Examinando por Autor "Sanchez, Ariel"

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A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential
(Elsevier, 2023) Sanchez, Ariel; Iagar, Razvan Gabriel
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential ∂tu = Δum + |x| −2up, (x, t) ∈ RN × (0,∞), in the range of exponents 1 ≤ p < m and dimension N ≥ 3. The self-similar solution is unbounded at x = 0 and has a logarithmic vertical asymptote, but it remains bounded at any x = 0 and t ∈ (0, ∞) and it is a weak solution in L1 sense, which moreover satisfies u(t) ∈ Lp(RN ) for any t > 0 and p ∈ [1, ∞). As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition u0, contrasting with previous results in literature for the critical limit p = m.
Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction
(2022-06-16) Sanchez, Ariel; Iagar, Razvan Gabriel
We prove existence and uniqueness of the branch of the so-called anomalous eternal solutions in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term ∂ t u = Δ u m + | x | σ u p , posed in R N with N ⩾ 3, where 1,$> 0 < m < m c = N − 2 N , p > 1 , and the critical value for the weight σ = 2 ( p − 1 ) 1 − m . The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a change of sign of both self-similar exponents at m = m s = ( N − 2)/( N + 2), leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a perfect equilibrium in the competition between the fast diffusion and the reaction effects.
Blow up pro les for a quasilinear reaction-diffusion equation with weighted reaction with linear growth
(Springer, 2019) Sanchez, Ariel; Iagar, Razvan Gabriel
We study the blow up profiles associated to the following second order reaction–diffusion equation with non-homogeneous reaction: ∂tu=∂xx(um)+|x|σu,with σ> 0. Through this study, we show that the non-homogeneous coefficient | x| σ has a strong influence on the blow up behavior of the solutions. First of all, it follows that finite time blow up occurs for self-similar solutions u, a feature that does not appear in the well known autonomous case σ= 0. Moreover, we show that there are three different types of blow up self-similar profiles, depending on whether the exponent σ is closer to zero or not. We also find an explicit blow up profile. The results show in particular that global blow up occurs when σ> 0 is sufficiently small, while for σ> 0 sufficiently large blow up occurs only at infinity, and we give prototypes of these phenomena in form of self-similar solutions with precise behavior. This work is a part of a larger program of understanding the influence of non-homogeneous weights on the blow up sets and rates. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
Instantaneous and finite time blow-up of solutions toareaction-diffusion equation with Hardy-type singular potential
(Elsevier, 2020) Sanchez, Ariel; Iagar, Razvan Gabriel
We deal with radially symmetric solutions to the reaction-diffusion equation with Hardy-type singular potential ut = Δum + K |x|2 um, posed in RN × (0, T), in dimension N ≥ 3, where m > 1 and 0 0. The proofs are based on a transformation mapping solutions to our equation into solutions to a non-homogeneous porous medium equation.
Radial equivalence and applications to the qualitative theory for a class of nonhomogeneous reaction-diffusion equations
(Wiley-Blackwell, 2023) Sanchez, Ariel; Iagar, Razvan Gabriel
Some transformations acting on radially symmetric solutions to the followingclass of nonhomogeneous reaction-diffusion equations|x|𝜎1𝜕tu=Δum+|x|𝜎2up,(x,t)∈RN×(0,∞),which has been proposed in a number of previous mathematical works as wellas in several physical models, are introduced. We consider herem≥1,p≥1,N≥1, and𝜎1,𝜎2real exponents. We apply these transformations in connec-tion to previous results on the one hand to deduce general qualitative propertiesof radially symmetric solutions and on the other hand to construct self-similarsolutions, which are expected to be patterns for the dynamics of the equations,strongly improving the existing theory. We also introduce mappings betweensolutions which work in the semilinear casem=1
Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction
(Springer, 2022) Sanchez, Ariel; Iagar, Razvan Gabriel
We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and unbounded weight: ∂tu = ∂x x (um) + |x| σ u p, posed for x ∈ R, t ≥ 0, where m > 1, 0 < p < 1 such that m+ p = 2 and σ > 2 completing the analysis performed in a recent work where this very interesting critical case was left aside. We show that finite time blow-up solutions in self-similar form exist for σ > 2. Moreover all the blow-up profiles have compact support and their supports are localized: there exists an explicit η > 0 such that any blow-up profile satisfies supp f ⊆ [0, η]. This property is unexpected and contrasting with the range m+ p > 2. We also classify the possible behaviors of the profiles near the origin.