César Ducruet
email: ducruet@parisgeo.cnrs.fr

Luis Carvalho

Stanislas Roussin

THE FLIGHT OF ICARUS? INCHEON'S TRANSFORMATION FROM PORT GATEWAY TO GLOBAL CITY

This chapter examines the relationship between freight and urban development in the case of the South Korean city of Incheon. Functionally linked to the capital Seoul as main gateway of the Gyeongin corridor, home to a major seaport (second after Busan) and new international airport, the local and national government have been especially aggressive in promoting all modes of freight transportation and logistics activity, as well as free economic zones, notably based on the "Pentaport" concept. The case of Incheon thus provides a site for exploring how (and with what consequences) cities in export-oriented Asian nations have gone about planning for heightened and elaborated flows of goods. Furthermore, it explores how logistic activities co-exist with the ambitions of Incheon -and South Korea as a whole -of becoming a Northeast Asia's knowledge and business hub. In light of the multiplication of other projects nationally and in Asia, and of the competition from Chinese manufacturers and transportation providers, this case provides some insights into what trends may emerge in the future.

INTRODUCTION

Devastated by a fratricide war (1950)(1951)(1952)(1953) followed by a decade of unstable economic policies, South Korea was in the early 1960s a land of deep poverty. Very rare were the experts able to predict that this rural country, deprived from natural resources, challenged by an unemployment rate above 20 per cent, continuously under the threat of a new War with its neighbour, and ruled by a military government, would have a bright future. South Korea is nowadays not only a stable democracy but also an economically dynamic and innovative Asian nation. It is the world's 13 th largest economy, excelling for example in shipbuilding, automotive, rolling-stock, steel, electronics and petrochemical industries. Underlying these developments was an export-oriented economic industrial policy based on strong alliances between the government, banks and a few entrepreneurs who built in less than 30 years some of the world's largest conglomerates. This success story has been dubbed the Han River miracle, backed by strong American, Japanese support and coordination of the South Korean people. More recently, South Korea could quickly overcome the severe financial crisis of 1997 without falling into a long term depression and even reach a GNP per capita of $20,000 (USD) in the mid 2000s.

South Korea is still, and more than ever, an export-oriented country. As for 2011, the Korea Institute for Industrial Economics and Trade forecast that exports would reach $562 billion (and imports $530 billion), raising the annual trade volume over $1 trillion for the first time in the country's history; its GDP in 2010 was also over $1 trillion. However due to the division of the Korean peninsula between North and South in 1953, South Korea effectively remains an island with a high dependence on maritime transport: 99 per cent of its foreign trade is shipped by sea [START_REF] Cullinane | Container terminals in South Korea: problems and panaceas[END_REF], thereby giving strategic importance to Korean port cities such as Busan, Gwangyang, and Incheon.

With a current population of over 2.7 million inhabitants in its Metropolitan Area, Incheon is the country's third largest city after Seoul and Busan. However, Incheon remains largely unexplored in the urban research and economic geography literature, namely when compared with Busan, the largest port [START_REF] Frémont | The emergence of a mega-port: from the local to the global, the case of Busan[END_REF], and the capital Seoul [START_REF] Hong | Seoul: a global city in a nation of rapid growth[END_REF].

In fact, Incheon is part of the so-called "capital region" together with Seoul and the Gyeonggi province, ranking third after Tokyo and Mexico City in demographic size. The Capital Region is the country's main engine: with 22 million inhabitants it concentrates about 45 per cent of South Korea's population, 84 per cent of public services, 65 per cent of universities, 91 per cent of corporate headquarters, and 46 per cent of the country's GDP [START_REF] Shin | Recent experience of and prospects for high-speed rail in Korea: implications of a transport system and regional development from a global perspective[END_REF]. Although territorial de-concentration policies of the early 1980s initially proved rather successful [START_REF] Smith | Semiperipheral urbanization? South Korea in the 1980s[END_REF], their impact since then remains limited and congestion effects in the capital region have become critical [START_REF] Choi | Inchon: gateway to Seoul[END_REF]. Simultaneously, Incheon hosts since the late 1990s the world's second largest sea-reclamation project after Zuiderzee in the Netherlands [START_REF] Wallace | Avec la nouvelle ville de Songdo, la Corée du Sud lutte pour le prestige régional[END_REF], and the largest private urban development project in the world [START_REF] Lee | New Songdo City and the value of flexibility: a case study of implementation and analysis of a mega-scale project[END_REF], with the creation of Incheon International Airport (Winged City) and the launch of the Incheon Free Economic Zone (IFEZ) strategy in 2001.

This chapter investigates the local embedding of such recent projects and initiatives in their relation with freight flows and transport systems within Incheon and the capital region. The developments taking place in Incheon can (and should) be interpreted bearing Seoul's dynamics in mind; they are in many ways an extension of those. Seoul faces enormous problems of congestion, lack of space and a real estate crisis which are related with the location choice of Incheon for many of the ongoing developments, namely in close vicinity and possessing some advantages such as waterside location, existing infrastructure, and industrial base. However, the proximity to Seoul was seen by [START_REF] Fujita | The role of ports in the making of major cities: self-agglomeration and hub-effect[END_REF] as a constraint to port-related urban growth. While Incheon's current development strategy to become a "world class global city" (IT Times, 2009) implicitly claims functional independence from Seoul, Incheon itself seems to have always been highly related with the capital city, their proximity being a catalyst or a constraint depending on the period considered:

1) Emergence of gateway and corridor (1870s-1960s): Incheon emerges as a major industrial and load centre in Korea following the Ganghwa Treaty (1876) and the opening of its port (1883) to international trade under Japanese pressure prior to the invasion (1910) and after the Korean War (1953);

2) Southward shift and relative stagnation (1960s-1990s): the southward shift resulted from major projects such as early free trade zones being located at other locations, partly due to physical limitations of Incheon's site, tensions with North Korea, and the national planning policy giving priority to Southern and Eastern regions;

3) Diversification and hub formation (1990s-2000s): in the 1990s Incheon's proximity to Seoul becomes once again an opportunity for economic development, illustrated by heavy investments in transport infrastructure, large scale urban development projects and new innovative sectors (e.g. Pentaport and free economic zone strategies -see next sections).

Nowadays Incheon particularly suffers from severe congestion problems due to high urban density, industrial sprawl, dominance of trucking flows, and scattering of development projects torn between different and overlapping logics and interests. This is combined with increasing unemployment and lowering economic welfare in recent years compared with the average score of all metropolitan cities in South Korea (see Figure 1). Other cities such as Busan exhibit similar trends but Incheon remains special due to the pace of change, which seems to contradict the projected benefits of current developments (even though many potentially high-impact projects are not yet completed). The analogy with the legend of Icarus alluded to in the chapter title is thus motivated by the apparent gap between Incheon's ambition to fly high among world cities and the internal tensions of its socio-economic and transport systems. Urban development, transport planning and its governance are thus key issues faced by Incheon where the smooth integration of transport and urban redevelopment is hindered by external pressures such as inter-city competition on national and international scales, dependence on private and foreign investment, and reliance on its gateway role with Seoul. 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006

SEOUL'S MARITIME GATEWAY AND THE GYEONGIN CORRIDOR

Industrial and port development in the early phase

By the late 19 th century, the Ganghwa and Pyongja treaties initiated the transformation of Incheon from a small fishing village of 4,700 dwellers to a strategic gateway for foreign trade [START_REF] Lee | A study on the development process of Inchon as industrial city and its on-going trend of transformation[END_REF][START_REF] Chung | Process of urbanization in the reclaimed land in Inchon city, Korea[END_REF]. This dynamic was reinforced by Japan's invasion of Korea (1910Korea ( -1945) ) and the need for improving Seoul's maritime access through the construction of the Gyeongin Road (1899) and the Gyeonginson Railway (1900), which finally crossed the Han River in 1917. The development of load centre functions between Southern and Northern peninsula by the Japanese fostered a series of developments through land reclamation (30 per cent of Incheon's current territory has been sea-reclaimed) such as the bridge to Wolmi Island (1904) and the expansion of the port (1906)(1907)(1908)(1909)(1910)(1911)(1912). Related economic and industrial developments were based on rice cleaning and brewery (1883-1910), weaving and milling (1911-1930), heavy industry and ammunitions (1931-1945), reconstruction, and manufacturing industries (1945-1960) A number of constraints soon appeared, however. Tensions with North Korea, proximity with Japan, and the strong regional ties of General Park Chung-Hee (Gelézeau, 2005) incited South Korean authorities to modernize south-eastern ports instead of Incheon, located too close to the DMZ (Demilitarized Zone). Naturally handicapped by high tidal ranges, Incheon quickly faced important limitations in terms of nautical accessibility. Busan notably benefitted from lesser war damage, deeper water depth, and population shifts, thereby facilitating industrial growth in the next decades [START_REF] Frémont | The emergence of a mega-port: from the local to the global, the case of Busan[END_REF]. Welcoming Korea's first container terminal in 1974 could not prevent traffic reorientation towards Busan, thus relegating Incheon to handling bulk cargoes (Rimmer, 2004a). Rapid urban growth, fuelled by rural exodus and industrial sprawl from Seoul, and the specialization in heavy industries and traditional activities resulted in the degradation of urban living environments and in the unplanned dispersal of activities around the port and along communication networks as in many other Korean port cities [START_REF] Jun | Port-city relations in Korea[END_REF].

Rising spatial planning problems

The aforementioned trends are long-lasting and are still visible nowadays on various levels, namely local, regional, and national. First, on the local level, while the whole city grew 24 per cent between 1990 and 2000, downtown areas lost 26 per cent of their population over the same period. This occurred as an effect of a suburban shift in context of intense real estate speculation (Incheon Metropolitan City, 2004) but also as a reaction to the ongoing dereliction of traditional industrial areas where living standards increasingly suffered from environmental degradation, deficiency in public services and cultural facilities (Kim, 2007), calling for revitalization [START_REF] Kim | Revitalization of the logistic function of the Free Economic Zone[END_REF]. The Bupyeong industrial district is emblematic of recent changes in Incheon's urban geography. Notably, [START_REF] Lee | Debt restructuring and the politics of exclusion: a case study of the Daewoo Motor Bupyeong plants in Incheon, South Korea[END_REF] examined the rather passive role of the local government during the debt restructuring of Daewoo Motor in Bupyeong. His in-depth study of power relations confirmed the privilege given to "new" economic functions and "new districts" (see next section) as well as the paramount importance of the national government shaping urban affairs: "The local government actively engaged in the politics of forgetting by withdrawing from its alliance with the union to pursue its economic interest in upgrading Incheon's economic and industrial bases" (p. 2407).

Second, on the regional level, significant increases in income, car ownership and shortage of available land combined with the multiplication of satellite cities and the scarcity of railway networks [START_REF] Kim | Critical issues in transformation of transportation policy in Korean metropolitan areas[END_REF]. More than 60 per cent of passenger flows occur by road in the Capital region nowadays [START_REF] Song | Industrial agglomeration and transport accessibility in metropolitan Seoul[END_REF]. In parallel, the greenbelt policy around Seoul had the effect of raising rents by limiting land supply [START_REF] Jun | The effects of Seoul's greenbelt on the spatial distribution of population and employment, and on the real estate market[END_REF] while fostering the search for available land for new developments, notably in the Incheon areawhich became linked with the local ambitions to develop new, post-industrial activities. However, local and regional authorities in Korea are still constrained by a long tradition of highly centralized government as most major projects are implemented top-down (Kim, 2007).

Third, on the national level, Incheon was de facto excluded from several industrial and planning policies. The first free-trade zones were located at southern locations (i.e. Masan, Gumi, and Iksan) for the aforementioned reasons of regional preference and proximity to Japan [START_REF] Ducruet | Free economic zones in North Korea and South Korea[END_REF]. The industrial de-concentration policy also favoured southern regions and second-tier cities [START_REF] Lee | The Masan Free Export Zone: conflict and attrition[END_REF][START_REF] Park | Kumi and Ansan: dissimilar Korean satellite platforms[END_REF]. Since 2003 these policies also apply to the tertiary sector through the concept of the "multifunctional administrative city" and the creation of innovative and "enterprise cities" outside the capital region to foster balanced regional development (Kim, 2007).

Incheon's present situation is thus in the middle of such spatial and functional pressures.

Traditional industries of the inner core become increasingly isolated from the rest of the city.

Both national and local governments concentrate new investments at the urban fringe where cheaper land is available, thereby reinforcing local disparities and tensions between old and new areas. However, the "second wave" of free economic zone planning of the late 1990s clearly reaffirmed the importance of coastal and urban locations as economic engines and, in particular, reaffirmed the importance of Incheon's proximity to Seoul and the Capital Region.

The question remains whether aiming to become the largest, urban-knowledge oriented free economic zone of the country as well as a major logistics hub in Asia can tackle, solve or at least minimize the long-term and complex inherited urban challenges of Incheon. We turn to these issues in the next sections.

CURRENT PROJECTS AND URBAN TRANSFORMATIONS

Current transformations taking place in Incheon directly reflect the goals stressed by the "2020 Incheon City Plan" designed by the Metropolitan Authority. Rimmer (2004a) notably underlined the strong reference of this plan to increasingly popular, western-focused concepts of "urban competitiveness" and "knowledge-based urban development", with all the caveats involved in the direct translation of those (per se problematic) notions to rather distinct contexts. This is in line with South Korea's post-1997 context favouring economic liberalization policies and openness to foreign capital [START_REF] Carvalho | Developing Locations in the Knowledge Economy: Incheon[END_REF]. Place marketing dimensions of the Songdo project are notably studied by [START_REF] Kim | Place promotion and symbolic characterization of New Songdo City, South Korea[END_REF] with a particular emphasis on the concepts of green city and ubiquitous-city.

Those plans and visions have blunt physical consequencestherefore, investigating their coupling with inherited transport and flow dynamics is of paramount importance. Two (potentially) complementary directions/visions are analysed. First, turning Incheon into a centre of international business and distribution for Northeast Asia (see also Rimmer, 2004b); second, and related with the first, the development of environmentally friendly and top-level urban amenities and modern atmospheres envisaging the attraction of talents, investors and visitors. Two main instruments, one operational and one more conceptual, are used to realize such goals: the Incheon Free Economic Zone (IFEZ) and the Pentaport concept, respectively.

Those are two sides of the same coin, i.e. the ambition to turn Incheon into a major business and logistics hub on regional and global scales. While the IFEZ proposes the development of three multi-functional new districts, the Pentaport notion focuses on the physical integration of transport functions within the city. These two dimensions are in line with current plans of Incheon city to introduce an intelligent, e-driven transportation system, home networking, tele-medicine, disaster prevention/monitoring, and a pollution controlling system by 2020 [START_REF] Sang | Towards ubiquitous city: concept, planning, and experiences in the Republic of Korea[END_REF]. We recall here the main directions and achievements made so far and discuss their ability to answer the city's wider socio-economic, flow-related challenges.

The Incheon Free Economic Zone (IFEZ) concept

The shift from port gateway to global city through the IFEZ strategy since 2003 is mostly based on the attraction of high-tech activities, which were previously developed at satellite towns around Seoul (e.g. Suwon and Paju). This is also a response to massive industrial shifts towards China and other Asian countries (e.g. Vietnam) in the manufacturing and electronics sectors [START_REF] Carvalho | Urban competitiveness, u-city strategies and the development of technological niches in Songdo, South Korea[END_REF]. Three sea-reclaimed and non-contiguous districts constitute the IFEZ: Songdo, Yeongjong, and Cheongna (see Figure 3).

Sea reclamation around Incheon is not totally new, as seen with housing projects in the 1980s and the choice of Yeongjong island in the 1990s to develop the new international airport (see Figure 3). The City of Incheon then launched a so-called tri-port strategy (airport, seaport and teleport) based on international accessibility and the attraction of new industries like IT. Such This complex and multifaceted project envisages a complete functional mix in dense and compact areas, integrating housing, working and leisure functions, as well as many amenities like parks, concert halls, hospitals, international schools, exhibition and convention centre, and golf courses. Spatially, IFEZ unfolds in a polynucleated area totalling 209 sq km based on three "new cities" having distinct vocations and ambitions (see also Table 1): Songdo is planned as a hub for international business, high-education, new high tech industries, and also container port facilities (see Figure 3); Yeongjong is the location of the international airport, focusing on air cargo and logistics, leisure resorts and (business) tourism; Cheongna is planned to focus on international finance, sports, leisure, and IT industry.

The development of IFEZ will take at least two decades to implement. Cheongna's development has barely begun, and in the face of the global economic downturn, there are concerns that this area will be removed from the original IFEZ plan altogether. In addition, Incheon currently faces high levels of municipal debt ($7 billion) due to the ambitious policy of former Mayor Ahn with huge spending on very large construction projects, and an unsuccessful Urban Future expo in 2009. Other difficulties are a consequence of the 2008 global financial crisis, notably resulting in delays for the construction of Songdo's highest tower due to unpaid building costs (Joongang Daily, 2011). So far, many apartment complexes remain empty, while potential residents wait for the completion of subway line 2.

The logistical and airport areas in Yeongjong are well developed, but all the other planned complexes are much more delayed. At present, few tourists visit the area and there seems to be little willingness to pay the bridge fee, thereby limiting the attraction of new residents.

Songdo is the most advanced of the three sub-areas, as seen with the new residential areas, amenities, convention centres and high-tech industries, including many high-tech research complexes (of which some relocated from other locations).The development of a large new international university campus is on-going and the University of Incheon as well as Yonsei University (Seoul) relocated some departments in Songdo.

The 'Pentaport' concept

The Pentaport concept is an extension of the aforementioned tri-port strategy proposed by the Ministry of Maritime Affairs and Fisheries (MOMAF) [START_REF] Kim | Long-term policy for the development of Korean ports in the 21st century[END_REF] and adopted by Incheon Metropolitan City to value the strategic situation of Incheon in Korea and Northeast Asia [START_REF] Chang | Korea's strategic plan to be Northeast Asia's logistics hub: towards the Pentaport approach[END_REF][START_REF] Rimmer | Incheon Pentaport -five ports in one: seeking its conceptual underpinning[END_REF]. Airport and seaport should combine with teleport (or technoport), business port, and leisure port, but the functional and spatial principles of such integration remain unclear in the discourses. [START_REF] Miller | Leisure port development in the Incheon area: concepts and benchmarks[END_REF] rightly state that "Pentaport/City-hub" would better express the urban grounding of the concept.

Although not clearly defined, the Pentaport idea has gradually been applied in several ways, besides the expansion of port facilities (North Incheon Port, Incheon Container Terminal).

The Gyeongin Canal, completed in late 2011 but still only used as a promenade, was designed to reduce trucking with Seoul and to lower logistics costs with China, using 5,000 tonne freighters and ferryboats directly connecting Seoul city centre (Ship-Technology, 2011).

Despite concerns from environmentalists about the real economic impact of the project [START_REF] Hankyoreh | Land Ministry moves forward with Gyeongin Canal construction: environmental groups express concerns about inconsistent reports on the economic efficiency of the plan[END_REF], the canal is expected to accommodate containers (about 970,000 TEUs per year) and other products (steel, cars, sand) as well as passengers. Among the seven Korean international airports, Incheon is the largest. It was ranked 4th in the world ranking of cargo tonnage in 2009 (39th for passengers), and handles more than 97 per cent of South Korea's air freight, although only 0.4 per cent of South Korea's trade occurs by air transport overall [START_REF] Roth | South Korea and Northeast Asian Logistics[END_REF]. Incheon International Airport (IIA) regularly receives praise for its service quality and performance by various international organizations.

The Pentaport concept also applies to the efficient interconnection of IFEZ districts, such as between Songdo and Cheongna (second highway belt of Seoul), Songdo and Yeongjong (second bridge connected with the planned third Gyeongin highway), Songdo and south Gyeonggi province (highway bridge, third Gyeongin highway), and Cheongna and Western Incheon (Gyeongin highway extension). Although the subway system is currently becoming more integrated with the extension of lines 1, 2, 4, and 7, together with a new railway link (Seoul-Yeongjong) and new subway stations, there is a tendency to favour the Seoul-IFEZ axis, as reflected in the limited integration of Incheon in the bus transport system of Seoul and Gyeonggi. Internally, the $ 772 million Wolmi Galaxy Rail (monorail) is currently suspended and about to be dismantled atr a cost of $ 23 million due to corruption and overestimation of local budgets.

In addition, congestion issues remain only partially answered. The relocation of some activities from Incheon old core and Seoul to IFEZ districts generated new commuting flows (e.g. airport-Songdo-Seoul), in addition to already existing ones, notably related with the port (see Figure 3). It was argued by [START_REF] Jun | An agenda for the Incheon Free Economic Zone -Developing Quad Ports and Quad Parks[END_REF] that the inclusion of neighbouring localities such as Shiheung, Ansan, Gimpo and Pyongtaek in the IFEZ project would have enhanced local integration. Pyongtaek, located about 40 km south of Incheon, is a direct rival of Incheon as it is a growing container port and the heart of the Yellow Sea Free Economic Zone. There is certainly some hope to absorb more commuting flows with the completion of the Suin Line (Suwon-Incheon) and of Line 2. The Songdo area is attractive for foreign companies due to its (still) light traffic congestion compared to Seoul's Central Business District. Official plans of Incheon Metropolitan City to redevelop and regenerate other (older) areas in the city based on real estate profits have yet to be implemented. The extent to which internal dynamics are reflected in the position of Incheon in the regional transport system are discussed in the next section.

THE CHANGING PATTERN OF PHYSICAL FLOWS

Incheon as Yellow Sea hub

The proximity with Northern China as well as the lower cost of land transport between Seoul and Incheon compared with Seoul and Busan gradually created an opportunity for Incheon to develop transhipment activities despite its physical limitations [START_REF] Choo | The role of Seoul port in strengthening economic cooperation between China and Korea[END_REF][START_REF] Lee | Trade reorientation and its effects on regional port systems: the Korea-China link along the Yellow Sea rim[END_REF]. Following the first Korea-China freight link (1989) excluding Incheon, several new services were launched during the 1990s mostly based on passenger traffic; it is only in the 2000s that containerized trade grew significantly at Yellow Sea ports, with a sustained bypass of Busan. Incheon increased its share in the Korean port system from 8 per cent in 1991 to 10 per cent in 2004, while Busan decreased from 67 per cent to 46 per cent.

This was mostly due to domestic competition from secondary ports such as Pyongtaek for short-sea shipping, and a twin-hub strategy favouring Gwangyang for deep-sea shipping [START_REF] Lee | Port challenge in Northeast Asia: Korea's two-hub port strategy[END_REF]. During that period, Incheon strengthened its centrality in the Northeast Asian liner shipping networks, by becoming a hub for several North China ports and becoming third most central port in 2006 after Busan and Hong Kong (Ducruet et al., 2009a).

Another dimension of hub formation has been the development of air-sea cargo transfer at Incheon airport to and from nearby Chinese port cities [START_REF] Park | An analysis for the competitive strength of Asian major airports[END_REF]. Because air transport infrastructure in many regional Chinese cities is lacking, an increasing amount of cargo (e.g.

textile) is shipped by some multinationals (e.g. NIKE) from China to Incheon port for being exported through the airport (Table 2). The evolution of this specific niche has been fast, although its remains secondary vis-à-vis total airport cargo throughput: from 33,000 tonnes in 2003 (1.8 per cent) to 60,000 tonnes in 2009 (2.6 per cent) [START_REF] Roth | South Korea and Northeast Asian Logistics[END_REF]. Such services have the advantage of reducing from three weeks to one week the delivery time between China and overseas importers. As presented in Table 2, most air-sea traffic is realized with Yellow Sea Chinese cities located in Shandong and Liaoning provinces, followed by Shanghai and Tianjin.

Concentration and dispersion of hinterland flows

Recent figures on the spatial distribution of port-related cargo flows confirm Incheon's lasting dependence upon the Capital Region, which concentrates about 90 per cent of Incheon's total traffic. Incheon itself accounts for about 60 per cent of tonnage due to the presence of very large industrial complexes and a large population (Figure 5). While hinterland patterns are comparable between imports and exports, we observe a spatial shrinkage of the import hinterland, which concerns heavier traffic due to Incheon's role as an import gateway.

CONCLUSION: WINGS OF DESTINY?

The examination of past and current developments in Incheon through the lense of urban and transport issues provide a contrasting picture of the situation. While Incheon may not, like in the legend of Icarus, be about to "fall" from its attempt to touch the shining model of the global city, there are certainly more efforts needed to better "ground" the future of this rapidly growing city. Also there is a need to better understand how different (except, perhaps, by the size) is Incheon's transformation from other waterfront redevelopment experiences and city models all over the world.

In When there is no further space in Seoul and while Busan is too far, Incheon appears as the logical choice for such policies, notably profiting from existing infrastructure, highly proactive and entrepreneurialism-based urban management, and all the advantages of developing new real estate and functions in sea-reclaimed "virgin" land. What remains problematic is the growing social and economic divide within the city, between the IFEZ as a new enclave (in terms of architecture and amenities, English speaking, level of schools and services within the new city) and the rest of the city, which does not seem to be included in the project mostly focussed on the domestic and foreign upper class. If the IFEZ succeeds in attracting and fixing new residents, it will, at least, help decongest Seoul and the capital region as a whole, but the way towards local integration (both physical, social, and economic) between old and new remains to be seen.

Figure 1 :

 1 Figure 1: Economic welfare and unemployment in Incheon

Figure 2 :

 2 Figure 2: Incheon and the Gyeongin corridor

  visions gained momentum in the aftermath of the 1997 Asian financial crisis, as they became linked with the economic reforms towards the attraction of external investment and progressive liberalization of service industries (e.g. finance). In 1998 the Foreign Investment Promotion Act resulted in the creation of new Free Economic Zones (FEZ) while in 2002 the (entrepreneurial and business-oriented) Mayor Ahn Sang-Soo undertook the realization of the IFEZ. Nine years after the opening of the new international airport (2001), the new Incheon Grand Bridge linked Yeongjong and Songdothe most emblematic face of IFEZin a 15 minutes car drive.

Figure 3 :

 3 Figure 3: Land occupation at IFEZ new districts and main projects

  The proximity with North Korea became another opportunity during the early 2000s, in the context of warmer North-South political relations and growing inter-Korean trade backed by the Gaeseong Industrial Park near the DMZ and a maritime agreement. Incheon gradually concentrated North-South shipping flows as well as a growing proportion of total North Korean traffic, especially with Nampo, Pyongyang's gateway (Figure4). Although more recent figures are unknown and might reveal a decline in inter-Korean traffic due to prolonged tensions on the peninsula and persistent crisis in North Korea, Incheon's share reached about 30 per cent of North Korea's totals in recent years(Ducruet et al., 2009b). The same figure for containers even reached 54 per cent in 2003 and 82 per cent in 2006, a stark contrast with the average of 5-10 per cent during the 1980s and 1990s. While such trends reveal ongoing North-South economic integration processes they also point at North Korea's growing economic difficulties.

Figure 4 :

 4 Figure 4: Inter-Korean maritime container flows, 1985-2005

  Between 2006 and 2009, Incheon and Seoul have increased their respective shares from 60.5 per cent to 64.6 per cent and from 14 per cent to 15.8 per cent. By contrast, distant regions such as Chungcheong (6.6 per cent to 4.6 per cent), Gangwon (2.3 per cent to 1.1 per cent), Gyeongsang (2.6 per cent to 1.7 per cent), and Jeolla (2.7 per cent to 0.6 per cent) have witnessed a regular decrease of both their import traffic volume and share in relation with Incheon port. Mainly because of cheaper land prices, many logistics and distribution centres moved from Incheon to Pyongtaek in recent years. As a result, export flows from outside the Capital Region have increased in proportion to import flows from 10 per cent to 26 per cent along the period. The geographic pattern of export hinterlands has remained rather stable during the same period, notwithstanding a regular increase of the traffic volume and share of Seoul (13.2 per cent to 20.1 per cent) but also Busan and Gyeongsang province, the two latter generating rather negligible volumes. The higher spatial flexibility of containers is reflected in their wider geographic coverage and the lower share of Incheon city (20-30 per cent) and the capital region (60 per cent) in total hinterland traffic, while the share of the rest of country has increased from 13 per cent in 2006 to 21 per cent in 2009. This underlines the specialization of Incheon in non-containerized freight flows as mentioned above. The capital region has slightly lost market share, from 65 per cent in 2006 to 60 per cent in 2009. This geographic expansion also confirms the improved landward accessibility of Incheon with regard to export flows towards Chinese forelands, with the aforementioned traffic shift towards Yellow Sea ports for short-sea shipping. The spatial shrinkage of total cargo hinterlands is, by contrast, better explained by local and national factors of congestion and reduced domestic demand, notably after the 2008 global financial crisis.

Figure 5 :

 5 Figure 5: Incheon's hinterland flows, 2006-2009

  terms of transport functions and options, the Pentaport concept has stimulated the development of diversity and integration, although the direct link between concept and practice remains questionable. Port and airport functions have increased in both absolute and relative terms on domestic and Northeast Asian levels since the early 2000s, with the modernisation and expansion of infrastructure. The rapid development of air-sea freight services with China as well as the planned completion of river canal shipping for ocean-river barges between China and Seoul through Incheon complement such dynamics. The growing density and connectedness of both the public transport system and the road system are very positive elements to reduce traffic congestion within Incheon as well as the mixing of commuting and trucking, which is typical of congested cities in developing countries. Such a picture is thus far from the one of Incheon as a specialized port gateway relying entirely on trucking. Nevertheless, more efforts are needed in terms of intermodalism to and from the port, despite the short distance between Seoul and Incheon hampering the establishment of rail services to further reduce truck flows along the main arteries. The shift of important logistics activities towards Pyongtaek is not the sign of improvement within Incheon itself.Yet, intense competition among South Korean cities and the pursuit of economic growth through the IFEZ strategy do not seem to delete entirely the main tensions associated with freight flows in a large port city that Incheon continues to be. The coexistence of old and new functions within a relatively narrow space still faces problems of urban sprawl, environmental degradation, and social exclusion. Indeed, transport-related issues were somewhat left behind other matters by the Ministry of Finance and Economy and by the City itself (leader in the IFEZ project), in their pursuits of "urban entrepreneurialism" strategies. The spatial shrinkage of Incheon port's hinterland around the capital region during recent years has a global cause (overall fall in demand) but also reflects the expansion of nearby competitors domestically such as Pyongtaek in Gyeonggi province. Just like for many other large port cities of the world, the redevelopment of existing sites within the urban area is more costly than greenfield development at the urban fringe. Diversifying the maritime accessibility of the capital region is certainly desirable to relieve congestion from the main axis and reduce logistics costs; however, there is a risk to further spread transport-related negative impacts across the entire region.The question whether Incheon becomes something else than a growing suburb or an edge city of an expanding Seoul remains difficult to answer since many developments are not yet completed. In many ways, Incheon's developments are simply "shifts" or "transfers" from Seoul, such as the international airport in Yeongjong island and the new university campus in Songdo district, among other examples. This is mostly explained by the lack of available space within Seoul and the peak rent values in a context of intense real estate speculation across the whole country. Instead of the emergence of a "new global city" in Incheon, we would rather talk about the expansion of the capital region (centred upon Seoul) through both spatial diffusion (within the capital region) and spatial concentration (within the country). In such transfers the local authority seems to follow the guidelines of the central government rather than to develop its own solutions and projects, which remain largely influenced by external views on what Incheon should become.

2007 Ratio based on Metropolitan Cities' average

	The remainder of this chapter is organized as follows. First we provide background on the
	emergence of Incheon and the Gyeongin (referring to Seoul-Incheon). corridor since the late
	nineteenth century, complemented by a look at the local impact of the "Han River miracle" on
	Incheon. Second, we focus on the policies and strategies pursued since the 1990s to turn
	Incheon into a new logistics hub and a prime location for innovative activities through the
	Free Economic Zone (FEZ) strategy. Third, by looking at the distribution of freight flows in
	recent years through Incheon airport and seaport, we provide evidence about Incheon's
	changing role as transport node at various geographic scales. In the conclusion, we propose to
	evaluate the gaps between the dynamics, the ambition and the action in the transformation of
	Incheon from port gateway to global city.
	GDP per capita (constant prices)	Unemployment rate

Table 1 : The new cities of IFEZ

 1

	New IFEZ cities	Area (Km 2 )	Planned population (2020)	Developers	Major projects (ongoing and planned, examples)
					Int'l business district, knowledge
				Incheon Metropolitan City,	and information industry complex,
	Songdo	53.4	253,000	Songdo Technopark, NSIC, Incheon Urban development	bio-complex, high-tech cluster, global university campus, leisure
				corporation, etc.	park (e.g. Universal), Incheon new
					port.
				Korea land corporation, airport	Airport, tourism complex, leisure
	Yeongjong	138.3	169,000	cooperation, Incheon Urban	complex, sky city, media-city,
				development corporation, etc.	logistics and distribution centre.
	Cheongna	17.8	90,000	Korea land corporation, Incheon Metropolitan City, Korea rural community & agriculture corporation, etc.	Global Finance/Business and leisure, sports, R&D centre, high-tech industry complex (automotive, "Robotland"), Floricultural complex.
	Total IFEZ 209.5	512,000
				Source: IFEZ (2010)

Table 2 : Air-sea cargo flows between Incheon and Chinese port cities (Unit: tons)

 2

	Port of origin	2003	2004	2005	2006	2007	2008	2009
	Qingdao				19,961.9 17,131.8 18,627.9 15,484.0
	Yantai				3,699.9 7,562.5 14,047.8 14,815.2
	Shanghai				8,930.2 9,674.6 12,005.7 7,917.3
	Weihai				8,104.7 7,292.1 8,577.5 7,857.8
	Shidao				197.2	112.3	2,312.6 4,782.1
	Tianjin				745.4	933.4	1,155.4 2,990.2
	Long Yan				-	585.2	332.0	2,441.1
	Dalian				2,399.1 2,151.9 2,730.4 2,121.8
	Qinhuangdao				-	226.6	-	401.9
	Lianyungang				-	-	-	347.5
	Rizhao				371.3	1,096.6 1,039.4	-
	Shekou				-	-	185.0	-
	Yingkou				140.5	-	-	-
	Total	33,437 39,415 44048 45,245.4 47,111.7 61,477.9 59,612.7
	Share (%) in total airport cargo traffic	1.81	1.85	2.05	1.95	1.86	2.52	2.58

ACKNOWLEDGEMENTS

Luis de Carvalho thanks the financial support of the Portuguese Foundation for Science and Technology. We thank Chung Tae Won and Lee Kwon-Hyung (IDI) for their valuable


Faouzia Aloui
email: aloui@ann.jussieu.fr

Alain Haraux
email: haraux@ann.jussieu.fr

Sharp ultimate bounds of solutions to a class of second order linear evolution equations with bounded forcing term

We establish a precise estimate of the ultimate bound of solutions to some second order evolution equations with possibly unbounded linear damping and bounded forcing term.

Introduction

Let H be a real Hilbert space. In the sequel we denote by (u, v) the inner product of two vectors u, v in H and by |u| the H-norm of u. Given f ∈ L ∞ (R, H), we consider the second order evolution equation with possibly unbounded and time-dependent damping operator B:

u

′′ + Au + Bu ′ = f (t) (0.1)
where A is a fixed linear, self-adjoint and positive operator in H. We assume that the domain of A is dense in H and A is coercive, in other terms:

∃λ > 0, ∀u ∈ D(A), (Au, u) ≥ λ|u| 2 . (0.2)
Obviously the set of λ satisfying (0.2) is closed. For our purpose the best possible is the largest one, ie. λ = inf u∈D(A),|u|=1

(Au, u) =: λ 1 (A).

We introduce V = D(A

2 ) endowed with the norm given by

∀u ∈ V, u = |A 1 2 u|.
This norm defined on V is equivalent to the graph norm of A 1 2 as a result of the coerciveness hypothesis on A.

In the sequel, B : V → V ′ may be a time-dependent continuous operator. When B is linear and time-independent, we write (0.1) in the following form:

U ′ + LU = F (t) (0.3) with U = (u, u ′ ), L = 0 -I A B and F = (0, f ). If B ∈ L(V, V ′ ) satisfies Bv, v ≥ 0 ∀v ∈ V
then it is not difficult to check (cf.e.g. [START_REF] Aloui | Compactness of trajectories to some nonlinear second order evolution equations and applications[END_REF][START_REF] Brezis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert North-Holland Mathematics Studies[END_REF][START_REF] Brezis | Equations et inéquations non linéaires dans les espaces vectoriels en dualité[END_REF]) that L is a maximal monotone operator with dense domain D(L) = {(u, v) ∈ V × V, Au + Bv ∈ H} in V × H. Then, by Hille -Yosida's Theorem (cf.e.g. [START_REF] Brezis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert North-Holland Mathematics Studies[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]), L generates a C 0 contraction semi-group S(t) that insures the existence and uniqueness of a mild solution u ∈ C(R + , V ) ∩ C 1 (R + , H) to (0.1) on R + for any pair of initial data u 0 = u(0) ∈ V ; u 1 = u ′ (0) ∈ H. Moreover, the two following properties are equivalent cf [START_REF] Haraux | Uniform decay and Lagrange stability for linear contraction semi-groups[END_REF]:

1) S(t) is exponentially damped on V ×H which means that for some constants

M ≥ 1, δ > 0 ∀t ≥ 0, S(t) L(H) ≤ M exp(-δt) 2) ∀F ∈ L ∞ (R + , H), any solution of (0.3) is bounded in V × H for t ≥ 0 .
In addition in this case we have

lim t→∞ U (t) ≤ M δ lim t→∞ F (t) H
In applications to infinite or even finite dimensional second order equations, this method does not give the best possible estimate because it is not easy to optimize on M and δ. This was already observed in [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF] and [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF] where precise estimates of lim t→∞ U (t) were given in the case of (0.1) with B = cI or B = cA

1 2 .
The main objective of this paper is to generalize the results of [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF][START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF] for B time independent and improve some of the results in the specific cases B = cI andB = cA 1 2 . We shall consider also the case B = cA which was not studied before.

The plan of the paper is the following: section 1 contains an improvement of the main result from [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF] in the general case B = β(t). Section 2 is devoted to the case where B = B(t) is linear and self-adjoint. Section 3 gives the precise statements when B = cA α with a special treatment in the case B = B 0 = cA 1 2 and Section 4 is devoted to the main concrete applications of Theorem 2.1. Finally Section 5 is devoted to slightly different examples and some additional remarks.

An ultimate bound valid for general time-dependent damping terms

We consider the equation:

u ′′ + β(t)u ′ + Au = f (t) (1.1)
where t ∈ R + . For this equation, we improve some general estimates obtained in [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF] when

β(t) : R + → C(V, V ′
) is a measurable family of possibly nonlinear continuous operators which satisfies the two hypotheses:

∃c > 0, ∀t ∈ R + , ∀v ∈ V, β(t)v, v ≥ c|v| 2 . (1.2) ∃C > 0, ∀t ∈ R + , ∀v ∈ V, β(t)v 2 * ≤ C β(t)v, v . (1.3)
It is immediate (cf. e.g. [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF]) that c ≤ Cλ 1 where λ 1 = λ 1 (A). Our main result is the following

Theorem 1.1. For any solution u ∈ W 1,∞ loc (R + , V ) ∩ W 2,∞ loc (R + , H) of (1.1)
we have the estimate :

max( lim t→∞ u(t) , lim t→∞ |u ′ (t)|) ≤ max( √ 12 C c , 3 c ) lim t→∞ |f (t)| (1.4)
Proof. For simplicity of the formulas, we drop the variable t whenever possible and we denote by z ′ the time derivative of a (scalar or vector) time-dependent function z. We consider for some α > 0 to be chosen later the following modified energy functional:

Φ = |u ′ | 2 + u 2 + α(u, u ′ ) - α 2 4 |u| 2 .
Then

Φ ′ = -2 βu ′ , u ′ + α|u ′ | 2 -α u 2 -α βu ′ , u + f, 2u ′ + αu - α 2 2 (u, u ′ ) = - α 2 (|u ′ | 2 + u 2 + α(u, u ′ )) -2 βu ′ , u ′ + 3α 2 |u ′ | 2 - α 2 u 2 -α βu ′ , u + f, 2u ′ + αu we set Ψ = |u ′ | 2 + u 2 + α(u, u ′ ) ≥ Φ.
Then, by using (1.2), we have:

Φ ′ ≤ - α 2 Φ - 1 2 βu ′ , u ′ -( 3c 2 - 3α 2 )|u ′ | 2 - α 2 u 2 -α βu ′ , u + f, 2u ′ + αu we have, from (1.3): | βu ′ , u | ≤ √ C βu ′ , u ′ 1 2
u By using Young's inequality we deduce :

|α βu ′ , u | ≤ αC βu ′ , u ′ + α u 2 4
Assuming αC ≤ 1 2 , then:

Φ ′ + α 2 Φ ≤ - 3 2 (c -α)|u ′ | 2 + 2 f, u ′ - α 4 u 2 + α f, u Assuming 3 2 (c -α) ≥ 1 2 c, then α ≤ 2 3 c.
We have, by using Young's inequality:

- 3 2 (c -α)|u ′ | 2 + 2 f, u ′ ≤ - c 2 |u ′ | 2 + 2 f, u ′ ≤ 2 c |f | 2 Moreover α f, u ≤ α √ λ 1 |f | u
Therefore, by Young's inequality:

- α 4 u 2 + α f, u ≤ α(- u 2 4 + 1 √ λ 1 |f | u ) ≤ α λ 1 |f | 2 ≤ αC c |f | 2 ≤ 1 2c |f | 2 Then Φ ′ + α 2 Φ ≤ 5 2c |f | 2
Then, we find that Φ is bounded with:

lim t→∞ Φ(t) ≤ 5 cα lim t→∞ |f (t)| 2 .
Moreover, we have:

-α(u, u ′ ) ≤ |u ′ | 2 + α 2 4 |u| 2
We set

F = lim t→∞ |f (t)| 2 .
In particular for any ǫ > 0 we have for t large enough

(1 - α 2 2λ 1 ) u(t) 2 ≤ u(t) 2 - α 2 2 |u(t)| 2 ≤ Φ(t) ≤ 5 cα F + ǫ 2 .
Now since α ≤ 2 3 c and α ≤ 1 2C , we have

α 2 2λ 1 ≤ c 6λ 1 C ≤ 1 6
Then we find lim

t→∞ u(t) 2 ≤ 6 cα F + 2ǫ Finally, by choosing α = inf( 2 3 c, 1
2C ), we obtain by letting ǫ → 0:

lim t→∞ u(t) ≤ max( 12C c , 3 c ) lim t→∞ |f (t)|.
In order to estimate u ′ , observe that for t large enough:

|u ′ (t)| 2 + λ 1 |u(t)| 2 + α(u, u ′ ) - α 2 4 |u(t)| 2 ≤ 5 cα F + ǫ 2 Since α ≤ 2 3 c ≤ c and α ≤ 1 2C ≤ λ 1 2c , then α 2 ≤ αc ≤ λ 1 2 .
Consequently for t large enough

5 6 |u ′ (t)| 2 + 2α 2 |u(t)| 2 + 1 6 |u ′ (t)| 2 + α(u, u ′ ) - α 2 4 |u(t)| 2 ≤ 5 cα F + ǫ 2 In other terms 5 6 |u ′ (t)| 2 + α 2 4 |u(t)| 2 + | 1 √ 6 u ′ + √ 3 √ 2 αu| 2 ≤ 5 cα F + ǫ 2 Then: lim t→∞ |u ′ (t)| 2 ≤ 6 cα F + 2ǫ Also assuming α = inf( 1 2C , 2 3
c) and letting ǫ → 0, we have:

lim t→∞ |u ′ (t)| ≤ max( 12C c , 3 c ) lim t→∞ |f (t)|. Remark 1.2. If β(t) = B 0 ∈ L(V, V ′ ), it is well known that the conditions (u 0 , u 1 ) ∈ D(A) × V and f ∈ C 1 (R + , V ) imply u ∈ C 1 (R + , V ) ∩ C 2 (R + , H)
. By density on (u 0 , u 1 , f ) we obtain easily the following

Corollary 1.3. Let β(t) = B 0 ∈ L(V, V ′ ). In this case any mild solution u ∈ C(R + , V ) ∩ C 1 (R + , H) of (1.1) satisfies (1.4).
Remark 1.4. In [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF], the following estimate was established sup{ lim

t→∞ u(t) , lim t→∞ |u ′ (t)|)} ≤ √ 3(C + 4 c ) lim t→∞ |f (t)| (1.5) Since 12C c ≤ √ 12 4 (C + 4 c ) = √ 3 2 (C + 4 c ) and 3 c ≤ 3 4 (C + 4 c ) ≤ √ 3 2 (C + 4 c ),
we can see that Theorem 1.1 improves the estimate (1.5) by a factor 2 for all values of c and

C. Moreover if C → ∞ with C c bounded, max( 12C c , 3 c
) remains bounded and (C + 4 c ) tends to infinity, therefore (1.4) improves (1.5) by an arbitrarily large amount. A typical case is :

β = cB 0 with c → ∞ since then C c is fixed and C → ∞.
2 The case of a linear self-adjoint damping operator

In this section, we study the equation (0.1) where B : R + -→ L(V, V ′ ) is a self-adjoint and possibly unbounded operator and satisfies the following hypotheses:

∃c > 0, ∀t ∈ R + , ∀v ∈ V, B(t)v, v ≥ c|v| 2 (2.1) ∃C > 0, ∀t ∈ R + , ∀v ∈ V, B(t)v, v ≤ C Av, v (2.2)
The following result, will give close to optimal estimates even when B is independent of time.

Theorem 2.1.

Any solution u ∈ W 1,∞ loc (R + , V ) ∩ W 2,∞ loc (R + , H
) of (0.1) satisfies the following estimate:

max( lim t→∞ u(t) , lim t→∞ |u ′ (t)|) ≤ max( 3C c , 3 √ 2c ) lim t→∞ |f (t)| (2.3) Proof. Considering again the energy functional Φ = |u ′ | 2 + u 2 + α(u, u ′ ) -α 2
4 |u| 2 we find:

Φ ′ = -2|B 1 2 u ′ | 2 + α|u ′ | 2 -α u 2 -α(Bu ′ , u) + (f, 2u ′ + αu) - α 2 2 (u, u ′ ) = - α 2 Ψ -(2 - 3α 2c )|B 1 2 u ′ | 2 - α 2 u 2 -α(Bu ′ , u) + (f, 2u ′ + αu) ≤ - α 2 Φ -(2 - 3α 2c )|B 1 2 u ′ | 2 - α 2 u 2 -α(Bu ′ , u) + (f, u ′ + αu) + (f, u ′ ) where Ψ = |u ′ | 2 + u 2 + α(u, u ′ ) ≥ Φ. By (2.1
) and Young's inequality, we have

(f, u ′ ) ≤ 1 2c |f | 2 + c 2 |u ′ | 2 ≤ 1 2c |f | 2 + 1 2 |B 1 2 u ′ | 2
Therefore by using (2.2), we obtain

Φ ′ ≤ - α 2 Φ -( 3 2 - 3α 2c )|B 1 2 u ′ | 2 - α 2C |B 1 2 u| 2 -α(Bu ′ , u) + 1 2c |f | 2 + (f, u ′ + αu) Assuming 3 2 - 3α 2c ≥ 1 2 and α 2 ≤ α C which means α ≤ 2 3 c and α ≤ 1 C we deduce Φ ′ ≤ - α 2 Φ - 1 2 |B 1 2 u ′ | 2 - α 2 2 |B 1 2 u| 2 -α(Bu ′ , u) + 1 2c |f | 2 + (f, u ′ + αu) ≤ - α 2 Φ - 1 2 |B 1 2 (u ′ + αu)| 2 + 1 2c |f | 2 + (f, u ′ + αu)
By using (2.1), we find

Φ ′ ≤ - α 2 Φ - c 2 |u ′ + αu| 2 + 1 2c |f | 2 + (f, u ′ + αu)
By using Young's inequality in the last term, we have

(f, u ′ + αu) ≤ 1 2c |f | 2 + c 2 |u ′ + αu| 2 Then Φ ′ ≤ - α 2 Φ + 1 c |f | 2
Then we find that Φ is bounded with

lim t→∞ Φ(t) ≤ 2 cα lim t→∞ |f (t)| 2
By setting F = lim t→∞ |f (t)| 2 we see that for t large enough and any ǫ > 0

|u ′ (t)| 2 + u(t) 2 + α(u(t), u ′ (t)) - α 2 4 |u(t)| 2 ≤ 2 cα F + ǫ 2
In other terms

u(t) 2 + |u ′ (t) + α 2 u(t)| 2 - α 2 2 |u(t)| 2 ≤ 2 cα F + ǫ 2 By using α ≤ 2
3 c and (2.1), we obtain for t large enough:

u(t) 2 - α 3 |B 1 2 u(t)| 2 ≤ 2 cα F + ǫ 2 now using α ≤ 1 C and (2.
2), for t large enough we obtain :

u(t) 2 ≤ 3 αc F + 2ǫ
Finally by selecting α = inf( 2 3 c, 1 C )and letting ǫ → 0 we find :

lim t→∞ u(t) ≤ max( 3C c , 3 √ 2c ) lim t→∞ |f (t)|
In order to estimate u ′ , for t large enough by using (0.2)

2 3 |u ′ (t)| 2 + λ 1 |u(t)| 2 + α(u(t), u ′ (t)) + 1 3 |u ′ (t)| 2 - α 2 4 |u(t)| 2 ≤ 2 cα F + ǫ 2 Since α ≤ 2 3 c ≤ c and α ≤ 1 C ≤ λ 1 c , we have α 2 ≤ αc ≤ λ 1 .
Therefore, for t large enough:

2 3 |u ′ (t)| 2 + α 2 |u(t)| 2 + α(u(t), u ′ (t)) + 1 3 |u ′ (t)| 2 - α 2 4 |u(t)| 2 ≤ 2 cα F + ǫ 2
Then, for t large enough

2 3 |u ′ (t)| 2 + 3α 2 4 |u(t)| 2 + α(u(t), u ′ (t)) + 1 3 |u ′ (t)| 2 ≤ 2 cα F + ǫ 2
In other terms

2 3 |u ′ (t)| 2 + | √ 3 2 αu(t) + 1 √ 3 u ′ (t)| 2 ≤ 2 cα F + ǫ 2
Hence, for t large enough 3 Applications when B = γA α , 0 ≤ α ≤ 1

|u ′ (t)| 2 ≤ 3 αc F + 2ǫ
In this section we consider the case of a time independent self-adjoint B proportional to some positive power of A. In order to garantee exponential damping of the associated semi-group the power will be taken ≤ 1.

3.3

The case B = γI :

we consider the equation:

u ′′ + γu ′ + Au = f (t) (3.5)
Applying Theorem 2.1 to (3.5) with C = γ λ 1 and c = γ we find :

max( lim t→∞ u(t) , lim t→∞ |u ′ (t))| ≤ max( 3 λ 1 , 3 √ 2γ ) lim t→∞ |f (t)| (3.6) Remark 3.1.
Let us compare our result on (3.6) with the estimates from [START_REF] Fitouri | Sharp estimates of bounded solutions to some semilinear second order dissipative equations[END_REF].

In [START_REF] Fitouri | Sharp estimates of bounded solutions to some semilinear second order dissipative equations[END_REF] it was shown that

lim t→∞ u(t) ≤ 4 γ 2 + 1 λ 1 lim t→∞ |f (t)|. (3.7)
If γ is fixed and λ 1 → ∞ we have:

max( 3 λ 1 , 3 √ 2γ ) = 3 √ 2γ and 4 γ 2 + 1 λ 1 ⋍ 2
γ therefore we find that (3.6) is worse than (3.7), hence Theorem 2.1 is weaker than the result of [START_REF] Fitouri | Sharp estimates of bounded solutions to some semilinear second order dissipative equations[END_REF] in this case.

If λ 1 is fixed and γ → ∞ we have:

max( 3 λ 1 , 3 √ 2γ ) = 3 λ 1 and 4 γ 2 + 1 λ 1 ⋍ 1
λ 1 therefore in this case Theorem 2.1 is also weaker than [START_REF] Fitouri | Sharp estimates of bounded solutions to some semilinear second order dissipative equations[END_REF].

Let us determine the values of γ and λ 1 for which condition (3.6) is better than (3.7).

To this end we can study the condition:

4 γ 2 + 1 λ 1 max( 3 λ 1 , 3 √ 2γ ) > 1
Therefore, we introduce:

g(γ, λ 1 ) = 4 + γ 2 λ 1 max( 3γ 2 λ 1 , 3 √
2 ) By setting r = γ √ λ 1 , we obtain:

g(γ, λ 1 ) = p(r) = √ 4 + r 2 max( √ 3r 2 , 3 √ 2 )
Introducing τ = r 2 , we have: 92 , 3τ ) A simple calculation shows that

P (τ ) = 4 + τ max(
P (τ ) > 1 ⇐⇒ τ ∈] 1 2 , 2[ ⇐⇒ r 2 ∈] 1 2 , 2[ ⇐⇒ r ∈] 1 √ 2 , √ 2[. Finally, we obtain that if γ ∈] λ 1 2 , √ 2λ 1 [, Theorem 2.
1 improves the result of [START_REF] Fitouri | Sharp estimates of bounded solutions to some semilinear second order dissipative equations[END_REF].

3.4

The case B = γA :

Let us consider the equation:

u ′′ + γAu ′ + Au = f (t) (3.8)
with γ > 0.

When we apply Theorem 2.1 to the equation (3.8) with C = γ and c = γλ 1 , we obtain immediately:

Corollary 3.2. Any solution of (3.8) satisfies the following hypotheses:

max( lim t→∞ u(t) , lim t→∞ |u ′ (t)|) ≤ max( 3 λ 1 , 3 √ 2γλ 1 ) lim t→∞ |f (t)| (3.9)
Remark 3.3. This result is new and was not obtained in [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF].

3.5

The case B = γA

1 2
In this subsection we consider the so-called structural damping (cf [START_REF] Chen | A mathematical model for elastic systems with structural damping[END_REF][START_REF] Chen | Proof of extensions of two conjectures on structural damping for elastic systems[END_REF][START_REF] Chen | Russell on Structural Damping for Elastic Systems[END_REF] for the terminology and main properties). Therefore we consider as in [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF] the equation:

u ′′ + γA 1 2 u ′ + Au = f (t) (3.10) with γ > 0. If we apply theorem (2.1) with c = γ √ λ 1 and C = γ √ λ 1 , we obtain max( lim t→∞ u(t) , lim t→∞ |u ′ (t)|) ≤ max( 3 λ 1 , 3 √ 2γ √ λ 1 ) lim t→∞ |f (t)| (3.11)
By comparaison with [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF], we remark that (2.1) gives a weaker result. We shall now recover the estimate on u from [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF] in the case of large damping by a method introduced by C. Fitouri (cf. [START_REF] Fitouri | Sharp estimates of bounded solutions to some semilinear second order dissipative equations[END_REF]) which is less complicated than the method of [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF]. We recall the main result from [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF].

then

Φ ′ ≤ - α 2 Ψ + α 2 |f | 2
We have:

α = γ -γ 2 -4 = 4 γ + γ 2 -4 ≤ 4 γ then, from (3.12) 0 < α 2 4 ≤ 4 γ 2 < 1 We have Ψ = |A 1 2 u ′ | 2 + α(Au, A 1 2 u ′ ) + |Au| 2 = |A 1 4 (A 1 4 u ′ + α 2 A 3 4 u)| 2 + (1 - α 2 4 )|Au| 2 ≥ λ 1 |A 1 4 u ′ + α 2 A 3 4 u)| 2 + (1 - α 2 4 )|A 3 4 u| 2 = λ 1 Φ Hence Φ ′ ≤ - α √ λ 1 2 Φ + α 2 |f | 2 since Φ is bounded, we have ∀t ∈ R, Φ(t) ≤ 1 √ λ 1 f (t) 2 ∞ which means ∀t ∈ R, |A 1 4 u ′ (t)| 2 + |A 3 4 u(t)| 2 + α(A 1 2 u(t), A 1 2 u ′ (t)) ≤ 1 √ λ 1 f (t) 2 ∞ Then ∀t ∈ R, λ 1 |A 1 2 u(t)| 2 + α 2 d dt |A 1 2 u(t)| 2 ≤ 1 √ λ 1 f (t) 2 ∞ Finally, since u is bounded in V on R, we obtain ∀t ∈ R, u(t) ≤ 1 √ λ 1 f (t) ∞ (3.13)
Remark 3.5. By this method, we do not recover the estimate of u ′ from [START_REF] Haraux | Sharp estimates of bounded solutions to a second-order forced equation with structural damping[END_REF] in the strongly damped case γ > 2.

Main examples

Let Ω be a bounded domain in R N and γ > 0.

Example 4.1. We consider the following equation

u tt -∆u + γu t = f u /∂Ω = 0 (4.1)
Then, as a consequence of (3.5) we have the following result valid for all mild solutions

lim t→∞ Ω ∇u 2 dx 1 2 ≤ max 3 λ 1 (Ω) , 3 √ 2γ lim t→∞ |f (t)|
This result improves on [START_REF] Haraux | Sharp estimates of bounded solutions to some second-order forced dissipative equations[END_REF] when

λ 1 (Ω) 2 < γ < 2λ 1 (Ω).
Example 4.2. We consider the equation

u tt -∆u -γ∆u t = f u /∂Ω = 0 (4.2)
We have the following result valid for all mild solutions

lim t→∞ Ω ∇u 2 dx 1 2 ≤ max 3 λ 1 (Ω) , 3 √ 2γλ 1 (Ω) lim t→∞ |f (t)|
Example 4.3. We consider the equation

u tt + ∆ 2 u -γ∆u t = f u = ∆u = 0 on ∂Ω (4.3)
Then, we have for all mild solutions

lim t→∞ Ω |∆u| 2 dx 1 2 ≤ 1 λ 1 (Ω) max 1, 2 γ lim t→∞ |f (t)|
This follows from Theorem 3.4 since here λ 1 (A) = λ 1 (Ω) 2

Example 4.4. We consider the equation

u tt + ∆ 2 u -γ∆u t = f u = |∇u| = 0 on ∂Ω (4.4)
Then, we shall establish

lim t→∞ Ω |∆u| 2 dx 1 2 ≤ max 3 λ 1 (Ω)λ 1 (A) , 3 √ 2γλ 1 (Ω) lim t→∞ |f (t)|
Indeed, in this example, we have

B = -γ∆ : H 1 0 → H -1 ; A = ∆ 2 with domain D(A) = {u ∈ H 2 (Ω)| u = |∇u| = 0 on ∂Ω} and cI ≤ B ≤ CA, with c = γλ 1 (Ω)
To get an estimate for C we observe that

(Bv, v) = γ Ω ∇u 2 dx = -γ Ω ∆v.v dx = γ( Ω |∆v| 2 dx) 1 2 ( Ω |v| 2 ds) 1 2 = γ λ 1 (A) ( Ω |∆v| 2 dx) = γ λ 1 (A) (Av, v)
Therefore, we can take C ≤ γ λ 1 (A) and this shows the claim. Remark 4.5. Actually, since we used a Cauchy-Schwarz inequality for two linearly independent functions it is clear that the optimal value of C is striclly less than γ λ 1 (A) . More precisely to obtain the optimum we need to evaluate

µ = inf Ω |∆v| 2 dx Ω |∇v| 2 dx , v ∈ H 2 0 (Ω), v = 0 = inf Ω |∆v| 2 dx, v ∈ H 2 0 (Ω), Ω |∇v| 2 = 1
By the Lagrange multiplier theory, there is v = 0 such that

∆ 2 v = -µ∆v v ∈ H 2 0 (Ω)
2 )(1 -cos(πλ) + 2 cos 2 ( πλ 2 )) = λπ ⇐⇒ 2 tan( πλ 2 ) = λπ ⇐⇒ tan( πλ 2 ) = πλ 2 Therefore πλ 2 > π =⇒ λ > 2 and µ = λ 2 > 4. πλ
Summarizing the 3 cases we conclude that the minimal possible value of µ is 4.

Corollary 4.7. Any mild solution u of

u tt + u xxxx -γu xxt = f u(t, 0) = u(t, π) = u x (t, 0) = u x (t, π) = 0 (4.5)
satisfies the asymptotic bound:

lim t→∞ Ω |u xx | 2 dx 1 2 ≤ max √ 3 2 , 3 √ 2γ lim t→∞ |f (t)| (4.6)
5 Additional results

5.1

The first eigenvalue of a square root.

At several places in this paper we used implicitely the property

λ 1 (A 1 2 ) = (λ 1 (A)) 1 2
where A is a self-adjoint coercive operator. This property is obvious when A has compact inverse, but it is natural to ask what happens in general. In the next subsection we shall derive a similar property for any positive power of A, but in the case of square roots an easier proof can be given. The result is as follows Proposition 5.1. Let A be as the introduction. Then A 1 2 is also coercive and λ 1 (A

2 ) = (λ 1 (A)) 1 2 . 1
The proof of this proposition relies on 2 simple lemmas : Lemma 5.2. Let B ∈ L(H) be symmetric and nonnegative. Then we have Proof. By definition it is clear that

B 2 = B 2 ∀v ∈ H, |Bv| 2 ≤ B (Bv, v) Proof.
λ 1 (A) = 1 A -1 2 2
Then the result follows from the previous Lemma.

Remark 5.6. For a general positive operator the conditions are not equivalent . For instance take V = H = C and for some α > 0, β ∈ R ∀v ∈ V, Bv = (α + iβ)v

In this case we have B = (α 2 + β 2 )

1 2
∀v ∈ V, (Bv, v) = α|v| 2 so that the optimal value of C in (5.2) is α. The optimal value of C in (5.3) is α 2 +β 2 α . As soon as β = 0 we have α < (α 2 + β 2 ) 1 2 < α 2 + β 2 α and therefore the three constants are all different.

case 3 :c 3 = -c 4 tan( πλ 2 ) c 2 =

 3322 If sin(πλ) = 0, then 2c 4 sin 2 ( -c 3 , c 1 = -λc 4 . and -λπc 4 + c 4 tan( πλ 2 )c 4 cos(πλ) tan( πλ 2 ) + c 4 sin(πλ) = 0. If c 4 = 0, then u = 0. If c 4 = 0, we can reduce to c 4 = 1, then we find tan(

2 Finally we have for any v ∈ H |Bv| 2 1 2Lemma 5 . 3 . 1 (

 221531 First we have B 2 ∈ L(H) and B 2 ≤ B 2 . The reverse inequality is also immediate since |Bu| 2 = (B 2 u, u) ≤ B 2 |u| v| 2 = B (Bv, v) Let A be a self-adjoint, positive, coercive operator. Then λ

5. 4 m|u| 2 + ∇u 2 dx 1 Example 5 . 8 .

 42158 Some more examples.Sometimes Theorem 2.1 can be applied to equations in unbounded domains. For brevity we give only 2 typical examples Example 5.7. Let Ω be a possibly unbounded domain in R N and m > 0, γ > 0. We consider the following equationu tt -∆u + mu + γu t = f u /∂Ω = 0 (5.4)Then, as a consequence of (3.5) we have the following result valid for all mild solutions lim t→∞ Ω Let Ω be a bounded domain in R N and γ > 0. We consider the cylinder C = Ω × R and the following equation inR + × C u tt -∆u + γu t = f u /∂C = 0 (5.5)Then, as a consequence of (3.5), since A = -∆ is coercive in C with λ 1 (C) = λ 1 (Ω)we have the following result valid for all mild solutions lim t→∞ Ω

  Indeed in this case (but not in general) the two inequalities (1.3) and (2.2) are equivalent, see Section 5 below.

	Finally by letting ǫ → 0
	lim t→∞	|u ′ (t)| ≤ max(	3C c	,	3 √ 2c	) lim

t→∞ |f (t)| By using Remark 1.2 we obtain Corollary 2.2. Let β

(t) = B 0 ∈ L(V, V ′ ). In this case any mild solution u ∈ C(R + , V ) ∩ C 1 (R + , H) of (0.1) satisfies (2.3).

Remark 2.3. When B is linear and self-adjoint, Theorem 2.1 improves the result (1.4) with β(t) = B(t) by a factor ∈ [ √ 2, 2] depending on the values of C and c.

The ODE case

We consider the equation:

We apply theorem 2.1 to (3.1) with c = γ and C = γ ω 2 , we find

By comparaison with the estimates in [START_REF] Haraux | On the double well Duffing equation with a small bounded forcing term[END_REF], we find that the result of theorem 2.1 is optimal up to a factor K(ω, γ) = 3π 4 √ 2 , if γ < 2ω and √ 3 if γ ≥ 2ω. More precisely, in [START_REF] Haraux | On the double well Duffing equation with a small bounded forcing term[END_REF] the exact minimum global bound for solutions bounded on the whole line is given, and the minimum turns out to be achieved on some periodic solutions (corresponding to a periodic source term) for which the ultimate bound of course coincides with the global bound on R.

The case

In this case (cf. Proposition 5.4 )we have c = γλ α 1 and

, then, by Theorem 2.1, we have the following estimates max( lim

Considering the special case H = R , A = ω 2 I we conclude that this result is always sharp up to a factor √ 3.

Theorem 3.4. The bounded solution of (3.10) satisfies the estimate

Proof. In the case of a small damping we refer to [START_REF] Balakrishnan | Fractional powers of closed operators and the semigroups generated by them[END_REF]. We now prove (3.4) when γ ≥ 2 (3.12)

We choose the energy functional

Then, we have:

by using Young's inequality, we obtain:

we remark that α = γγ 2 -4 is a solution of the equation:

and

Then we have C = 1 µ . To illustrate this we consider the one dimensional case.

Proposition 4.6.

Proof. In order to compute C we need to find the minimal value of µ when

Then, setting λ = √ µ, we have

Therefore u = c 2 (1 -cos(2kx)) = 2c 2 sin 2 kx. In this case µ = 4k 2 and therefore µ ≥ 4.

Proof of Proposition 5.1. We first show that ,H) is onto and by Banach Theorem, A -1 2 ∈ L(H, V ). By Lemma 5.2 we find that that A 1 2 is coercive. Then λ 1 (A

and the result follows from a last application of Lemma 5.2

5.2

The first eigenvalue of a fractional power.

Let A be a self-adjoint coercive operator. The fractional power A α with α ∈ (0, 1) is defined as the inverse of the operator

with domain equal to the range of A -α . D(A α ) is also the closure of D(A) under the seminorm p α (u) := |A α (u)| (cf. e.g. [START_REF] Balakrishnan | Fractional powers of closed operators and the semigroups generated by them[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). We now generalize Proposition 5.1 to any positive power by relying on the above formula.

Proposition 5.4. For any α ∈ (0, 1), A α is also coercive and

Proof. By homogeneity it is clearly sufficient to establish the result when λ 1 (A) = 1 Then applying the result to A 1 = λ 1 (A) -1 A gives the general case. First we show that

Indeed we have

and then Lemma 5.3 gives the result. Now we have for any u ∈ D(A)

The relationship between the two main results.

In Remark 2.3 we said that when B is linear and self-adjoint, the two inequalities (1.3) and (2.2) are equivalent. This is a consequence of the following Proposition 5.5. Let A be as the introduction and ∈ L(V, V ′ ) be symmetric and nonnegative. Then the 3 following conditions are equivalent

Proof. We proceed in 3 steps 1) Proof of (5.1) =⇒ (5.2). Assuming (5.1) we have

Hence B ≤ CA.

2) Proof of (5.3) =⇒ (5.1). Assuming (5.3) we have

Hence, either Bu = 0 or Bu * ≤ C u and we have (5.1).

3) Proof of (5.2) =⇒ (5.3). Since B ≥ 0 we have

In this formula we choose v = A -1 (Bu) Then

so that we find

by using (5.2). Now

and if Bu = 0 we obtain (5.1) on dividing through by Bu 2 * .

We conclude this section by giving 2 examples of application for Theorem 1.1 and Theorem 2.1: a case where B is not selfadjoint and a case where B is non linear.

Example 5.9. Let V = H = C. Then any solution u of the ODE


Audrey Chapelle
email: chapelle@chimie.ups-tlse.fr

Mohd H Yaacob

Isabelle Pasquet

Lionel Presmanes

Antoine Barnabé

Philippe Tailhades

Johan Du Plessis

Kourosh Kalantar-Zadeh

Structural and gas-sensing properties of CuO-Cu x Fe 3-x O 4 nanostructured thin films

Keywords:

Nanocrystalline CuO-Cu x Fe 3-x O 4 thin films were developed using a radio-frequency sputtering method followed by a thermal oxidation process. Thin films were deposited applying two very different conditions by varying the argon pressure and the target-to-substrate distance. Structural, microstructural and gas-sensing characteristics were performed using grazing incidence X-ray diffraction (GXRD), Raman spectroscopy, atomic force microscopy (AFM), X-ray photoelectron spectroscopy (XPS), transmission electron microscopy (TEM), scanning electron microscopy (SEM), and electrical measurements. Their sensing properties were examined using hydrogen gas in dry synthetic air. The shortest response and recovery times were observed between 280 and 300 • C independently of the deposition conditions.

Introduction

Hydrogen is a colorless and odorless gas, and such, its leakage cannot be noticed easily. As hydrogen is highly flammable, any potential leak can present an explosion hazard. Thus, there is a great demand and subsequent challenges for monitoring hydrogen levels for the purpose of control and safety applications in domestic and industrial fields [START_REF] Yamazoe | Toward innovations of gas sensor technology (review)[END_REF]. Metal oxide semiconductors, such as CuO and its composites, have been the subject of interest for a many years for the detection of reducing gases such as H 2 [START_REF] Patil | CuO-modified tin titanate thick film resistors as H2-gas sensors[END_REF][START_REF] Dandeneau | Thin film chemical sensors based on p-CuO/n-ZnO heterocontacts[END_REF][START_REF] Hoa | Synthesis of p-type semiconducting cupric oxide thin films and their application to hydrogen detection[END_REF], C 2 H 5 OH [START_REF] Wang | Surface accumulation conduction controlled sensing characteristic of p-type CuO nanorods induced by oxygen adsorption[END_REF], CO [START_REF] Nakamura | Gas sensitivity of CuO/ZnO hetero-contact[END_REF][START_REF] Nakamura | Selective CO gas sensing mechanism with CuO/ZnO heterocontact[END_REF][START_REF] Liao | Multifunctional CuO nanowire devices: p-type field effect transistors and CO gas sensors[END_REF][START_REF] Kim | CuO nanowire gas sensors for air quality control in automotive cabin[END_REF], and H 2 S [START_REF] Zhou | Study on sensing mechanism of CuO-SnO2 gas sensors[END_REF][START_REF] Chen | H2S detection by vertically aligned CuO nanowire array sensors[END_REF]. The gas sensing properties of these metal oxide semiconductors are influenced by many factors such as their operating temperatures, morphology and chemical composition. CuO is especially an interesting metal oxide for gas sensing. It is an intrinsically p-type semiconductor with a bandgap in the range of 1.2-1.4 eV [START_REF] Marabelli | Optical gap of CuO[END_REF][START_REF] Ghijsen | Electronic structure of Cu2O and CuO[END_REF][START_REF] Koffyberg | A photoelectrochemical determination of the position of the conduction and valence band edges of p-type CuO[END_REF] and its properties can be altered upon mixing with other metals and metal oxides. The interest for nanocomposites and p-n heterojunctions is growing in the field of gas sensing. Some authors have reported that by the association of a metal oxide with CuO, and forming mixtures such as CuO/ZnO, CuO/SnO 2 , high sensitivity and selectivity to reducing gas species can be exhibited [START_REF] Jun | CO gas-sensing properties of ZnO/CuO contact ceramics[END_REF][START_REF] Ishihara | Mixed oxide capacitor of BaTiO3-PbO as a new type CO2 gas sensor[END_REF][START_REF] Aygün | Hydrogen sensitivity of doped CuO/ZnO heterocontact sensors[END_REF][START_REF] Patil | Heterocontact type CuO-modified SnO2 sensor for the detection of a ppm level H2S gas at room temperature[END_REF]. Among such metal oxide mixtures, spinels present interesting semiconducting properties due to the mixed valence of the elements. Spinels in general are known for their catalytic properties which enable changes in the electronic behaviors. There are several reports on spinel oxide composites capabilities for gas sensing [START_REF] Choi | Gas sensor having superior sensitivity and selectivity for carbon monoxide gas and method for manufacturing such gas sensor[END_REF][START_REF] Yu | Current-voltage characteristics and selective CO detection of Zn2SnO4 and ZnO/Zn2SnO4, SnO2/Zn2SnO4 layered-type sensors[END_REF][START_REF] Vijaya | Preparation and VOC gas sensing properties of Sr(II)-added copper aluminate spinel composites[END_REF]. In this work, Cu x Fe 3-x O 4 has been chosen for its n-type semiconducting property, as well as the possibility of forming into nanocomposites thin films with CuO [START_REF] Mugnier | Thin films preparation by RF-sputtering of copper/iron ceramic targets with Cu/Fe = 1: from nanocomposites to delafossite compounds[END_REF][START_REF] Mugnier | Nanocomposites of metallic copper and spinel ferrite films: growth and selfassembly of copper particles[END_REF]. By changing the Cu and Fe contents, the material can be tuned in a wide range of conductivities and altered from n-to p-type. Subsequently, the band-gap can also be tuned within a broad range.

Radio-frequency (RF) sputtering is a reliable and well controlled method in the preparation of nanocomposite thin films with a good homogeneity and crystallinity at low substrate temperature [START_REF] Presmanes | Magnetic and semi-conducting nano-composite films of spinel ferrite and cubic zinc oxide[END_REF][START_REF] Fernández | Radio frequency sputter deposition of high-quality conductive and transparent ZnO:Al films on polymer substrates for thin film solar cells applications[END_REF]. It has been shown that the physical and chemical properties of the growing layer depend on the deposition conditions such as pressure, magnetron, target-to-substrate distance and substrate temperature [START_REF] Mugnier | Thin films preparation by RF-sputtering of copper/iron ceramic targets with Cu/Fe = 1: from nanocomposites to delafossite compounds[END_REF][START_REF] Windischmann | Intrinsic stress in sputter-deposited thin films[END_REF]. Consequently, changing the sputtering conditions alters the microstructure and the stoichiometry of the deposited thin films [START_REF] Mauvernay | Elaboration and characterization of Fe1-xO thin films sputter deposited from magnetite target[END_REF][START_REF] Capdeville | Microstructure and electrical properties of sputter-deposited Zn0.87Fe2.13O4 thin layers[END_REF].

In the present paper, we report the structural and gas-sensing properties of CuO-Cu x Fe by RF-sputtering. Mixing of different metal oxides and the formation of binary compounds can make the resulting mixture selective to target gas species [START_REF] Ishihara | Application of mixed oxide capacitor to the selective carbon dioxide sensor[END_REF][START_REF] Galatsis | Sol-gel prepared MoO3-WO3 thin-films for O2 gas sensing[END_REF]. As a result, there have been a large number of reports on the formation of such binary metal oxide and reports as how the change of concentration of one metal oxide with reference to other can be used for tuning the gas sensing properties [START_REF] Liao | Study on CuO-BaTiO3 semiconductor CO2 sensor[END_REF][START_REF] Galatsis | Comparison of single and binary oxide MoO3, TiO2 and WO3 sol-gel gas sensors[END_REF][START_REF] Cantalini | Cr2O3, WO3 single and Cr/W binary oxide prepared by physical methods for gas sensing applications[END_REF]. The structures and morphologies of the films were studied by grazing incidence X-ray diffraction (GXRD), Raman spectroscopy, atomic force microscopy (AFM), scanning electron microscopy (SEM) and X-ray photoelectron spectroscopy (XPS). Device's resistance was investigated as a function of operating temperatures and different H 2 gas concentrations.

Experimental

Synthesis of CuO-Cu x Fe 3-x O 4 nanocomposites

Cu-Cu x Fe 3-x O 4 thin films were deposited with Alcatel A450 apparatus using a custom-made sintered ceramic target of pure CuFeO 2 with a relative density of 70%. The RF power was set at 200 W and the pressure inside the deposition chamber was lower than 5 × 10 -4 Pa prior to the deposition. All studied films were deposited on pre-cleaned substrates which were placed on a water cooled sample holder. No additional heating was required during the deposition. The target was pre-sputtered in an argon plasma for 10 min to remove surface contamination prior to the process. The depositions were carried out with various argon pressures (P) and target-to-substrate distances (d) ranging from 0.5 Pa to 2 Pa and 5 cm to 8 cm, respectively. The deposition conditions are summarized in Table 1. Films with the thicknesses in the range of 100 nm to 600 nm were deposited onto substrates for structural characterizations. The deposition rates of thin films at different conditions are presented in Table 1.

The CuO-Cu x Fe 3-x O 4 nanostructured films were obtained by the thermal treatment of the as-deposited layers in air at 450 • C for 12 h. Comb-type interdigitated electrodes were patterned on the top of the annealed thin films via the sputter deposition of 50 nm thick gold through a metal shadow mask (as shown in Fig. 1). Afterwards, two gold wires were bonded to the gold electrode using silver epoxy.

Characterization and gas sensing property measurements

The GXRD patterns (grazing angle ˛= 1 • ) were obtained with a Siemens D5000 diffractometer using the Cu K␣ radiation. The surface morphology of the thin films were determined by a Veeco Dimension 3000 AFM equipped with a super sharp TESP-SS Nanoworld © tip (nominal resonance frequency 320 kHz, nominal radius curvature 2 nm) and JEOL 2100 TEM operating at 200 kV equipped with a X-ray energy dispersive spectroscopy (X-EDS) system. The SEM images were obtained using a field emission gun Jeol JSM 6400. The Raman spectra were collected under ambient conditions using a modular Olympus BX41 Raman microscope (Horiba Jobin Yvon) fitted with a 50× objective lens, through which spectra were excited using a fiber-coupled 532 nm laser and power of 1.1 mW. The scattered signal was gathered by the same 50× objec- tive lens in a backscattering configuration and was then coupled to a spectrometer (Jobin Yvon Triax 320), which was fitted with a thermoelectrically cooled CCD detector. In all cases, 5 accumulations of 30 s each were averaged together. A Thermo Scientific K-Alpha apparatus was used for XPS measurements using a monochromatic Cu K␣ radiation with a 400 m spot size. Survey and peaks were scanned at 200 and 50 eV pass energy, respectively. Flood gun was used to minimize charging effects.

For the gas sensing measurements, the CuO-Cu x Fe 3-x O 4 based sensors were mounted on a heater in a Teflon chamber. The heater was controlled by a regulated DC power supply providing different operating temperatures. The sensor output resistance as a function of time was recorded using a multimeter (Keithley 2001), while the gases were continuously flown through the chamber at a total flow rate of 200 sccm. A computerized gas calibration system, with mass flow controllers, was used for exposing the sensor to the desired concentrations of H 2 gas coupled with dry synthetic air used as the reference gas. The sensor response S was defined as S (%) = ((R -R 0 )/R 0 ) × 100, where R 0 is the resistance of the sensor in dry air, and R is the resistance of the sensor under hydrogen flow.

Results and discussion

Structural analysis

Sample P 0.5 d 5 was deposited at low pressure (0.5 Pa) and low target-to-substrate distance (5 cm), while sample P 2 d 8 was pre- pared at higher pressure (2 Pa) and higher target-to-substrate distance (8 cm). The deposition conditions were adjusted [START_REF] Mugnier | Thin films preparation by RF-sputtering of copper/iron ceramic targets with Cu/Fe = 1: from nanocomposites to delafossite compounds[END_REF] in order to obtain either much reduced phases than that of the target (CuFeO 2 ) or phases with the same oxygen content.

Fig. 2 shows the GXRD patterns of the as-deposited samples using the two different deposition conditions. A broad peak is observed at 2 = 35.5 • , i.e d hkl ≈ 2.5 Å, which could be attributed to various mixed oxide phases such as Cu 2 O (d 111 = 2.46 Å), CuO (d 11-1 = 2.52 Å), CuFeO 2 (d 012 = 2.51 Å), Cu x Fe 3-x O 4 (0 < x < 1) (d 311 = 2.50 Å-2.52 Å) as reported previously [START_REF] Chapelle | CO2 sensing properties of semiconducting copper oxide and spinel ferrite nanocomposite thin film[END_REF]. At low pressures, the mean free path of the particles ejected from the target is higher and they are more energetic, as these particles face a smaller number of collisions during the transfer from the target to the substrate. As a result for sample P 0.5 d 5 , the particles arriving on the growing layer are more energetic and tend to re-bombard the growing layer (especially the oxygen atoms in the layer, which are light elements). The consequence of the re-bombardment is a layer, which is more reduced than the target. In this case, the equilibrium diagram of the Cu-Fe-O system [START_REF] Jacob | Activities in the spinel solid solution, phase equilibria and thermodynamic properties of ternary phases in the system Cu-Fe-O[END_REF] predicts a mixture of copper (Cu) and spinel ferrite (Cu x Fe 3-x O 4 ). However, the sputtering technique provides an out-of-equilibrium reaction and thus the reduction is only partial. Consequently, we obtain a thin film composed of both metallic copper and an oxide mixture of spinel ferrite and delafossite phase (CuFeO 2 ). Copper tends to crystallize much easier than the two other oxides, which explain the GXRD pattern (Fig. 2a). The two other oxides are nanocrystalline, which give the GXRD pattern a broad peak. Inversely for the P 2 d 8 deposition, as the mean free paths of particles leaving the target are lower, they are less energetic. Therefore, the stoichiometry of the growing layer is close to the target and no metallic copper is detected by the GXRD (Fig. 2b).

The GXRD patterns of the annealed samples in air for 12 h at 450 • C are shown in Fig. 2c andd. This treatment leads to the formation of two phases predicted by the Cu-Fe-O system [START_REF] Jacob | Activities in the spinel solid solution, phase equilibria and thermodynamic properties of ternary phases in the system Cu-Fe-O[END_REF], which are associated well to copper oxide and spinel ferrite.

The SEM and AFM were performed on as-deposited samples and annealed samples in air at 450 • C for 12 h. Fig. 3 shows the morphologies obtained at different conditions for the 300 nm thick sample. For P 0.5 d 5 (Fig. 3a andb), the average particle diameter is approximately 60 nm with a relatively high roughness of almost 7 nm, which is due to the presence of metallic crystallized copper particles, tending to increase the roughness. On the other hand, for P 2 d 8 (Fig. 3e andf) the thin film shows a smoother surface (Ra = 1 nm) with the average particle diameter of approximately 35 nm. In confirmation, the metallic copper was also not detected by the GXRD pattern (Fig. 2b), which explains why the surface roughness is smaller.

The thermal treatments in air of the as deposited layers lead to the formation of increasingly crystalline CuO-Cu x Fe 3-x O 4 composite as confirmed by the GXRD patterns (Fig. 2c andd), where the copper oxide originates from either the oxidation of the metallic copper or the decomposition of the CuFeO 2 phase. However, their microstructures remain different. Fig. 3c andd shows an agglomeration of small grains involving increasingly compact structure for P 0.5 d 5 and Fig. 3g and h exhibits larger grains within P 2 d 8 with a higher roughness and seems to be more porous than P 0.5 d 5 . It has been demonstrated [START_REF] Oudrhiri-Hassani | Microstructure, porosity and roughness of RF sputtered oxide thin films: characterization and modelization[END_REF] that films prepared with a sputtering argon pressure of 0.5 Pa and a target-to-substrate distance of 5 cm are usually very dense. It has also been reported that increasing the pressure and/or target-to-substrate distance augment the porosity of the films [START_REF] Oudrhiri-Hassani | Microstructure, porosity and roughness of RF sputtered oxide thin films: characterization and modelization[END_REF]. As it is generally assumed, the porosity results from shadowing effects lead to a small mean free path for the atoms in the plasma. At high pressure deposition, the atoms reach the substrate in an oblique incidence, forming crystallites that produce localized shadow zones, which hinders the homogenous deposition of the films, hence enhances the porosity. The angle of incidence also becomes higher by increasing the target-to-substrate distance, which affects the deposited film similarly.

The TEM was performed for the as-deposited samples with a thickness of 30 nm on carbon covered nickel grid (P 0.5 d 5 -Fig. 4a). The black dots were associated to metallic copper by X-EDS, which were also confirmed by the GXRD patterns. It can be seen that the sample obtained at low pressure-low distance is a nanocomposite of many metallic copper particles in an oxide matrix. The TEM image of the film obtained at high pressure-high distance (P 2 d 8 -Fig. 4b) is mostly made of an oxide matrix. However, some large metallic particles were also observed even though no metallic copper was detected by the GXRD.

The Raman spectra of the as-deposited and annealed samples at different deposition conditions are presented in Fig. 5. All spectra were taken with a collection time of 2 min 30 s. The results in Fig. 5a andc show that the as-deposited thin films for both conditions are characterized by a broad band at 630 cm -1 . Copper is not expected to appear in these spectra since it is Raman silent as Cong et al. demonstrated [START_REF] Gong | Atomic force microscopy and Raman spectroscopy studies on the oxidation of Cu thin films[END_REF]. The Raman spectra of annealed thin films at 450 • C are shown in Fig. 5b andd. Three bands were observed at 293, 340, and 623 cm -1 and a broad, but relatively strong feature appeared at 1100 cm -1 ,which were all assigned to CuO [START_REF] Gong | Atomic force microscopy and Raman spectroscopy studies on the oxidation of Cu thin films[END_REF][START_REF] Chrzanowski | Raman scattering from cupric oxide[END_REF][START_REF] Irwin | Raman scattering from single crystals of cupric oxide[END_REF]. The peak at 293 cm -1 seems to be more intense for P 2 d 8 than P 0.5 d 5 , which means that this sample contains more copper oxide. It seems that the copper content in the sample with larger particle morphologies (P 0.5 d 5 ) not completely turned into CuO after annealing, while the 450 • C annealing temperature appears to be more effective in transforming small particles of P 2 d 8 into CuO.

Gas sensing results

The sensor responses to 0.125, 0.25 and 0.5% of H 2 (in dry air) at a temperature ranging from 220 to 310 • C are displayed in Fig. 6a andb. In our case, gas sensing experiments were performed in dry air atmosphere to avoid any interference from humidity. The maximum response of the sensor occurs at the temperature of 280 and 295 • C for sensors with P 0.5 d 5 and P 2 d 8 layers (thickness 100 nm), respectively. As can be seen in Fig. 7, the sensors showed good repeatability as the initial baseline was regained upon exposure to dry air.

The increase in the electrical resistance of the sensors upon exposure to a reducing gas such as H 2 indicates the presence of that the obtained films had p-type semiconducting properties. This implies the domination of CuO over Cu x Fe 3-x O 4 , which is an n-type material.

Hoa et al. [START_REF] Hoa | Synthesis of porous CuO nanowires and its application to hydrogen detection[END_REF] suggested that behaviour of CuO based sensor originates from the interaction between the chemisorbed H 2 and the adsorbed oxygen on the sensor surfaces. They showed that when CuO is exposed to hydrogen, the H 2 molecules interact with CuO through the oxygen ions preadsorbed onto the CuO surface as described by Eqs. ( 1) and ( 2):

H 2(ads) + O - (ads) ↔ H 2 O + e - (1)
H 2(ads) + O 2- (ads) ↔ H 2 O + 2e - (2)
At 0.125% H 2 , the characteristic response and recovery times of the sensor at the operating temperature of 280 • C for P 0.5 d 5 were 150 and 320 s, respectively, with a response S of 37% and at 295 • C for P 2 d 8 were 190 and 400 s, respectively, with a response (S) of 40%. The difference in gas response between P 0.5 d 5 and P 2 d 8 can be ascribed to the morphologies of microstructures and difference in porosity, as describe in the previous section, as well as the nature of the phases in presence. Further sensing measurements are found in the "Supporting information": Table of sensor responses, response and recovery times at different temperatures calculated from the dynamic responses for P 0.5 d 5 and P 2 d 8 samples.

The XPS spectra of CuO-Cu x Fe 3-x O 4 after H 2 testing are shown in Fig. 8. The survey scan spectrum Fig. 8a shows photoelectron lines for Cu2p, C1s, O1s, Fe2p, Au4f, Na1s and the Auger lines LMM. The XPS peak for Na observed in the spectrum imply that chemical reactions occur at the interface between the film and the glass substrate and that sodium migrates from the glass substrate into the thin film. The element C in the film is attributed to residual carbon. The XPS peaks for Au come from the gold electrode sputtered at the surface of the thin film for the formation of electrodes (the surface analysis is close to the distance between two gold electrodes). High-resolution scan of Cu2p includes four peaks at 934, 942, 954, and 962 eV (Fig. 8b).

Two peaks at 933.98 and 954.08 eV are due to the spin-orbit doublet of Cu2p and are assigned to the bonding energies of Cu2p 3/2 and Cu2p 1/2 in CuO, respectively. The two other peaks on the higher binding energy side of both Cu2p 3/2 and Cu2p 1/2 are satellite structures. These satellites can be attributed to shake-up transitions by ligand-to-metal 3d charge transfer [START_REF] Wallbank | 2p and 2s shake-up satellites in solid compounds of 3d ions[END_REF][START_REF] Kim | Charge transfer transition accompanying X-ray photoionization in transition-metal compounds[END_REF][START_REF] Watts | An Introduction to surface analysis by XPS and AES[END_REF]. This charge transfer can occur for copper present in the Cu 2+ form (3d 9 configuration) but cannot take place if it is present as a metallic or in a Cu + state (3d 10 configuration) because of their completely filled 3d shells.

Conclusion

The CuO-Cu x Fe 3-x O 4 thin films were deposited on glass substrates using the RF-sputtering method followed by a thermal annealing in air at 450 • C. The microcharacterization of samples showed that they were made of well dispersed particles with good homogeneity. Using TEM and AFM characterizations, it was found that before annealing surface of P 0.5 d 5 was rougher than P 2 d 8 due to the presence of metallic copper. However after annealing, the surface of P 2 d 8 became relatively rougher compared to P 0.5 d 5 . CuO-Cu x Fe 3-x O 4 nanocomposite thin films showed p-type semiconductor behaviours with increasing electrical resistance upon exposure to hydrogen gas. The best sensing performance was observed for P 2 d 8 at an operating temperature of 295 • C with responses of 40, 69, and 100% for H 2 concentrations of 0.125, 0.25, and 0.5%, respectively. The response and recovery times obtained at the lowest concentration (0.125% H 2 ) were 190 and 400 s, respectively. P 2 d 8 's better performance can be attributed to its rougher surface in comparison to P 0.5 d 5 . Further investigations such as testing the effect of the selectivity and atmosphere humidity are required to optimize sensing parameters.

Fig. 1 .

 1 Fig. 1. Picture of the device used for electrical testing of the CuO-CuxFe3-xO4 thin films.

Fig. 2 .

 2 Fig. 2. Grazing angle X-ray diffraction patterns of as-deposited samples at different deposition conditions, (a) P0.5d5, (b) P2d8 and annealed samples in air at 450 • C (c) P0.5d5, (d) P2d8.

Fig. 3 .

 3 Fig. 3. SEM and AFM images of samples obtained at different deposition conditions, (a and b) P0.5d5 as-deposited, (c and d) P0.5d5 annealed in air at 450 • C, (e and f) P2d8 as-deposited, (g and h) P2d8 annealed in air at 450 • C. Ra represents the roughness.

Fig. 4 .

 4 Fig. 4. TEM images of as-deposited samples at (a) P0.5d5 and (b) P2d8.

Fig. 5 .

 5 Fig. 5. Raman spectra of thin films: (a) P0.5d5 as-deposited, (b) P0.5d5 annealed at 450 • C in air, (c) P2d8 as-deposited and (d) P2d8 annealed at 450 • C in air.

2 Fig. 6 .Fig. 7 .Fig. 8 .

 2678 Fig. 6. Response of the sensors at different operating temperatures upon exposure to different concentration of H2 for a 100 nm thin samples: (a) P0.5d5 and (b) P2d8.

Table 1

 1 3-x O 4 nanocomposite thin films obtained Sputtering deposit parameters.

	Target	CuFeO2
	Magnetron	No
	RF power (W)	200
	Argon pressures (Pa) P	0.5	2
	Target-to-substrate distance (cm) d	5	8
	Deposition rates (nm/min)	6.8	3.5
	Substrate	Glass

doi:10.1016/j.snb.2010.10.018

Acknowledgements

This work was completed while the author was visiting the Micro/Nano-electronics and Sensors Group at RMIT University (Melbourne, Australia). This work was supported by a fellowship, ATUPS, from Paul Sabatier University (Toulouse, France).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.snb.2010.10.018.