Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

The chemotaxis system u t = Δ u − ∇ ⋅ ( u ∇ v ) , v t = Δ v − u v , is considered under the boundary conditions ∂ u ∂ ν − u ∂ v ∂ ν = 0 and v = v ⋆ on ∂Ω, where Ω ⊂ R n is a ball and v ⋆ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case n = 2, while global weak solutions are constructed when n ∈ {3, 4, 5}. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

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

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 Weak Solutions for the Weakly Dissipative Dullin-Gottwald-Holm Equation

In this paper, we are concerned with the existence and uniqueness of global weak solutions for the weakly dissipative Dullin-Gottwald-Holm equation describing the unidirectional propagation of surface waves in shallow water regime: ut − α2uxxt + c0ux + 3uux + γuxxx + λ(u − α2uxx) = α2(2uxuxx + uuxxx).Our main conclusion is that on c0 = − γ/α2 and λ ≥ 0, if the initial data satisfies certain sign conditions, then we show that the equation has corresponding strong solution which exists globally in time, finally we demonstrate the existence and uniqueness of global weak solutions to the equation.

Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

In this paper, we study the nonexistence of global weak solutions to higher-order time-fractional evolution inequalities with subcritical degeneracy. Using the test function method and some integral estimates, we establish sufficient conditions depending on the parameters of the problems so that global weak solutions cannot exist globally.

Remarks on Parabolicity in a One-Dimensional Interdiffusion Model with the Vegard Rule

Until 1948 the interdiffusion theory was based on the Onsager phenomenology, namely thermodynamics of irreversible processes, and a drift was not included. Its main limitation is practical impossibility of the experimental as well as theoretical determination of mobilities (diffusivities) in multicomponent systems ($$r > 2$$ r > 2 ). After experimental discovery of the drift by Smigelskas and Kirkendall (Trans AIME 171:130–142, 1947), Darken (Trans AIME 175:184–201, 1948) formulated his famous model for the binary system. Consequently, the bi-velocity approach dominates interdiffusion studies (e.g. in more than 500 papers in 2020). In this paper, we consider the diffusional transport in a one-dimensional r-component solid solution. The model is expressed by the nonlinear system of strongly coupled evolution differential equations with initial and nonlinear coupled boundary conditions. We present a non-trivial proof of a theorem called the criterion of parabolicity, which implies the generalized parabolicity condition formulated without a proof in our previous works. This condition is a key in the proofs of our previous theorems on existence, uniqueness and properties of global weak solutions of the differential problem studied. The criterion of parabolicity works if diffusion coefficients are not too dispersed, and it is true in many physical systems. The numerical simulations consistent with real experiments for which our criterion works are given.

Nonexistence of solutions for quasilinear hyperbolic inequalities

In this paper, we study the Cauchy problems for quasilinear hyperbolic inequalities with nonlocal singular source term and prove the nonexistence of global weak solutions in the homogeneous and nonhomogeneous cases by the test function method.