Examinando por Autor "Iagar, Razvan Gabriel"

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A critical non-homogeneous heat equation with weighted source
(Cambridge University Press, 2025-03-06) Iagar, Razvan Gabriel; Sánchez, Ariel
Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted source |x|−2∂tu=Δu+|x|σup,(x,t)∈RN×(0,T), are obtained, in the range of exponents p>1 , σ≥−2 . More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as t→∞ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case σ=−2, we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher–KPP equation is derived and employed in order to deduce these properties.
A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions
(Elsevier, 2025-03-01) Iagar, Razvan Gabriel; Munteanu, Diana-Rodica
This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption ∂tu =Δum−|x|σup, posed for (x, t) ∈ RN × (0, ∞), N ≥ 1, and in the range of exponents 1 0. We give a complete classification of (singular) self-similar solutions of the form u(x,t)=t−αf(|x|t−β), α= σ +2 σ(m−1)+2(p−1) , β= p −m σ(m−1)+2(p−1) , showing that their form and behavior strongly depends on the critical exponent pF(σ)=m+ σ+2 N . For p ≥ pF(σ), we prove that all self-similar solutions have a tail as |x| →∞of one of the forms 1/(p−1) u(x,t) ∼ C|x|−(σ+2)/(p−m) or u(x,t) ∼ 1 p −1 |x|−σ/(p−1), while for m
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.
Blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension
(2024) Iagar, Razvan Gabriel; Latorre, Marta; Sánchez, Ariel
We classify all the blow-up solutions in self-similar form to the following reaction-diffusion equation \partial_t u=\Delta u^m+|x|^{sigma}u^p, with m>1, 1<= p0.
Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential
(2024) Iagar, Razvan Gabriel; Latorre, Marta; Sánchez, Ariel
We prove existence and uniqueness of self-similar solutions with exponential form u(x,t)=e^{alpha t}f(|x|e^{-beta t}), alpha, beta>0, to the quasilinear reaction-diffusion equation \partial_t u=Delta u^m+|x|^{sigma}u^p, with m>1, 10 if u_0 is compactly supported.
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.
Optimal existence, uniqueness and blow-up for a quasilinear diffusion equation with spatially inhomogeneous reaction
(2024) Iagar, Razvan Gabriel; Latorre, Marta; Sánchez, Ariel
Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source partial_t u=Delta u^m+|x|^{sigma}u^p, with exponents 10, are established. More precisely, we identify the optimal class of initial conditions u_0 for which (local in time) existence is ensured and prove non-existence of solutions for the complementary set of data. We establish then (local in time) uniqueness and a comparison principle for this class of data. We furthermore prove that any non-trivial solution to the Cauchy problem blows up in a finite time T and finite speed of propagation holds true for t0, then u(t) is compactly supported for t
Programa docente Matemáticas II
(2023-01-16) Iagar, Razvan Gabriel
Este programa docente incluye los objetivos de aprendizaje, los contenidos y su desarrollo por semanas y temas, de la asignatura Matemáticas II que se imparte en el Grado en Ingeniería Electrónica Industrial y Automática de la Escuela Superior de Ciencias Experimentales y Tecnología, Universidad Rey Juan Carlos.
Qualitative properties of solutions to a reaction-diffusion equation with weighted strong reaction
(2023-11) Iagar, Razvan Gabriel; Muñoz Montalvo, Ana Isabel; Sánchez, Ariel
We study the existence and qualitative properties of solutions to the Cauchy problem associated to the quasilinear reaction-diffusion equation ∂tu = ∆u^m + (1 + |x|)^σ u^p, posed for (x, t) ∈ R^N × (0, ∞), where m > 1, p ∈ (0, 1) and σ > 0. Initial data are taken to be bounded, non-negative and compactly supported. In the range when m + p ≥ 2, we prove existence of local solutions with a finite speed of propagation of their supports for compactly supported initial conditions. We also show in this case that, for a given compactly supported initial condition, there exist infinitely many solutions to the Cauchy problem, by prescribing the evolution of their interface. In the complementary range m + p < 2, we obtain new Aronson-Bénilan estimates satisfied by solutions to the Cauchy problem, which are of independent interest as a priori bounds for the solutions. We apply these estimates to establish infinite speed of propagation of the supports of solutions if m + p < 2, that is, u(x, t) > 0 for any x ∈ R^N , t > 0, even in the case when the initial condition u0 is compactly supported.
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 patterns for a reaction-diffusion equation with weighted reaction in general dimension
(2023-11) Iagar, Razvan Gabriel; Muñoz Montalvo, Ana Isabel; Sánchez, Ariel
We classify the finite time blow-up profiles for the following reaction-diffusion equation with unbounded weight: ∂tu = ∆u^m + |x|^σu^p, posed in any space dimension x ∈ R^N , t ≥ 0 and with exponents m > 1, p ∈ (0, 1) and σ > 2(1−p)/(m−1). We prove that blow-up profiles in backward self-similar form exist for the indicated range of parameters, showing thus that the unbounded weight has a strong influence on the dynamics of the equation, merging with the nonlinear reaction in order to produce finite time blow-up. We also prove that all the blow-up profiles are compactly supported and might present two different types of interface behavior and three different possible good behaviors near the origin, with direct influence on the blow-up behavior of the solutions. We classify all these profiles with respect to these different local behaviors depending on the magnitude of σ. This paper generalizes in dimension N > 1 previous results by the authors in dimension N = 1 and also includes some finer classification of the profiles for σ large that is new even in dimension N = 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.
Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential
(2023-11) Iagar, Razvan Gabriel; Muñoz Montalvo, Ana Isabel; Sánchez, Ariel
We prove existence and uniqueness of a global in time self-similar solution growing up as t → ∞ for the following reaction-diffusion equation with a singular potential ∂tu = ∆u^m + |x|^σ u^p posed in dimension N ≥ 2, with m > 1, σ ∈ (−2, 0) and 1 1 and p > 1, showing an interesting effect induced by the singular potential |x|^σ . This result is also applied to reaction-diffusion equations with general potentials V (x) to prevent finite time blow-up via comparison.