Symmetrization and anti-symmetrization in parabolic equations

Rossi, Luca

doi:10.1090/proc/13391

Symmetrization and anti-symmetrization in parabolic equations
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Proc. Amer. Math. Soc. 145 (2017), 2527-2537 Request permission

Abstract:

We derive some symmetrization and anti-symmetrization properties of parabolic equations. First, we deduce from a result by Jones (1983) a quantitative estimate of how far the level sets of solutions are from being spherical. Next, using this property, we derive a criterion providing solutions whose level sets do not converge to spheres for a class of equations including linear equations and Fisher-KPP reaction-diffusion equations.

References

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
H. Berestycki, The Influence of Advection on the Propagation of fronts in Reaction-Diffusion Equations, Nonlinear PDE’s in Condensed Matter and Reactive Flows, NATO Science series C: Mathematical and physical Sciences, 569:11–48, 2002.
Arnaud Ducrot, On the large time behaviour of the multi-dimensional Fisher-KPP equation with compactly supported initial data, Nonlinearity 28 (2015), no. 4, 1043–1076. MR 3333711, DOI 10.1088/0951-7715/28/4/1043
Paul C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), no. 4, 335–361. MR 442480, DOI 10.1007/BF00250432
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Christopher K. R. T. Jones, Asymptotic behaviour of a reaction-diffusion equation in higher space dimensions, Rocky Mountain J. Math. 13 (1983), no. 2, 355–364. MR 702830, DOI 10.1216/RMJ-1983-13-2-355
L. Rossi, The Freidlin-Gärtner formula for general reaction terms, Preprint, 2016.
Violaine Roussier, Stability of radially symmetric travelling waves in reaction-diffusion equations, Ann. Inst. H. Poincaré C Anal. Non Linéaire 21 (2004), no. 3, 341–379 (English, with English and French summaries). MR 2068306, DOI 10.1016/S0294-1449(03)00042-8
James Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318. MR 333220, DOI 10.1007/BF00250468
K\B{o}hei Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ. 18 (1978), no. 3, 453–508. MR 509494, DOI 10.1215/kjm/1250522506
Hiroki Yagisita, Nearly spherically symmetric expanding fronts in a bistable reaction-diffusion equation, J. Dynam. Differential Equations 13 (2001), no. 2, 323–353. MR 1829601, DOI 10.1023/A:1016632124792
A. Zlatoš, Propagation of reactions in inhomogeneous media, Comm. Pure Appl. Math., to appear.