Abstract
We consider a reaction–diffusion–advection equation of the form: \(u_t=u_{xx}-\beta (t)u_x+f(t,u)\) for \(x\in (g(t),h(t))\), where \(\beta (t)\) is a T-periodic function representing the intensity of the advection, f(t, u) is a Fisher–KPP type of nonlinearity, T-periodic in t, g(t) and h(t) are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both \(\beta \) and f are independent of t) was recently studied by Gu et al. (J Funct Anal 269:1714–1768, 2015). In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing–spreading dichotomy result holds when \(\beta \) is small; a vanishing–transition–virtual spreading trichotomy result holds when \(\beta \) is a medium-sized function; all solutions vanish when \(\beta \) is large. Here the partition of \(\beta (t)\) depends not only on the “size” \(\bar{\beta }:= \frac{1}{T}\int _0^T \beta (t) dt\) of \(\beta (t)\) but also on its “shape” \(\tilde{\beta }(t) := \beta (t) - \bar{\beta }\).
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Notes
In [5], \(f\in C^2\) is assumed and \(v\rightarrow U_1\) is taken in \(H^2 ([0,\ell ])\). Note that for our problem (3.3), the assumption for f and k is sufficient to guarantee that the omega limit set of \(v(t,\cdot )\) in the topology \(C^2 ([0,\ell ])\) is not empty, and then a similar zero number argument as in [5] gives the convergence \(v\rightarrow U_1\) in \(C^2 ([0,\ell ])\). Moreover, the zero number properties we used here are those in Angenent [1], where the coefficient k(t) of \(v_z\) is assumed to be in \(W^{1,\infty }([0,T])\). We remark that for our problem, the condition \(k\in C^{\nu /2}([0,T])\) is sufficient to proceed the zero number argument. In fact, denote \(K(t):= \int _0^t k(s)ds\), \(y:= z+K(t)\) and \(w(t,y):= v(t, y-K(t))\), then the equation of v is converted into \(w_t = w_{yy} + f(t,w)\). This equation has no the first order term, though it is considered in a moving frame \(K(t)< y < K(t) +\ell \), the zero number properties as in [1] remain valid (cf. [11, 17]). In Sect. 5.2, we use the zero number properties in the same way.
Note that [19, Proposition 2.3] holds for the equation \(u_t =u_{xx} + au\) with \(a>0\) being a constant. If \(a=a(t)\) is a periodic function with \(\bar{a} >0\), then the function \(w:= ue^{-\int _0^t [a(s) -\bar{a}] ds}\) satisfies \(w_t =w_{xx} + \bar{a}w\), and so the conclusions in [19, Proposition 2.3] hold for w.
References
Angenent, S.B.: The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390, 79–96 (1988)
Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein J.A. (eds) Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 446, pp. 5–49. Springer, Berlin, Heidelberg (1975). doi:10.1007/BFb0070595
Aronson, D.G., Weinberger, H.F.: Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76 (1978)
Berestycki, H., Hamel, F., Nadin, G.: Asymptotic spreading in heterogeneous diffusive excitable media. J. Funct. Anal. 255, 2146–2189 (2008)
Brunovský, P., Poláčik, P., Sandstede, B.: Convergence in general periodic parabolic equations in one space dimension. Nonlinear Anal. 18, 209–215 (1992)
Bunting, G., Du, Y., Krakowski, K.: Spreading speed revisited: analysis of a free boundary model. Netw. Heterog. Media 7, 583–603 (2012)
Chen, X., Friedman, A.: A free boundary problem arising in a model of wound healing. SIAM J. Math. Anal. 32, 778–800 (2000)
Du, Y., Guo, Z.M., Peng, R.: A diffusion logistic model with a free boundary in time-periodic environment. J. Funct. Anal. 265, 2089–2142 (2013)
Du, Y., Lin, Z.: Spreading-vanishing dichtomy in the diffusive logistic model with a free boundary. SIAM J. Math. Anal. 42, 377–405 (2010)
Du, Y., Lou, B.: Spreading and vanishing in nonlinear diffusion problems with free boundaries. J. Eur. Math. Soc. 17, 2673–2724 (2015)
Du, Y., Lou, B., Zhou, M.: Nonlinear diffusion problems with free boundaries: convergence, transition speed and zero number arguments. SIAM J. Math. Anal. 47, 3555–3584 (2015)
Du, Y., Matsuzawa, H., Zhou, M.: Sharp estimate of the spreading speed determined by nonlinear free boundary problems. SIAM J. Math. Anal. 46, 375–396 (2014)
Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eugen. 7, 335–369 (1937)
Friedman, A., Hu, B.: Bifurcation from stability to instability for a free boundary problem arising in a tumor model. Arch. Ration. Mech. Anal. 180, 293–330 (2006)
Ge, J., Kim, K., Lin, Z., Zhu, H.: A SIS reaction–diffusion–advection model in a low-risk and high-risk domain. J. Differ. Equ. 259, 5486–5509 (2015)
Gu, H., Lin, Z., Lou, B.: Different asymptotic spreading speeds induced by advection in a diffusion problem with free boundaries. Proc. Am. Math. Soc. 143, 1109–1117 (2015)
Gu, H., Lou, B., Zhou, M.: Long time behavior of solutions of Fisher–KPP equation with advection and free boundaries. J. Funct. Anal. 269, 1714–1768 (2015)
Hamel, F.: Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity. J. Math. Pures Appl. 89, 355–399 (2008)
Hamel, F., Nolen, J., Roquejoffre, J., Ryzhik, L.: A short proof of the logarithmic Bramson correction in Fisher–KPP equations. Netw. Heterog. Media 8, 275–289 (2013)
Hamel, F., Roques, L.: Uniqueness and stability properties of monostable pulsating fronts. J. Eur. Math. Soc. 13, 345–390 (2011)
Hess, P.: Periodic-parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics, vol. 247. Longman Scientific and Technical, Harlow (1991)
Hilhorst, D., van der Hout, R., Peletier, L.A.: Diffusion in the presence of fast reaction: the case of a general monotone reaction term. J. Math. Sci. Univ. Tokyo 4, 469–517 (1997)
Kolmogorov, A.N., Petrovski, I.G., Piskunov, N.S.: A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. Bull. Moscow Univ. Math. Mech. 1, 1–25 (1937)
Li, F., Liang, X., Shen, W.: Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete Contin. Dyn. Syst. 36, 3317–3338 (2016)
Nadin, G.: Existence and uniqueness of the solution of a space–time periodic reaction–diffusion equation. J. Differ. Equ. 249, 1288–1304 (2010)
Wang, M.X.: A diffusive logistic equation with a free boundary and sign-changing coefficient in time-periodic environment. J. Funct. Anal. 270, 483–508 (2016)
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Communicated by P. Rabinowitz.
This research was partly supported by NSFC (No. 11671262).
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Sun, N., Lou, B. & Zhou, M. Fisher–KPP equation with free boundaries and time-periodic advections. Calc. Var. 56, 61 (2017). https://doi.org/10.1007/s00526-017-1165-1
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DOI: https://doi.org/10.1007/s00526-017-1165-1