Abstract
We consider the degenerate parabolic equation with nonlocal source given by
which has been proposed as a model for the evolution of the density distribution of frequencies with which different strategies are pursued in a population obeying the rules of replicator dynamics in a continuous infinite-dimensional setting.
Firstly, for all positive initial data u0 ∈ C0(ℝn) satisfying u0 ∈ Lp(ℝn) for some p ∈ (0, 1) as well as \(\int_{{\mathbb{R}^n}} {{u_0} = 1} \), the corresponding Cauchy problem in ℝn is seen to possess a global positive classical solution with the property that \(\int_{{\mathbb{R}^n}} {u(\cdot,t) = 1} \) for all t > 0.
Thereafter, the main purpose of this work consists in revealing a dependence of the large time behavior of these solutions on the spatial decay of the initial data in a direction that seems unexpected when viewed against the background of known behavior in large classes of scalar parabolic problems. In fact, it is shown that all considered solutions asymptotically decay with respect to their spatial H1 norm, so that
always grows in a significantly sublinear manner in that (0.1) \({{{\cal E}(t)} \over t} \to 0\;\;\;\;{\rm{as}}\;t \to \infty;\) the precise growth rate of ℰ, however, depends on the initial data in such a way that fast decay rates of u0 enforce rapid growth of ℰ. To this end, examples of algebraic and certain exponential types of initial decay are detailed, inter alia generating logarithmic and arbitrary sublinear algebraic growth rates of ℰ, and moreover indicating that (0.1) is essentially optimal.
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References
D. G. Aronson, The porous medium equation, in Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Mathematics, Vol. 1224, Springer, Berlin, 1986, pp. 1–46.
M. Bertsch, R. Dal Passo and M. Ughi, Discontinuous “viscosity” solutions of a degenerate parabolic equation, Transactions of the American Mathematical Society 320 (1990), 779–798.
I. M. Bomze, Dynamical aspects of evolutionary stability, Monatshefte für Mathematik 110 (1990), 189–206.
R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Advances in Computational Mathematics 5 (1996), 329–359.
M. Fila, J. R. King, M. Winkler and E. Yanagida, Linear behaviour of solutions of a superlinear heat equation, Journal of Mathematical Analysis and Applications 340 (2008), 401–409.
M. Fila, J. L. Vázquez, M. Winkler and E. Yanagida, Rate of convergence to Barenblatt profiles for the fast diffusion equation, Archive for Rational Mechanics and Analysis 204 (2012), 599–625.
M. Fila and M. Winkler, Slow growth of solutions of superfast diffusion equations with unbounded initial data, Journal of the London Mathematical Society 95 (2017), 659–683.
M. Fila and M. Winkler, A Gagliardo-Nirenberg-type inequality and its applications to decay estimates for solutions of a degenerate parabolic equation, preprint, https://doi.org/abs/1710.11312.
N. Kavallaris, J. Lankeit and M. Winkler, On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics, SIAM Journal on Mathematical Analysis 49 (2017), 954–983.
D. Kravvaritis, V. G. Papanicolaou and A. N. Yannacopoulos, Similarity solutions for a replicator dynamics equation, Indiana University Mathematics Journal 57 (2008), 1929–1946.
D. Kravvaritis, V. G. Papanicolaou, A. Xepapadeas and A. N. Yannacopoulos, On a class of operator equations arising in infinite dimensional replicator dynamics, Nonlinear Analysis. Real World Applications 11 (2010), 2537–2556.
J. Lankeit, Equilibration of unit mass solutions to a degenerate parabolic equation with a nonlocal gradient nonlinearity, Nonlinear Analysis. Theory, Methods & Applications 135 (2016), 236–248.
T. Y. Lee and W.-M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Transactions of the American Mathematical Society 333 (1992), 365–378.
S. Luckhaus and R. Dal Passo, A degenerate diffusion problem not in divergence form, Journal of Differential Equations 69 (1987), 1–14.
J. Maynard Smith, Evolution and The Theory of Games, Cambridge University Press, Cambridge, 1982.
J. Oechssler and F. Riedel, Evolutionary dynamics on infinite strategy spaces, Economic Theory 17 (2001), 141–162.
V. G. Papanicolaou and G. Smyrlis, Similarity solutions for a multi-dimensional replicator dynamics equation, Nonlinear Analysis. Theory, Methods & Applications 71 (2009), 3185–3196.
V. G. Papanicolaou and K. Vasilakopoulou, Similarity solutions of a replicator dynamics equation associated to a continuum of pure strategies, Electronic Journal of Differential Equations (2015), paper no. 231.
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Basel, 2007.
P. D. Taylor and L. B. Jonker, Evolutionary stable strategies and game dynamics, Mathematical Biosciences 40 (1978), 145–156.
J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and Its Applications, Vol. 33, Oxford University Press, Oxford, 2006.
M. Wiegner, A degenerate diffusion equation with a nonlinear source term, Nonlinear Analysis. Theory, Methods & Applications 28 (1997), 1977–1995.
M. Winkler, On the Cauchy problem for a degenerate parabolic equation, Zeitschrift für Analysis und ihre Anwendungen 20 (2001), 677–690.
M. Winkler, Oscillating solutions and large ω-limit sets in a degenerate parabolic equation, Journal of Dynamics and Differential Equations 20 (2008), 87–113.
M. Winkler, Slowly traveling waves and homoclinic orbits in a nonlinear parabolic equation of super-fast diffusion type, Mathematische Annalen 355 (2013), 519–549.
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Lankeit, J., Winkler, M. Counterintuitive dependence of temporal asymptotics on initial decay in a nonlocal degenerate parabolic equation arising in game theory. Isr. J. Math. 233, 249–296 (2019). https://doi.org/10.1007/s11856-019-1900-8
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DOI: https://doi.org/10.1007/s11856-019-1900-8