The self-consistent chemotaxis-fluid system $$\left\{\begin{array}{cc}{n}_t+u·\nabla n=\Delta n-\nabla ·\left(n\nabla c\right)+\nabla ·\left(n\nabla \phi \right),\hfill & x\in \Omega ,t>0,\hfill \\ c_t+u·\nabla c=\Delta c-nc,\hfill & x\in \Omega ,t>0,\hfill \\ u_t+\kappa (u·\nabla )u+\nabla P=\Delta u-n\nabla \phi +n\nabla c,\hfill & x\in \Omega ,t>0,\hfill \\ \nabla ·u=0,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$ is considered under no-flux boundary conditions for $n,c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $\Omega \subset {ℝ}^N$$(N=2,3...$

We discuss several global approximate controllability properties for the semilinear heat equation with superlinear reaction-convection term, governed in a bounded domain by locally distributed controls. First, based on the asymptotic analysis in vanishing time, we study the steering of the projections of its solution on any finite dimensional space spanned by the eigenfunctions for the truncated linear part. We show that, if the control-supporting area is properly chosen, then they can approximately...

We examine the parabolic system of three equations $u_t$ - Δu = $v^p$, $v_t$ - Δv = $w^q$, $w_t$ - Δw = $u^r$, x ∈ ${ℝ}^N$, t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.

The Fujita type global existence and blow-up theorems are proved for a reaction-diffusion system of m equations (m>1) in the form $u_{it}=\Delta u_i+u_{i+1}^{p_i},i=1,...,m-1,$$u_{mt}$

A model of chemotaxis is analyzed that prevents blow-up of solutions. The model consists of a system of nonlinear partial differential equations for the spatial population density of a species and the spatial concentration of a chemoattractant in n-dimensional space. We prove the existence of solutions, which exist globally, and are L∞-bounded on finite time intervals. The hypotheses require nonlocal conditions on the species-induced production of the chemoattractant.

In this paper we consider a model of chemorepulsion. We prove global existence and uniqueness of smooth classical solutions in space dimension n = 2. For n = 3,4 we prove the global existence of weak solutions. The convergence to steady states is shown in all cases.

We study the global existence and long-time behavior of solutions for a class of semilinear degenerate parabolic equations in an arbitrary domain.

Si considerano equazioni di Ginzburg-Landau complesse del tipo $u_t-\alpha \mathrm{\Delta }u+P\left({\left|u\right|}^2\right)u=0$ in ${\mathbb{R}}^N$ dove $P$ è polinomio di grado $K$ a coefficienti complessi e $\alpha$ è un numero complesso con parte reale positiva $\mathrm{\Re }\alpha$. Nell'ipotesi che la parte reale del coefficiente del termine di grado massimo $P$ sia positiva, si dimostra l'esistenza e la regolarità di una soluzione globale nel caso $\left|\alpha \right|<C\mathrm{\Re }\alpha$, dove $C$ dipende da $K$ e $N$.