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In this paper we consider radially symmetric solutions of the following parabolic-elliptic cross-diffusion system { u t = Δ u − ∇ (u f (| ∇ v | 2) ∇ v) , 0 = Δ v − μ (t) + g (u) , μ (t) = 1 | Ω | ∫ Ω g (u (⋅ , t)) d x u (x , 0) = u 0 (x) , in Ω × (0 , ∞) , with Ω a ball in R N , N ≥ 1 under homogeneous Neumann boundary conditions, g (u) a regular function with the prototype g (u) = u k , u ≥ 0 , k > 0 . The function f (ξ) = k f (1 + ξ) − α , k f > 0 , describes gradient-dependent limitation of cross diffusion fluxes. Under suitable conditions on the data, we prove that the solution is global in time. If N ≥ 3 , under conditions on f , g and initial data, we prove that if the solution u (x , t) blows up in L ∞ -norm at finite time T m a x then for some p > 1 it blows up also in L p -norm. Moreover a lower bound of blow-up time is derived. [ABSTRACT FROM AUTHOR]

In this note we consider degenerate chemotaxis systems with porous media type diffusion and a source term satisfying the Hadamard growth condition. We prove the Hölder regularity for bounded solutions to parabolic-parabolic as well as for parabolic-elliptic chemotaxis systems. [ABSTRACT FROM AUTHOR]

We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system u t = ∇ ⋅ (u ∇ u u 2 + | ∇ u | 2 ) − χ k f ∇ ⋅ (u ∇ v (1 + | ∇ v | 2) α ) , x ∈ Ω , t > 0 , 0 = Δ v − μ + u , x ∈ Ω , t > 0 under no flux boundary conditions in a ball B = Ω ⊂ R N and initial condition u (x , 0) = u 0 (x) > 0 , χ > 0 , α > 0 , k f > 0 and μ = 1 | Ω | ∫ Ω u 0 d x. Under suitable conditions on α and u 0 it is shown that the solution blows up in L ∞ -norm at a finite time T m a x and for some p > 1 it blows up also in L p -norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates. [ABSTRACT FROM AUTHOR]

This article is concerned with three optimization problems. In the first problem, a functional is maximized with respect to a set that is the weak closure of a rearrangement class; that is, a set comprising rearrangements of a prescribed function. Questions regarding existence, uniqueness, symmetry, and local minimizers are addressed. The second problem is of maximization type related to a Poisson boundary value problem. After defining a relevant function, we prove it is differentiable and derive an explicit formula for its derivative. Further, using the co-area formula, we establish a free boundary result. The third problem is the minimization version of the second problem. [ABSTRACT FROM AUTHOR]

We consider blow-up solutions to parabolic systems, coupled through their nonlinearities under various boundary conditions with nonlinearities depending on the gradient solution. To obtain a lower bound to blow up time t* for the vector solution, Sobolev-type inequalities are introduced to make use of a differential inequality technique. In addition for Dirichlet systems sufficient conditions are introduced to derive an upper bound for t* and to have a criterion for the global existence of the vector solution. [ABSTRACT FROM AUTHOR]

We investigate continuous dependence on initial data for solutions of a nonlinear parabolic problem, when Robin conditions are prescribed on the boundary ∂ Ω × (t > 0), Ω a bounded R2 domain. [ABSTRACT FROM AUTHOR]

We study blow-up solutions of a nonlinear hyperbolic system of fourth order with time dependent coefficients under Dirichlet or Navier boundary conditions. We prove that under some restrictions on the data there exists a safe interval of existence of the solution and a lower bound of the lifespan is derived. The results are extended to a more general class of systems, where powers of the gradient of the solution are introduced. The proofs are based on some inequalities and coupled estimates techniques. [ABSTRACT FROM AUTHOR]