Theorem 2.1. Their normal derivatives on ∂BN are given as follows

∂ru1(t, 1) = 2[(b 1 -γa 1 )(T -t)-(γ-1)b 1 ] (T -t+1) γ+1 , ∂ru2(t, 1) = 2[(b 2 -γa 2 )(T -t)-(γ-1)b 2 ] (T -t+1) γ+1 , b1 = N -1/2 = b2, b1 -γa1 = N -1/2 -2N -2 + O(N -3 ), b2 -γa2 = -4N + O(1). (46) Obviously for N large ∂ru1(t, 1) ≤ 0, ∀t ∈ [τ0, T ], τ0 := (T -γ + 1) + (< T ), ∂ru2(t, 1) ≤ 0, ∀t ∈ [0, T ]. ( 47
)
We now introduce the solutions βi, i = 1, 2 of

   ∂tβi -di∆βi = 0 in (τ0, T ) × BN , ∂ν βi = -∂rui(t, 1) on (τ0, T ) × ∂BN , βi(τ0, •) ≡ 0. ( 48
)
By the C ∞ -property of ∂rui(t, 1), these solutions are also in C ∞ (QT ). Moreover, thanks to (47), and by maximum principle, we have βi(t, x) ≥ 0 on [τ0, T ) × BN . We now introduce for i = 1, 2 vi(t, x) := ui(t, x) + βi(t, x) ≥ 0 so that ∂ν vi = ∂rvi(t, 1) = 0 on ΣT .

And according to Theorem 2.1 and the definition of βi, we have

∂tvi -di∆vi = ∂tui -di∆ui = fi(u1, u2) = gi(t, x, v1, v2) in QT , i = 1, 2,
where we set gi(t, x, s1, s2) := fi(s1 -β1(t, x), s2 -β2(t, x)).

Here the functions fi are extended by 0 outside [0, +∞) 2 and with the help of (29), we see that they are C ∞ on R 2 . The functions gi defined in this way satisfy (P ), (M λ ), (G ) since, by nonnegativity of βi:

g1(t, x, 0, s2) = f1(-β1(t, x), s2 -β2(t, x)) = 0, g2(t, x, s1, 0) = f2(s1 -β1(t, x), -β2(t, x)) = 0, (g1 + λg2)(t, x, s1, s2) = (f1 + λf2)(s1 -β1(t, x), s2 -β2(t, x)) ≤ 0 [using (6)], |gi(t, x, s1, s2)| = |fi(s1 -β1(t, x), s2 -β2(t, x))| ≤ C0 + C1[(s1 -β1(t, x)) + + (s2 -β2(t, x) + ] d ⇒ |gi(t, x, s1, s2)| ≤ C 0 + C 1 |s1 + s2| d , where C 0 , C 1 depend on C0, C1, d, βi L ∞ (Q T )
. Finally, the L m (BN )-norm of vi(t) blows up as t → T -together with the L m (BN )-norm of ui(t). This ends the proof of Theorem 2.3.

7. About global weak solutions. Let us prove here that the solutions u1, u2 obtained in Theorem 2.1 may be extended to global weak solutions of the same system. For this we will use the approach of the previous section and, as a first step, we extend the functions v1, v2 to weak solutions on [0, +∞). This will provide extensions of u1, u2 as well.

Let us start by extending the functions βi as the solutions on [0, +∞) of

       ∂tβi -di∆βi = 0 in (τ0, +∞) × BN , ∂ν βi = -∂rui(t, 1) on (τ0, T ) × ∂BN -∂rui(T, 1) on (T, +∞) × ∂BN , βi(τ0, •) ≡ 0. (49)
These extensions are C ∞ on [τ0, +∞) × BN . We also extend the nonlinear function gi as follows for (t, x) ∈ (τ0, +∞) × BN gi(t, x, s1, s2) := fi(s1 -β1(t, x), s2 -β2(t, x)), ∀(s1, s2) ∈ [0, +∞) 2 .

They satisfy the same properties (P ), (M λ ), (G ) on [τ0, +∞). We now apply Theorem 5.9 in [START_REF] Pierre | Global Existence in Reaction-Diffusion Systems with Dissipation of Mass : a Survey[END_REF] to the system

       ∂tv1 -d1∆v1 = g1(t, x, v1, v2) in (τ0, +∞) × BN , ∂tv2 -d2∆v2 = g2(t, x, v1, v2) in (τ0, +∞) × BN , ∂ν v1 = 0 = ∂ν v2 on (τ0, +∞) × BN , vi(τ0, •) = v 0 i = ui(τ0, •) + β(τ0, •). (50)
Indeed, thanks to the property (M λ ), any good approximation of this system with bounded nonlinearities g n i in place of gi and the same regular data at t = τ0, will provide approximate regular solutions v n i such that, for all S ∈ [τ0, +∞) sup

n≥1 (τ 0 ,S)×B N |g n i (t, x, v n 1 (t, x), v n 2 (t, x))| < +∞, i = 1, 2,
We may for instance choose

g n i (t, x, s1, s2) := gi(t, x, s1, s2) 1 + n -1 [(|g1| + |g2|)(t, x, s1, s2)] ,
so that |g n i | ≤ n. These approximate functions g n i satisfy also (P ), (M λ ) and it is easy to prove that these properties imply the above L 1 ((τ0, S) × BN )-bounds (see e.g. Proposition 5.1 in [START_REF] Pierre | Global Existence in Reaction-Diffusion Systems with Dissipation of Mass : a Survey[END_REF]). As a consequence, and as proved in [START_REF] Pierre | Global Existence in Reaction-Diffusion Systems with Dissipation of Mass : a Survey[END_REF], these approximate solutions v n i converge to a weak solution of (50) which means that vi(t) = Si(t -τ0)v 0 i + that since all inequalities are strict in this example, we could slightly modify it to obtain a blowing up example with growth 3 -σ for some small σ > 0.

8.2. Blow up with a 7/2-growth in dimension N = 2. An example with such a growth was already mentioned in [START_REF] Pierre | Blow up in reaction-diffusion systems with dissipation of mass[END_REF]. We give here another one, easier to compute. Let us choose the following parameters with > 0 small enough. Thus this provides a blow up example with growth γ = 7/2 and an L m (BN )-norm blowing up for m ≥ 5/2. Again, since all inqualities are strict, we could find another example with growth 7/2 -σ for some σ > 0. This provides a blow up example with growth γ = 6 and with an L m (BN )-norm blowing up for m ≥ 5/2. As above again, we could improve it to a lower growth 6 -σ for some σ > 0.

4 )

 4 and α + β = 2 + η with η > 0 as small as we want. The boundary conditions of the examples provided in Theorem 2.1 are nonhomogeneous boundary conditions. We also describe blowing up examples for systems with homogeneous Neumann boundary conditions, up to replacing f i , i = 1, 2 by more general (still regular) nonlinearities

Remark 2 . 2 .

 22 The solutions u 1 , u 2 appearing in this result are the same as those in Theorem 2.1. Only the interpretation of the nonlinear part changes. In particular, they blow up in L m (B N ) as t → T -for the same values of m.

4 .

 4 Proof of Theorem 2.1. Theorem 2.1 is a consequence of Lemmas 3.1, 3.2, 3.3. We will prove these lemmas below. But let us first show how these lemmas imply Theorem 2.1.

1

 1 and the L m (BN )-norm of ui(t) blows up. This ends the proof of the main Theorem 2.1, assuming Lemmas 3.1, 3.2, 3.3.

N 2 +

 2 (1-γ)m , up to a constant, as t → T -and therefore tends to +∞. The conclusion is the same in the equality case N = 2m(γ -1) since then B N u m (t, x)dx behaves like | log(T -t)| as t → T -.

t τ 0 8 . 1 .

 081 Si(t -s)gi (s, •, v1(s, •), v2(s, •)) ds, i = 1, 2, where Si(t) denotes the semigroup generated by the operator -di∆ on BN with homogeneous Neumann boundary conditions. Blow up with a cubic growth in dimension N = 3. Here we choose with ∈ (0, 1) small enough,γ = 3/2, a = 10, b = 1, E = 1, G = .Then, aE -bG = 10 -= 0 andA = (γ -1)a -6E + 6γG = 5 -6 + 9 = -1 + 9 < 0, C = -a + γb + 2(γ -1)(3 -2γ)E = -10 + 3/2 + 0 = -17/2 < 0, B = (γ -2)a + γb + [6(γ -2) + 8γ]E -2γ[2(γ + 1) -3]G = 11/2 -) = -15/4 + O( ) < 0.Thus (51) holds so that this provides a blowing up example with growth γ = 3. By Lemma 3.1, for m ≥ 3, the L m (BN )-norm of the solution blows up as t → T -. Note

γ = 7 / 5 ,

 75 a = 4, b = , E = 1, G = . then aE -bG = 4 -2 5568] + O( ) = -448 125 + O( ) < 0.

8. 3 .

 3 Blow up with a 6-growth in dimension N = 1. Here we choose γ = 6/5, a = 5, b = 1/2, E = 1, G = 1/10. then aE -bG = 5 -1/20 = 0 and A = 1 -2 + 6/25 = -19/25 < 0, C = -5 + 3/5 + 2(1/5)(1 -12/5) = -22/5 -14/25 = -124/25 < 0, B = -4 + 3/5 + 8 -12.17/250 = 473/125. 125 2 [B 2 -4AC] = 473 2 -4.95.620 = 223729 -235600 = -11871 < 0.

These weak solution coincides on the interval [τ0, T ) with the classical solution of [START_REF] Ya | Solvability in the large of a system of reaction-diffusion equations with the balance condition[END_REF] as found in Theorem 2.3. Indeed, since v 0 i ∈ L ∞ (BN ), it is classical to prove that the approximate solution (v n 1 , v n 2 ) stays uniformly bounded, independently of n, on some interval [τ0, τ1] ⊂ [τ0, T ]. To see it, we might for instance use the inequalities

Thus the limit (v1, v2) is therefore a classical solution on [τ0, τ1]. By uniqueness of classical solutions for the reaction-diffusion system [START_REF] Ya | Solvability in the large of a system of reaction-diffusion equations with the balance condition[END_REF], both solutions coincide at least on [τ0, τ1],

and subsequently on the whole interval [τ0, T ] by a classical continuity argument.

Finally, if we now set ui(t, x) := vi(t, x) -βi(t, x), i = 1, 2 on the whole domain [τ0, +∞) × BN , we find a global weak solution of the initial system (1) on [0, +∞) which extends the (classical) solution obtained on [0, T ) in Theorem 2.1. It remains to get convinced that ui ≥ 0 for all t. This can be seen by noticing that

Thus u n i ≥ 0 and this is preserved at the limit.

8. Some more explicit examples in small dimensions. We use here Lemma 3.1 and Lemma 3.2. All blow up examples we are going to describe here are defined as in ( 9) so that they blow up as t → T -. Moreover by Lemma 3.2, we know that we can then find P1, P2 so that (19) holds, namely

The goal is then to choose all parameters involved in the definition (9) well enough so that (24) holds, namely

where A, B, C are defined in Lemma 3.1. Then, exactly as in the proof of Theorem 2.1, we prove that the previous inequalities implies that P1 + P2 ≤ 0 and that P1, P2 can then slightly be modified into functions f1, f2 satisfying (P), (M) and [START_REF] Cañizo | Improved duality estimates and applications to reaction-diffusion equations[END_REF]. Moreover the growth at infinity of f1, f2 is at most γ .

The following technical proposition is very useful to help finding examples of blow up.

Notations are as in Lemma 3.1.

Proposition 8.1. Let a, b, E, G ∈ (0, +∞). Then we can find ai, bi, di, i = 1, 2 ∈ (0, +∞)

Proof of Proposition 8.1. Let us choose d1, d2 such that

Then, given the four parameters a, b, E, G, a solution to (52) is given by

We easily check that, with this choice, the relations (52), (53) are satisfied and all parameters are positive. The relation (53) will be used to check (18).