Accepted for publication in JNEEA

On a class of higher order equations with higher gradient convolution nonlinear term


B. Ahmad

NAAM Research Group, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

M. Kirane

Université de La Rochelle, Faculté des Sciences et Technologies, Avenue M. Crépeau, 17042 La Rochelle Cedex 1, France

M. Qafsaoui

ESTACA - Groupe ISAE, Parc Universitaire Laval Changé, Rue G. Charpak, 53061 Laval Cedex 9, France

Received on November 9, 2017, revised version: April 11, 2018
Accepted on April 12, 2018

Communicated by Gaston M. N'Guérékata

Abstract.  This paper is devoted to the decay rates of solutions to the nonlinear parabolic equations
ut + (−1)m Σ|α|=|β|=m aαβ Dα+βu = ∇μ.( u( φ( ∇νNt+1) ∗ u )) ,     (1)
where u(0) = u0 ∈ L1(Rn), n > 2m, μ ∈ N* and ν ∈ N with max(μ, ν) ≤ m − 1. The symbol ∗ stands for the convolution operator in the space variable and the higher order nabla differential operator ∇θ denotes the vector (Dγ)|γ|=θ with γ = (γ1, · · · , γn) ∈ Nn. The vectorial function φ represents a nonlinearity term such that |φ(X)| ≤ C |X|M for some real M > (2m−μ)/(n+ν) and Nt : (x, t) ↦ N(x, t) stands for the heat kernel related to the homogeneous operator with positive constant coefficients aαβ.
Keywords: Higher operators, generalized aggregation equations, decay in Lp, large time behavior.
2010 Mathematics Subject Classification:   35B40, 35K25, 35K57, 35G05, 35A08.

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