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Dynamics of time-periodic reaction-diffusion equations with compact initial support on \(\mathbb{R}\). (English. French summary) Zbl 1439.35266

The authors consider the Cauchy problem for the reaction-diffusion equation in the unknown \(u=u(x,t)\), \[ \left\{ \begin{array}{ll} u_t = u_{xx} + f(t,u), & x\in\mathbb{R},\ t>0, \\ u(x,0)=u_0(x), & x\in\mathbb{R}. \end{array} \right. \tag{1} \] The initial data \(u_0\in L^\infty(\mathbb{R})\) is assumed to be nonnegative and having compact support. For positive solutions \(u\), the nonlinear term is assumed to be periodic in \(t\), \(f(t+T,,u)=f(t,u)\), and satisfying \(f(\cdot,0)=0\) in addition to being Hölder continuous on \(\mathbb{R}\times[0,\infty)\) and \(\mathcal{C}^1\) in \(u\). These assumptions guarantee a local in time solution bounded in \(L^\infty(\mathbb{R}).\)
This article presents some new results on the asymptotic behavior of the bounded solutions of (1). First, the authors show that \(\omega\)-limit solutions must be spatially constant or symmetrically decreasing and that the set of \(\omega\)-limit solutions either consists of a single time-periodic solution or of multiple time-periodic solutions and any heteroclinic connections between them. Under an additional assumption, the authors prove that the \(\omega\)-limit set is a singleton and hence, any \(\omega\)-limit solution converges to a time-periodic solution. The results are applied to (1) where \(f\) is a bistable nonlinearity or a combustion nonlinearity.

MSC:

35K57 Reaction-diffusion equations
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
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[1] Alikakos, N. D.; Bates, P. W.; Chen, X., Periodic traveling waves and locating oscillating patterns in multidimensional domains, Trans. Am. Math. Soc., 351, 2777-2805 (1999) · Zbl 0929.35067
[2] Angenent, S. B., The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390, 79-96 (1988) · Zbl 0644.35050
[3] Aronson, D. G.; Weinberger, H. F., Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, (Partial Differential Equations and Related Topics. Partial Differential Equations and Related Topics, Lecture Notes in Math., vol. 446 (1975), Springer: Springer Berlin), 5-49 · Zbl 0325.35050
[4] Aronson, D. G.; Weinberger, H. F., Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30, 33-76 (1978) · Zbl 0407.92014
[5] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029
[6] Brunovsky, P.; Poláčik, P.; Sandstede, B., Convergence in general periodic parabolic equations in one space dimension, Nonlinear Anal., 18, 209-215 (1992) · Zbl 0796.35009
[7] Chen, X.-Y., Uniqueness of the \(ω\)-limit point of solutions of a semilinear heat equation on the circle, Proc. Jpn. Acad. Ser. A, 62, 335-337 (1986) · Zbl 0641.35028
[8] Chen, X.-Y.; Matano, H., Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differ. Equ., 78, 160-190 (1989) · Zbl 0692.35013
[9] Chen, X.-Y.; Poláčik, P., Asymptotic periodicity of positive solutions of reaction-diffusion equations on a ball, J. Reine Angew. Math., 472, 17-51 (1996) · Zbl 0839.35059
[10] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0064.33002
[11] Contri, B., Pulsating fronts for bistable on average reaction-diffusion equations in a time periodic environment, J. Math. Anal. Appl., 437, 90-132 (2016) · Zbl 1391.35248
[12] Du, Y.; Lou, B., Spreading and vanishing in nonlinear diffusion problems with free boundaries, J. Eur. Math. Soc., 17, 2673-2724 (2015) · Zbl 1331.35399
[13] Du, Y.; Matano, H., Convergence and sharp thresholds for propagation in nonlinear diffusion problems, J. Eur. Math. Soc., 12, 279-312 (2010) · Zbl 1207.35061
[14] Du, Y.; Poláčik, P., Locally uniform convergence to an equilibrium for nonlinear parabolic equations on \(R^N\), Indiana Univ. Math. J., 64, 787-824 (2015) · Zbl 1336.35207
[15] Ducrot, A.; Giletti, T.; Matano, H., Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations, Trans. Am. Math. Soc., 366, 5541-5566 (2014) · Zbl 1302.35209
[16] Eckmann, J.-P.; Rougemont, J., Coarsening by Ginzburg-Landau dynamics, Commun. Math. Phys., 199, 441-470 (1998) · Zbl 1057.35508
[17] Fašangová, E., Asymptotic analysis for a nonlinear parabolic equation on \(R\), Comment. Math. Univ. Carol., 39, 525-544 (1998) · Zbl 0963.35080
[18] Feireisl, E., On the long time behavior of solutions to nonlinear diffusion equations on \(R^N\), Nonlinear Differ. Equ. Appl., 4, 43-60 (1997) · Zbl 0872.35014
[19] Feireisl, E.; Poláčik, P., Structure of periodic solutions and asymptotic behavior for time-periodic reaction-diffusion equations on \(R\), Adv. Differ. Equ., 5, 583-622 (2000) · Zbl 0987.35079
[21] Hale, J. K.; Raugel, G., Convergence in gradient-like systems with applications to PDE, Z. Angew. Math. Phys., 43, 63-124 (1992) · Zbl 0751.58033
[26] Matano, H., Convergence of solutions of one-dimensional semilinear parabolic equations, J. Math. Kyoto Univ., 18, 221-227 (1978) · Zbl 0387.35008
[27] Matano, H.; Poláčik, P., Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences, Commun. Partial Differ. Equ., 41, 785-811 (2016) · Zbl 1345.35052
[28] Poláčik, P., Threshold solutions and sharp transitions for nonautonomous parabolic equations on \(R^N\), Arch. Ration. Mech. Anal., 199, 69-97 (2011) · Zbl 1262.35130
[29] Poláčik, P., Examples of bounded solutions with nonstationary limit profiles for semilinear heat equations on \(R\), J. Evol. Equ., 15, 281-307 (2015) · Zbl 1319.35097
[30] Poláčik, P., Threshold behavior and non-quasiconvergent solutions with localized initial data for bistable reaction-diffusion equations, J. Dyn. Differ. Equ., 28, 605-625 (2016) · Zbl 1456.35039
[31] Shen, W., Traveling waves in time almost periodic structures governed by bistable nonlinearities I. Stability and uniqueness, J. Differ. Equ., 159, 1-54 (1999) · Zbl 0939.35016
[32] Shen, W.; Shen, Z., Stability, uniqueness and recurrence of generalized traveling waves in time heterogeneous media of ignition type, Trans. Am. Math. Soc., 369, 4, 2573-2613 (2017) · Zbl 1357.35079
[33] Xin, J. X., Front propagation in heterogeneous media, SIAM Rev., 42, 161-230 (2000) · Zbl 0951.35060
[34] Zelenyak, T. I., Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differ. Uravn., 4, 17-22 (1968), (in Russian) · Zbl 0232.35053
[35] Zlatoš, A., Sharp transition between extinction and propagation of reaction, J. Am. Math. Soc., 19, 251-263 (2006) · Zbl 1081.35011
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