Global weak solutions to fully cross-diffusive systems with nonlinear diffusion and saturated taxis sensitivity

Abstract

Systems of the type u t = ∇ ⋅ ( D 1 ( u ) ∇ u − S 1 ( u ) ∇ v ) + f 1 ( u , v ) , v t = ∇ ⋅ ( D 2 ( v ) ∇ v + S 2 ( v ) ∇ u ) + f 2 ( u , v ) ( ⋆ ) can be used to model pursuit-evasion relationships between predators and prey. Apart from local kinetics given by f 1 and f 2, the key components in this system are the taxis terms −∇ ⋅ (S 1(u)∇v) and +∇ ⋅ (S 2(v)∇u); that is, the species are not only assumed to move around randomly in space but are also able to partially direct their movement depending on the nearby presence of the other species. In the present article, we construct global weak solutions of (⋆) for certain prototypical nonlinear functions D i , S i and f i , i ∈ {1, 2}. To that end, we first make use of a fourth-order regularisation to obtain global solutions to approximate systems and then rely on an entropy-like identity associated with (⋆) for obtaining various a priori estimates.

Global existence and stabilization in a diffusive predator-prey model with population flux by attractive transition

“…pointwise convergence given in (3.24) we can conclude that ∇ϕ 1+εuε converges to ∇ϕ in L 4 (Ω × (0, ∞)) as ε = ε j ց 0. Combining this with (3.24) then yields that uε∇ϕ 1+εuε converges to u∇ϕ in L 4 3 (Ω×(0, ∞)), which in turn combined with (3.27) implies that…”

“…According to Corollary 3.7, the family (u ε ) ε∈(0,1) is bounded in L 4 3 ((0, T ); W 1, 4 3 (Ω)) for all T > 0 and, according to Lemma 3.8, the family (u εt ) ε∈(0,1) is bounded in L 1 ((0, T ); (W 1,4 (Ω)) * ) for all T > 0. Thus applying the Aubin-Lions lemma to the triple of spaces W 1, 4 3 (Ω) ⊂⊂ L 4 3 (Ω) ⊂ (W 1,4 (Ω)) * as well as the weak compactness property of bounded sets in Sobolev spaces for all T ∈ N combined with a diagonal sequence argument yields a null sequence (ε j ) j∈N and u : Ω × [0, ∞) → R such that (3.25) holds and such that u ε → u in L Similarly according to Lemma 2.2 and Corollary 3.7, the family (v ε ) ε∈(0,1) is bounded in the space L 4 ((0, T ); W 1,4 (Ω)) for all T > 0 and, according to Lemma 3.8, the family (v εt ) ε∈(0,1) is bounded in the space L 1 ((0, T ); (W 1,4 (Ω)) * ) for all T > 0. Given this, we can again apply the Aubin-Lions lemma to the triple of spaces W 1,4 (Ω) ⊂⊂ L 4 (Ω) ⊂ (W As we have now constructed our solution candidates, it remains only to be shown that they in fact fulfill our desired weak solution property and thus prove the second half of Theorem 1.1.…”

“…Moving yet slightly further away from the model we are interested in but staying in the realm of predator-prey modeling and analysis, there have been some efforts made to analyze predator-taxis systems, which model prey fleeing from its predators as opposed to the predators chasing the prey ( [27]). When this dynamic is combined with the already discussed prey-taxis, the resulting system seems to be rather challenging as there are to our knowledge thus far only some constructions of weak solutions available ( [15] considering the one dimensional case, [3] featuring nonlinear diffusion) and classical solutions seem to have only been constructed if both taxis terms are mediated by a regularizing indirection ( [25]) or the initial data is already very close to the equilibria ( [2]). There has also been some discussion of models featuring multiple predators attracted by the same prey ( [17]).…”

Global dynamics of Bazykin-type cross-diffusive model with indirect predator-taxis

“…Tao and Winkler [30,31] prove that (5) possesses at least one global weak solutions in one dimensional case. In addition, there are many researches on system (5), which can be refer to the details in [6,8,20].…”