J. Nonl. Evol. Equ. Appl. 2018 (6), pp. 75-84, published on August 28, 2019: | Journal of Nonlinear Evolution Equations and Applications

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

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

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

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 equationsu_t + (−1)^m Σ_\|α\|=\|β\|=m a_αβ D^α+βu = ∇^μ.( u( φ( ∇^νN_t+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 (D^γ)_\|γ\|=θ with γ = (γ₁, · · · , γ_n) ∈ 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 N_t : (x, t) ↦ N(x, t) stands for the heat kernel related to the homogeneous operator with positive constant coefficients a_αβ.
Keywords:	Higher operators, generalized aggregation equations, decay in L^p, large time behavior.
2010 Mathematics Subject Classification: 35B40, 35K25, 35K57, 35G05, 35A08.