On a cross-diffusion population model deduced from mutation and splitting of a single species

Well-Posedness of a Cross-Diffusion Population Model with Nonlocal Diffusion

“…where v is the solution of the heat problem (7)- (9). Similar results are obtained for nonlinear heat equations or p−Laplacian diffusion operators, see [3].…”

“…Regarding the problem of uniqueness of solutions of the SKT problem, Amman [1] proved the result in the triangular case. In [9], uniqueness of the full system is proven for weak solutions under the assumption ∇u i ∈ L ∞ (Q T ). More recently, Chen and Jüngel [6] have proven the weakstrong uniqueness property for renormalized solutions under several parameter restrictions.…”

On a cross-diffusion system arising in image denoising

“…Since generally no maximum principle holds for parabolic systems, the proof of L ∞ bounds is a challenging problem [20], and ad hoc techniques must be employed to deduce the existence of solutions of particular problems. A useful methodology avoiding the L ∞ bounds requirement of Amman's results was introduced in [10] and later generalized in a series of papers, see [7,8,11,9,12,20] and the references therein. However, this technique relies on the introduction of a Lyapunov functional needing of a particular crossdiffusion structure which is not satisfied by system (3) The nonlinear instability of these type of strongly coupled systems, as induced only by cross diffusion terms, has been also an active area of research, see [15,26,6,16] and the references therein.…”

Qualitative behavior of solutions to cross-diffusion systems from population dynamics

“…Global classical solutions for constant A ij were shown to exist in [17]. Galiano [10] proved the uniqueness of bounded weak solutions to the SKT model with either diagonal diffusion matrix or the regularity assumption ∇u i ∈ L ∞ . Uniqueness of strong solutions was shown by Amann [1] in the triangular case (a 21 = 0 in (4)).…”