Examinando por Autor "Sánchez, Ariel"

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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.
Equivalence and finite time blow-up of solutions and interfaces for two nonlinear diffusion equations
(Elsevier, 2020) Hernández-Bermejo, Benito; Iagar, Razvan G; Gordoa, Pilar R; Pickering, Andrew; Sánchez, Ariel
In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation u_t = (u^m)_{xx} + a(x)(u^m)_x + b(x)u^m, posed for x ∈ R, t ≥ 0 and m > 1, where a, b are two continuous real functions, and the solutions to the nonhomogeneous diffusion equation of porous medium type f(y)θ_τ = (θ^m)_{yy}, posed in the half-line y ∈ [0, ∞) with τ ≥ 0, m > 1 and suitable density functions f(y). We apply this correspondence to the case of constant coefficients a(x) = 1 and b(x) = K > 0. For this case, we prove that compactly supported solutions to the first equation blow up in finite time, together with their interfaces, as x → −∞. We then establish the large time behavior of solutions to a homogeneous Dirichlet problem associated to the first equation on a bounded interval. We also prove a finite time blow-up of the interfaces for compactly supported solutions to the second equation when f(y) = y^{−γ} with γ > 2.
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.
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
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.
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 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.