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...

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}$

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 $N$.

We prove global existence and stability results for a semilinear parabolic equation, a semilinear functional equation and a semilinear integral equation using an inequality which may be viewed as a nonlinear singular version of the well known Gronwall and Bihari inequalities.

We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and Debye-Hückel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method due to S. Montgomery-Smith.

The article deals with a nonlinear generalized Ginzburg-Landau (Allen-Cahn) system of PDEs accounting for nonisothermal phase transition phenomena which was recently derived by A. Miranville and G. Schimperna: Nonisothermal phase separation based on a microforce balance, Discrete Contin. Dyn. Syst., Ser. B, 5 (2005), 753–768. The existence of solutions to a related Neumann-Robin problem is established in an $N\le 3$-dimensional space setting. A fixed point procedure guarantees the existence of solutions...

Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.

We study the chemotaxis system with singular sensitivity and logistic-type source: $u_t=\Delta u-\chi \nabla ·(u\nabla v/v)+ru-\mu u^k$, $0=\Delta v-v+u$ under the non-flux boundary conditions in a smooth bounded domain $\Omega \subset {ℝ}^n$, $\chi ,r,\mu >0$, $k>1$ and $n\ge 1$. It is shown with $k\in (1,2)$ that the system possesses a global generalized solution for $n\ge 2$ which is bounded when $\chi >0$ is suitably small related to $r>0$ and the initial datum is properly small, and a global bounded classical solution for $n=1$.