Singular integral equations with applications to travelling waves for doubly nonlinear diffusion

Gárriz, Alejandro

doi:10.48550/arxiv.2001.11109

Cited by 1 publication

(3 citation statements)

References 17 publications

(29 reference statements)

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“…Further information about these results on wavefronts and many others can be found in [25] for the case p = 2 and in [4,5,15,22] for p = 2.…”

Section: Notationmentioning

confidence: 96%

“…The wavefront for the speed c * satisfies U c * < 1 (this is guaranteed by condition (1.3); see [22]) and is finite (we are using the slow diffusion regime assumption here); that is, there exists a value ξ 0 such that U c * (ξ) = 0 for all ξ ≥ ξ 0 and U c * (ξ) > 0 for all ξ < ξ 0 . Moreover,…”

Section: Notationmentioning

confidence: 99%

“…The problems that we are considering have a unique travelling wave with a monotonic front connecting 1 to 0 and supported in (−∞, 0]; see for instance [25] for the case p = 2 and [4,5,22] for p = 2. For p ≥ 2 we will prove that the free boundary of any spreading radial solution with bounded and compactly supported initial datum approaches as t → ∞ the sphere of radius c * t − (N − 1)c # log t − r 0 , where c * is the velocity of the unique travelling wave mentioned above, c # > 0 is a constant independent of the solution, and r 0 ∈ R is a constant that depends on the initial datum.…”

Section: Introductionmentioning

confidence: 99%

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Travelling-wave behaviour in doubly nonlinear reaction-diffusion equations

Du¹,

Gárriz²,

Quirós³

2020

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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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“…Further information about these results on wavefronts and many others can be found in [25] for the case p = 2 and in [4,5,15,22] for p = 2.…”

Section: Notationmentioning

confidence: 96%

Section: Notationmentioning

confidence: 99%