Symmetrization and anti-symmetrization in parabolic equations

Rossi, Luca

doi:10.1090/proc/13391

Cited by 7 publications

(5 citation statements)

References 10 publications

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“…In the semilinear case similar results hold when a > 0; see [18,32,47]. When a = 0 one has to distinguish between the so called pushed case, c * > 2h ′ (0), for which we have also analogous results [46,47], and the more involved pulled case, c * = 2h ′ (0), for which there is a logarithmic term (known as Bramson's correction term) in the description of the large-time behaviour of level sets even for N = 1 [8,14,23,27,41,42]. In all cases the first term in this description had been given in [3].…”

Section: Introductionmentioning

confidence: 67%

“…Of course, for this question to make sense we have to consider all possible balls, not just the ones centred at the origin. We expect a negative answer, in view of the counterexamples for bistable and concave monostable nonlinearities in the semilinear case given in [42,43,50]. The best we expect is to show that there exists a Lipschitz function s ∞ defined on the unit sphere such that u(x, t) approaches, as t goes to infinity, the function…”

Section: 4mentioning

confidence: 97%

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Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

Du¹,

Gárriz²,

Quirós³

2020

Preprint

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We study a family of reaction-diffusion equations that present a doubly nonlinear character given by a combination of the p-Laplacian and the porous medium operators. We consider the so-called slow diffusion regime, corresponding to a degenerate behaviour at the level 0, in which nonnegative solutions with compactly supported initial data have a compact support for any later time. For some results we will also require p ≥ 2 to avoid the possibility of a singular behaviour away from 0.Problems in this family have a unique (up to translations) travelling wave with a finite front. When the initial datum is bounded, radially symmetric and compactly supported, we will prove that solutions converging to 1 (which exist, as we show, for all the reaction terms under consideration for wide classes of initial data) do so by approaching a translation of this unique traveling wave in the radial direction, but with a logarithmic correction in the position of the front when the dimension is bigger than one. As a corollary we obtain the asymptotic location of the free boundary and level sets in the non-radial case up to an error term of size O(1). In dimension one we extend our results to cover the case of non-symmetric initial data, as well as the case of bounded initial data with supporting sets unbounded in one direction of the real line. A main technical tool of independent interest is an estimate for the flux.Most of our results are new even for the special cases of the porous medium equation and the p-Laplacian evolution equation.

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Section: Introductionmentioning

confidence: 67%

Section: 4mentioning

confidence: 97%

Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

Du¹,

Gárriz²,

Quirós³

2020

Preprint

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“…the initial data at (q(T ), ξ(T )) being nonnegative and sufficiently large so that v(T, r, Θ) ≥ |u i (T, r, Θ)|. Then, for any t ≥ T , q(t) ≥ 0 and ξ(t) ≥ 0 and we shall prove that v defined by (19) with ξ and q verifying (20) is a supersolution to (18). We have…”

Section: The O(t ε ) Estimatementioning

confidence: 91%

Sharp large time behaviour in $N$-dimensional reaction-diffusion equations of bistable type

Roquejoffre¹,

Roussier-Michom²

2021

Preprint

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We study the large time behaviour of the reaction-diffsuion equation ∂ t u = ∆u + f (u) in spatial dimension N , when the nonlinear term is bistable and the initial datum is compactly supported. We prove the existence of a Lipschitz function s ∞ of the unit sphere, such that u(t, x) converges uniformly in R N , as t goes to infinity, to, where U c * is the unique 1D travelling profile.This extends earlier results that identified the locations of the level sets of the solutions with o t→+∞ (t) precision, or identified precisely the level sets locations for almost radial initial data.

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“…[2,9,31,36,37,55] for extinction/invasion results in terms of the size and/or the amplitude of the initial condition u 0 for various reaction terms f , and to [9,11,34,35,42] for general local convergence and quasiconvergence results at large time. For the invading solutions u (that is, those converging to 1 locally uniformly in R N as t → +∞) with localized initial conditions, further estimates on the location and shape at large time of the level sets have been established in [13,18,27,45,49,51,53]. Lastly, equations of the type (1.1) set in unbounded domains Ω instead of R N and notions of spreading speeds and persistence/invasion in such domains have been investigated in [5,50].…”

Section: Two Main Questionsmentioning

confidence: 99%

Spreading speeds and spreading sets of~reaction-diffusion equations

Hamel¹,

Rossi²

2021

Preprint

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This paper is devoted to the study of the large time dynamics of bounded solutions of reaction-diffusion equations with unbounded initial support in R N . We first prove a general Freidlin-Gärtner type formula for the spreading speeds of the solutions in any direction. This formula holds under general assumptions on the reaction and for solutions emanating from initial conditions with general unbounded support, whereas most of earlier results were concerned with more specific reactions and compactly supported or almost-planar initial conditions. We also prove some results of independent interest on some conditions guaranteeing the spreading of solutions with large initial support and the link between these conditions and the existence of traveling fronts with positive speed. Furthermore, we show some flattening properties of the level sets of the solutions if initially supported on subgraphs. We also investigate the special case of asymptotically conical-shaped initial conditions. For Fisher-KPP equations, we prove some asymptotic one-dimensional symmetry properties for the elements of the Ω-limit set of the solutions, in the spirit of a conjecture of De Giorgi for stationary solutions of Allen-Cahn equations. Lastly, we show some logarithmicin-time estimates of the lag of the position of the solutions with respect to that of a planar front with minimal speed, for initial conditions which are supported on subgraphs with logarithmic growth at infinity. The proofs use a mix of ODE and PDE methods, as well as some geometric arguments. The paper also contains some related conjectures and open problems. * This work has been carried out in the framework of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the "Investissements d'Avenir" French Government program managed by the French National Research Agency (ANR). The research leading to these results has also received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 -ReaDi -Reaction-Diffusion Equations, Propagation and Modelling, from the ANR projects NONLOCAL (ANR-14-CE25-0013) and RESISTE (ANR-18-CE45-0019).

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Symmetrization and anti-symmetrization in parabolic equations

Cited by 7 publications

References 10 publications

Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

Sharp large time behaviour in $N$-dimensional reaction-diffusion equations of bistable type

Spreading speeds and spreading sets of~reaction-diffusion equations

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