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Dynamics of nonnegative solutions of one-dimensional reaction-diffusion equations with localized initial data. Part I: A general quasiconvergence theorem and its consequences. (English) Zbl 1345.35052
Summary: We consider the Cauchy problem $u_1 = u_{xx} + f(u), \;x\in \mathbb R, t>0,$
$u(x,0)=u_0 (x), \;x \in \mathbb R,$ where $$f$$ is a locally Lipschitz function on $$\mathbb R$$ with $$f(0)=0$$, and $$u_0$$ is a nonnegative function in $$C_0(\mathbb{R})$$, the space of continuous functions with limits at $$\pm$$ equal to $$0$$. Assuming that the solution $$u$$ is bounded, we study its large-time behavior from several points of view. One of our main results is a general quasiconvergence theorem saying that all limit profiles of $$u(\cdot,t)$$ in $$L^{\infty}_{loc} (\mathbb R)$$ are steady states. We also prove convergence results under additional conditions on $$u_0$$. In the bistable case, we characterize the solutions on the threshold between decay to zero and propagation to a positive steady state and show that the threshold is sharp for each increasing family of initial data in $$C_0(\mathbb R)$$.

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [1] DOI: 10.1515/crll.1988.390.79 · Zbl 0644.35050 [2] Berestycki H., Arch. Rational Mech. Anal 82 pp 313– (1983) [3] DOI: 10.1081/PDE-120016128 · Zbl 1021.35013 [4] DOI: 10.1002/cpa.3160420502 · Zbl 0685.35054 [5] DOI: 10.1007/s002080050202 · Zbl 0990.35028 [6] DOI: 10.1016/0022-0396(89)90081-8 · Zbl 0692.35013 [7] DOI: 10.1007/s10884-006-9053-y · Zbl 1166.35005 [8] DOI: 10.1088/0951-7715/5/6/004 · Zbl 0757.35059 [9] DOI: 10.1080/03605309908821497 · Zbl 0940.35107 [10] DOI: 10.4171/JEMS/198 · Zbl 1207.35061 [11] DOI: 10.1512/iumj.2015.64.5535 · Zbl 1336.35207 [12] DOI: 10.1090/S0002-9947-2014-06105-9 · Zbl 1302.35209 [13] DOI: 10.1007/s002200050508 · Zbl 1057.35508 [14] Fašangová E., Commentat. Math. Univ. Carolinae 39 pp 525– (1998) [15] DOI: 10.1007/PL00001410 · Zbl 0872.35014 [16] Feireisl E., Differ. Integr. Eqs 10 pp 181– (1997) [17] Feireisl E., Adv. Differ. Eqs 5 pp 583– (2000) [18] DOI: 10.1007/BF00250432 · Zbl 0361.35035 [19] DOI: 10.1007/BF01048791 · Zbl 0684.34055 [20] DOI: 10.1016/j.jde.2011.04.002 · Zbl 1263.35035 [21] DOI: 10.1023/A:1016624010828 · Zbl 1003.35085 [22] DOI: 10.1007/s10884-014-9376-z · Zbl 1338.35045 [23] Matano H., J. Math. Kyoto Univ 18 pp 221– (1978) · Zbl 0387.35008 [24] DOI: 10.1007/s00030-013-0220-7 · Zbl 1433.35173 [25] DOI: 10.1007/s00205-010-0316-8 · Zbl 1262.35130 [26] DOI: 10.1007/s00028-014-0260-4 · Zbl 1319.35097 [27] Polácik P., Prog. Nonlinear Differ. Eqs. Appl pp 405– (2016) [28] Zelenyak T. I., Differ. Eqs 4 pp 17– (1968) [29] DOI: 10.1090/S0894-0347-05-00504-7 · Zbl 1081.35011
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