Articles Récents / Recent Papers

ARTICLES RECENTS / RECENT PAPERS

A Hardy-Hénon equation in ℝ^N with sublinear absorption
pdf

with Razvan Gabriel Iagar (Madrid)

Consider m>1, N≥1 and max{-2,-N} < σ < 0. The Hardy-Hénon equation with sublinear absorption
- Δ v(x) - |x|^σ v(x) + v^1/m(x)/(m-1) = 0, x ∈ℝ^N,
is shown to have at least one solution v ∈ H¹(ℝ^N) ∩ L^(m+1)/m(ℝ^N), which is non-negative and radially symmetric with a non-increasing profile. In addition, any such solution is compactly supported, bounded and enjoys the better regularity v ∈ W^2,q(ℝ^N) for q ∈ [1,N/|σ|). A key ingredient in the proof is a particular case of the celebrated Caffarelli-Kohn-Nirenberg inequalities, for which we obtain the existence of an extremal function which is non-negative, bounded, compactly supported and radially symmetric with a non-increasing profile.
A by-product of these results is the existence of compactly supported separate variables solutions to a porous medium equation with a spatially dependent source featuring a singular coefficient.
Well-posedness of the growth-coagulation equation with singular kernels
pdf

with Ankik Kumar Giri (Roorkee) and Saroj Si (Roorkee)

The well-posedness of the growth-coagulation equation is established for coagulation kernels having singularity near the origin and growing atmost linearly at infinity. The existence of weak solutions is shown by means of the method of the characteristics and a weak L¹-compactness argument. For the existence result, we also show our gratitude to Banach fixed point theorem and a refined version of the Arzelà-Ascoli theorem. In addition, the continuous dependence of solutions upon the initial data is shown with the help of the DiPerna-Lions theory, Gronwall's inequality and moment estimates. Moreover, the uniqueness of solution follows from the continuous dependence. The results presented in this article extend the contributions made in earlier literature.
Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles
pdf

with Razvan Gabriel Iagar (Madrid) and Ariel Sánchez (Madrid)

Existence of a specific family of eternal solutions in exponential self-similar form is proved for the following porous medium equation with strong absorption
∂_t u = Δ u^m - |x|^σu^q in (0,∞) × ℝ^N,
with m>1, q ∈ (0,1) and σ = 2(1-q)/(m-1). Looking for solutions of the form
u(t,x) = e^-αt f(|x|e^βt), α = 2β/(m-1),
it is shown that, for m+q>2, there exists a unique exponent β_* ∈ (0,∞) for which there exists a one-parameter family of compactly supported profiles presenting a dead core. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.
The Redner--ben-Avraham--Kahng cluster system without growth condition on the kinetic coefficients
pdf

Existence of global mild solutions to the infinite dimensional Redner--ben-Avraham--Kahng cluster system is shown without growth or structure condition on the kinetic coefficients, thereby extending previous results in the literature. The key idea is to exploit the dissipative features of the system to derive a control on the tails of the infinite sums involved in the reaction terms. Classical solutions are also constructed for a suitable class of kinetic coefficients and initial conditions.

Port. Math., to appear
Analytic semigroups in weighted L₁-spaces on the half-line generated by singular or degenerate operators
pdf

with Patrick Guidotti (Irvine) and Christoph Walker (Hannover)

Ranges of the real-valued parameters α, a, b, and m are identified for which the operator
A_α(a,b)f(x) = x^α (f''(x) + a f'(x)/x + b f(x)/x²), x>0,
generates an analytic semigroup in L₁((0,∞),x^mdx).
Second order asymptotics and uniqueness for self-similar profiles to a singular diffusion equation with gradient absorption
pdf

with Razvan Gabriel Iagar (Madrid)

Solutions in self-similar form presenting finite time extinction to the singular diffusion equation with gradient absorption
∂_t u - div(ǀ∇uǀ^p-2∇ u) + ǀ∇uǀ^q=0 in (0,∞) × ℝ^N
are studied when N≥1 and the exponents (p,q) satisfy
p_c = 2N/N+1 < p < 2, p-1 < q < p/2.
Existence and uniqueness of such a solution are established in dimension N=1. In dimension N≥2, existence of radially symmetric self-similar solutions is proved and a fine description of their behavior as ǀxǀ→∞ is provided.
Global existence and boundedness of solutions to a fully parabolic chemotaxis system with indirect signal production in ℝ⁴
pdf

with Tatsuya Hosono (Sendai)

Global existence and boundedness of solutions to the Cauchy problem for the four dimensional fully parabolic chemotaxis system with indirect signal production are studied. We prove that solutions with initial mass below (8π)² exist globally in time. This value (8π)² is known as the four dimensional threshold value of the initial mass determining whether blow-up of solutions occurs or not. Furthermore, some condition on the initial mass guaranteeing that the solution remains uniformly bounded is also obtained.

J. Differential Equations, to appear
Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles
pdf

with Razvan Gabriel Iagar (Madrid)

Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption
∂_t u = Δ u^m - |x|^σu^q
posed for (t,x) ∈ (0,∞) × ℝ^N, with m > 1, q ∈ (0,1) and σ = σ_c := 2(1-q)/(m-1) is proved. Looking for radially symmetric solutions of the form
u(t,x) = e^-αt f(|x|e^βt), α = 2 β/(m-1),
we show that there exists a unique exponent β^* ∈ (0,∞) for which there exists a one-parameter family (u_A)_A>0 of solutions with compactly supported and non-increasing profiles (f_A)_A>0 satisfying f_A(0)=A and f_A^'(0)=0. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when σ ∈ (0,σ_c).

Proc. Roy. Soc. Edinburgh Sect. A, to appear