Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour

Gárriz, Alejandro

doi:10.48550/arxiv.1805.10955

Cited by 3 publications

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“…Passing to the nonlinear diffusion setting, the existence of free boundaries was already observed in Porous Medium setting (m > 1 and p = 2) in [20] for Fisher-KPP reactions and only more recently in [38] and, later, in [34] for reactions of type C with time delay. See also the recent preprint [30] where the author studies the long time behaviour of solutions to a Porous Medium reaction-diffusion equation with a larger class of reaction functions. Part (i) of Theorem 1.1 extends the results of [38,34] to the doubly nonlinear setting with reaction satisfying (1.2) (here we do not consider reactions with time delay).…”

Section: Resultsmentioning

confidence: 99%

Bistable reaction equations with doubly nonlinear diffusion

Audrito¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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Reaction-diffusion equations appear in biology and chemistry, and combine linear diffusion with different kind of reaction terms. Some of them are remarkable from the mathematical point of view, since they admit families of travelling waves that describe the asymptotic behaviour of a larger class of solutions 0 ≤ u(x, t) ≤ 1 of the problem posed in the real line. We investigate here the existence of waves with constant propagation speed, when the linear diffusion is replaced by the "slow" doubly nonlinear diffusion. In the present setting we consider bistable reaction terms, which present interesting differences w.r.t. the Fisher-KPP framework recently studied in [7]. We find different families of travelling waves that are employed to describe the wave propagation of more general solutions and to study the stability/instability of the steady states, even when we extend the study to several space dimensions. A similar study is performed in the critical case that we call "pseudo-linear", i.e., when the operator is still nonlinear but has homogeneity one. With respect to the classical model and the "pseudo-linear" case, the travelling waves of the "slow" diffusion setting exhibit free boundaries. Finally, as a complement of [7], we study the asymptotic behaviour of more general solutions in the presence of a "heterozygote superior" reaction function and doubly nonlinear diffusion ("slow" and "pseudo-linear").We anticipate that significant differences from the Fisher-KPP setting can be found in both the ODEs analysis (see Theorem 1.1) and in the asymptotic behaviour of the solutions (see Theorem 1.2 and 1.3), where "threshold effects" and "non-saturation" phenomena appear.We recall that the p-Laplacian is a nonlinear operator defined for all 1 ≤ p < ∞ by the formulaand we consider the more general diffusion term ∆ p u m := ∆ p (u m ) = ∇ · (|∇(u m )| p−2 ∇(u m )), that we call "doubly nonlinear" operator since it presents a double power-like nonlinearity. Here, ∇ is the spatial gradient while ∇· is the spatial divergence. The doubly nonlinear operator (which can be thought as the composition of the m-th power and the p-Laplacian) is much used in the elliptic and parabolic literature (see the interesting applications presented in [17,26,39]) and allows to recover the Porous Medium operator choosing p = 2 or the p-Laplacian operator choosing m = 1. Of course, choosing m = 1 and p = 2 we obtain the classical Laplacian. W.r.t. the Porous Medium setting or the p-Laplacian one, problem (1.1) with doubly nonlinear diffusion is less studied. However, the basic theory of existence, uniqueness and regularity is known. Results about existence of weak solutions of the pure diffusive problem and its generalizations, can be found in the survey [35] and the large number of references therein. The problem of uniqueness was studied later, see for instance [22,23,40,54,56]). For what concerns the regularity, we refer to [53,54] for the Porous Medium setting, while for the p-Laplacian case we suggest [21,41] and the references therein....

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Section: Resultsmentioning

confidence: 99%

Bistable reaction equations with doubly nonlinear diffusion

Audrito¹

2019

Discrete &Amp; Continuous Dynamical Systems - A

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“…for instance with [11,25,29] for the linear diffusion framework). In the nonlinear setting much less has been done (we quote the work [21] for the Fisher-KPP setting and Porous Medium diffusion and [27] for the bistable counterpart). The p-Laplacian case is completely open.…”

Section: Comments and Open Problemsmentioning

confidence: 99%

Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

Audrito

Vázquez

2020

Journal of Differential Equations

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For a fixed bounded domain D ⊂ R N we investigate the asymptotic behaviour for large times of solutions to the p-Laplacian diffusion equation posed in a tubular domainwith p > 2, i.e., the slow diffusion case, and homogeneous Dirichlet boundary conditions on the tube boundary. Passing to suitable re-scaled variables, we show the existence of a travelling wave solution in logarithmic time that connects the level u = 0 and the unique nonnegative steady state associated to the re-scaled problem posed in a lower dimension, i.e. in D ⊂ R N .We then employ this special wave to show that a wide class of solutions converge to the universal stationary profile in the middle of the tube and at the same time they spread in both axial tube directions, miming the behaviour of the travelling wave (and its reflection) for large times.The first main feature of our analysis is that wave fronts are constructed through a (nonstandard) combination of diffusion and absorbing boundary conditions, which gives rise to a sort of Fisher-KPP long-time behaviour. The second one is that the nonlinear diffusion term plays a crucial role in our analysis. Actually, in the linear diffusion framework p = 2 solutions behave quite differently.

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“…for some ℓ − , ℓ + ∈ R such that c(ℓ − ) = c(ℓ + ) = 0 is said to be a wavefront profile from ℓ − to ℓ + (the associated travelling wave is a wavefront solution). Wavefronts are already known to be important in the description of the large-time behaviour of more general solutions for wide classes of initial data when p = 2, a(u) = u m , m ≥ 1, and b ≡ 0 for several reaction nonlinearities c; see for instance [10,19,9,1,20,5,4,6,11]. They are also expected to give the large time behaviour when p = 2.…”

Section: Introductionmentioning

confidence: 99%

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

Gárriz¹

2020

Preprint

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We consider a family of singular Volterra integral equations that appear in the study of monotone travelling-wave solutions for a family of diffusionconvection-reaction equations involving the p-Laplacian operator. Our results extend the ones due to B. Gilding for the case p = 2. The fact that p = 2 modifies the nature of the singularity in the integral equation, and introduces the need to develop some new tools for the analysis. The results for the integral equation are then used to study the existence and properties of travellingwave solutions for doubly nonlinear diffusion-reaction equations in terms of the constitutive functions of the problem.

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Propagation of solutions of the Porous Medium Equation with reaction and their travelling wave behaviour

Cited by 3 publications

References 0 publications

Bistable reaction equations with doubly nonlinear diffusion

Bistable reaction equations with doubly nonlinear diffusion

Travelling wave behaviour arising in nonlinear diffusion problems posed in tubular domains

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

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