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Two-dimensional reaction-diffusion equations with memory

Monica Conti, Stefania Gatti, Maurizio Grasselli, Vittorino Pata

2010
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Quarterly of Applied Mathematics
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Reverts to public domain 28 years from publication 607 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf 608 M. CONTI, S. GATTI, M. GRASSELLI, AND V. PATA of finite fractal dimension. Convergence to equilibria is also examined. Finally, all the results are reinterpreted within the original framework. License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf
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... SION EQUATIONS WITH MEMORY 609 for some given ε ∈ (0, 1], called relaxation time, equation (1.2) can be formally transformed into the hyperbolic equation which, provided that φ is bounded from below, becomes dissipative for small values of ε. However, if k reduces to the Dirac mass at 0 (i.e., the past memory is lost), we recover (1.1). This suggests that (1.1) is a good approximation of (1.2) when the system keeps a very short memory. If so, then it is reasonable to work with (1.1) in place of (1.2), with the great advantage of handling a much simpler equation. Concerning concrete applications, (1.2) has been proposed as a model of viscoelastic fluids [8, 11, 35] . Alternatively, it can be viewed as a semilinear (hyperbolic) heat equation based on the Gurtin-Pipkin heat conduction law [18, 22, 34] , or as a simple model for chemosensitive movements [28, 29] . Its relevance has been further underlined in a series of papers (see [10, 32, 33] and references therein). In those contributions, the authors refer to (1.3) as a hyperbolic reaction-diffusion equation, and they show how it can be used to describe a number of phenomena of biological and chemical interest, such as chemically reacting systems, gene selection, population dynamics, and forest fire propagation. Despite these attractive features, very few rigorous mathematical results are available so far. Regarding existence and uniqueness, some theorems can be found in [1], where (1.2) is incorporated in a phase-field system and φ is allowed to be a maximal monotone graph. In [6, 18], the infinite time delay version of (1.2) is shown to generate a dynamical system in the history space framework. In particular, considering the rescaling k ε (s) = ε −1 k(s/ε) of the original kernel k, with a small parameter ε > 0, the paper [6] establishes the convergence on finite time intervals of the solutions to (1.2) to the solutions to (1.1) in the limit ε → 0, provided that the initial data are regular enough. Moreover, in the one-dimensional case, [21] proves the existence of the global attractor for small values of ε, exploiting the precompact pseudometric technique [23], but without any regularity result. We remark that, in [6, 21] , the phase space for u is L 2 (Ω). In this functional setting, it seems particularly hard to find satisfactory global asymptotic results, such as the existence of regular global attractors or of exponential attractors. Nonetheless, if we work in the smaller phase space H 1 (Ω), as [18] does, then we can appeal to a more familiar hyperbolic formulation, close to (1.3). In which case, a small relaxation time ε is needed in order to have dissipativity. This is motivated by the recent papers [15, 36, 37] , focused on (1.3) in different space dimensions, with 1 + εφ (u) replaced by a generic damping coefficient σ(u) ≥ σ 0 > 0. These papers shed a new light on the approach devised in [18] , based on the reformulation of equation (1.2) as a hyperbolic equation similar to (1.3), but also containing a convolution term. The present work is devoted to a detailed investigation of the two-dimensional case, which is of particular interest for biological and chemical applications. The main goal is the analysis of the longtime behavior of solutions, with special regard to their dependence on ε, in the spirit of [6, 9, 14, 24] . More precisely, we compare in a quantitative way the closeness of the global dynamics of (1.2) and (1.1). As a consequence, it will be clear that the latter equation can be viewed as a good approximation of (1.2) when k is rapidly fading (short memory),

doi:10.1090/s0033-569x-2010-01167-7
fatcat:i7muflxlzvhx5mqqrccbfrao3y