Articles Récents / Recent Papers

ARTICLES RECENTS / RECENT PAPERS

A parabolic Hardy-Hénon equation with quasilinear degenerate diffusion
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with Razvan Gabriel Iagar (Madrid)

Local and global well-posedness, along with finite time blow-up, are investigated for the following Hardy-Hénon equation involving a quasilinear degenerate diffusion and a space-dependent superlinear source featuring a singular potential
∂_t u=Δ u^m + |x|^σ u^p, t>0, x∈ℝ^N,
when m>1, p>1 and σ∈ (max{-2,-N},0). While the superlinear source induces finite time blow-up when σ=0, whatever the value of p>1, at least for sufficiently large initial conditions, a striking effect of the singular potential |x|^σ is the prevention of finite time blow-up for suitably small values of p, namely, 1 < p ≤ p_G := [2-σ(m-1)]/2. Such a result, as well as the local existence of solutions for p>p_G, is obtained by employing the Caffarelli-Kohn-Nirenberg inequalities. Another interesting feature is that uniqueness and comparison principle hold true for generic non-negative initial conditions when p>p_G, but their validity is restricted to initial conditions which are positive in a neighborhood of x=0 when p∈ (1,p_G), a range in which non-uniqueness holds true without this positivity condition. Finite time blow-up of any non-trivial, non-negative solution is established when p_G < p ≤ p_F:=m+(σ+2)/N, while global existence for small initial data in some critical Lebesgue spaces and blow-up in finite time for initial data with a negative energy are proved for p>p_F. Optimal temporal growth rates are also derived for global solutions when p∈ (1,p_G]. All the results are sharp with respect to the exponents (m,p,σ) and conditions on u₀.
Global boundedness induced by asymptotically non-degenerate motility in a fully parabolic chemotaxis model with local sensing
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with Jie Jiang (Wuhan)

A fully parabolic chemotaxis model of Keller--Segel type with local sensing is considered. The system features a signal-dependent asymptotically non-degenerate motility function, which accounts for a repulsion-dominated chemotaxis. Global boundedness of classical solutions is proved for an initial Neumann boundary value problem of the system in any space dimension. In addition, stabilization towards the homogeneous steady state is shown, provided that the motility is monotone non-decreasing. The key steps of the proof consist of the introduction of several auxiliary functions and a refined comparison argument, along with uniform-in-time estimates for a functional involving nonlinear coupling between the unknowns and auxiliary functions.
Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles
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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.

Anal. Appl. (Singap.), to appear
The Redner--ben-Avraham--Kahng cluster system without growth condition on the kinetic coefficients
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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
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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).
Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part I: Radially non-increasing profiles
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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