ARTICLES RECENTS / RECENT PAPERS

• A sharp mass threshold for boundedness in a critical quasilinear chemotaxis model with indirect signal production: The radially symmetric case
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  with Christoph Walker (Hannover)

  A critical quasilinear chemotaxis model with indirect signal production in space dimensions higher than two is studied under radial symmetry. Exploiting the variational structure of the system, the existence of a sharp mass threshold M_c > 0 is established: all solutions with mass below M_c are global and bounded, while those with mass above M_c and sufficiently negative energy are global but blow up in infinite time. Moreover, the value of the critical mass M_c is related to the best constant of a variant of a Hardy-Littlewood-Sobolev inequality.

• A sharpened Carlson's integral inequality
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  A sharpened version of Carlson's integral inequality is established, which features a remainder term involving a relative distance to the set of extremal functions. A similar approach leads to an alternative derivation of Carlson's and Landau's discrete inequalities and provides another characterization of the optimal constant in Carlson's inequality for finite sums.

  Math. Scand., to appear

• Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
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  with Tatsuya Hosono (Osaka)

  Global existence and boundedness of weak solutions for a fully parabolic degenerate chemotaxis system with indirect signal production are proved for any initial data in the subcritical case and under smallness conditions in the critical and supercritical cases. To construct weak solutions, a time discrete scheme is set up, for which the first equation has a gradient flow structure with respect to the 2-Wasserstein distance, while the other two equations feature an L²-variational structure. The proof relies in particular on the flow interchange method and discrete maximal regularity.

• Convergence to self-similarity for a degenerate parabolic equation with fast-growing spatially-dependent absorption
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  with Razvan Gabriel Iagar (Madrid)

  The large time behavior of non-negative solutions to the absorption-diffusion equation
       ∂_t u = Δ u^m - |x|^σ u^m,    (t,x) ∈ (0,∞) × ℝ^N,
  with m>1 and σ>σ₀:=N(m-1)/(m+1) is identified. It is shown that all solutions approach a unique stationary solution in self-similar variables, which also provides a universal upper bound (friendly giant), strongly contrasting to the standard case σ=0. On the one hand, the convergence proof exploits the variational structure of the equation and a suitable Caffarelli-Kohn-Nirenberg inequality, along with the Bénilan-Crandall homogeneity regularizing effect. On the other hand, the detailed study of the stationary problem combines elliptic estimates, Moser iteration and techniques from ordinary differential equations.

• Dynamics of the reversible Gray-Scott model and convergence to its irreversible limit
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  with Christoph Walker (Hannover)

  Well-posedness of a reversible variant of the Gray-Scott model is shown, along with the convergence of each trajectory to one of the two spatially homogeneous steady states. The principle of linearized stability provides the local attractivity at an exponential rate of the stable steady state, while the long-term limit is identified with the help of center manifold theory. Finally, convergence to the classical Gray-Scott model is proved for an appropriate choice of parameters.

• Sharp non-existence threshold for a parabolic Hardy-Hénon equation with quasilinear diffusion
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  with Razvan Gabriel Iagar (Madrid)

  Optimal conditions for initial data leading to non-existence of non-negative solutions to the Cauchy problem for the parabolic Hardy-Hénon equation
       ∂_t u = Δ u^m + |x|^σ u^p,    (t,x) ∈(0,∞) × ℝ^N,
  with m > 0, σ > 0 and p > max{1,m}, are identified. Assuming that the initial condition satisfies
       u₀∈ L^∞(ℝ^N),    lim_|x|→∞ |x|^γ u₀(x) = L ∈ (0,∞),    u₀ ≥ 0,
  it is shown that non-existence of solution occurs for
       γ < (σ+2)/(p-m) - 2max{p-p_G,0}/[(p-1)(p-m)]
  with
       p_G:=1 +[σ(1-m)/2].
  The above threshold for non-existence is optimal, in view of the existence of self-similar solutions for the limiting value of γ.

  Rev. Mat. Complut., to appear

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