Articles Récents / Recent Papers

ARTICLES RECENTS / RECENT PAPERS

Large time convergence for a chemotaxis model with degenerate local sensing and consumption
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Convergence to a steady state in the long term limit is established for global weak solutions to a chemotaxis model with degenerate local sensing and consumption, when the motility function is C¹-smooth on [0,∞), vanishes at zero, and is positive on (0,∞). A condition excluding that the large time limit is spatially homogeneous is also provided. These results extend previous ones derived for motility functions vanishing algebraically at zero and rely on a completely different approach.
Global bounded classical solutions to a parabolic-elliptic chemotaxis model with local sensing and asymptotically unbounded motility
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with Jie Jiang (Wuhan)

Global existence and boundedness of classical solutions are shown for a parabolic-elliptic chemotaxis system with local sensing when the motility function is assumed to be unbounded at infinity. The cornerstone of the proof is the derivation of L^∞-estimates on the second component of the system and is achieved by various comparison arguments.
Long term spatial homogeneity for a chemotaxis model with local sensing and consumption
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Global weak solutions to a chemotaxis model with local sensing and consumption are shown to converge to spatially homogeneous steady states in the large time limit, when the motility is assumed to be positive and C¹-smooth on [0,∞). The result is valid in arbitrary space dimension n≥ 1 and extends a previous result which only deals with space dimensions n∈ {1,2,3}.
Well-posedness of the discrete collision-induced breakage equation with unbounded fragment distribution
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with Mashkoor Ali (Roorkee) and Ankik Kumar Giri (Roorkee)

A discrete version of the nonlinear collision-induced breakage equation is studied. Existence of solutions is investigated for a broad class of unbounded collision kernels and daughter distribution functions, the collision kernel a_i,j satisfying a_i,j ≤ Aij for some A>0. More precisely, it is proved that, given suitable conditions, there exists at least one mass-conserving solution for all times. A result on the uniqueness of solutions is also demonstrated under reasonably general conditions. Furthermore, the propagation of moments, differentiability, and the continuous dependence of solutions are established, along with some invariance properties and the large-time behaviour of solutions.
Finite time extinction for a diffusion equation with spatially inhomogeneous strong absorption
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with Razvan Gabriel Iagar (Madrid)

The phenomenon of finite time extinction of bounded and non-negative solutions to the diffusion equation with strong absorption
∂_t u - Δ u^m + |x|^σu^q = 0, (t,x)∈ (0,∞) × ℝ^N,
with m≥ 1, q∈(0,1) and σ>0, is addressed. Introducing the critical exponent σ^* := 2(1-q)/(m-1) for m>1 and σ^*=∞ for m=1, extinction in finite time is known to take place for σ∈ [0,σ^*) and an alternative proof is provided therein. When m>1 and σ≥ σ^*, the occurrence of finite time extinction is proved for a specific class of initial conditions, thereby supplementing results on non-extinction that are available in that range of σ and showing their sharpness.

Differential Integral Equations, to appear
Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption
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with Razvan Gabriel Iagar (Madrid) and Ariel Sánchez (Madrid)

We study the dynamics of the following porous medium 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>2(1-q)/(m-1). Considering the Cauchy problem with non-negative initial condition u₀ ∈ L^∞(ℝ^N), instantaneous shrinking and localization of supports for the solution u(t) at any t>0 are established. With the help of this property, existence and uniqueness of a non-negative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.
Bounded weak solutions to a class of degenerate cross-diffusion systems
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with Bogdan-Vasile Matioc (Regensburg)

Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence (E_n)_{n≥ 2} of Liapunov functionals such that E_n is equivalent to the L_n-norm for each n≥ 2 and E_n^1/n controls the L_∞-norm in the limit n→∞. Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.