We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

We consider discontinuous as well as continuous Galerkin methods for the time discretization of a class of nonlinear parabolic equations. We show existence and local uniqueness and derive optimal order optimal regularity a priori error estimates. We establish the results in an abstract Hilbert space setting and apply them to a quasilinear parabolic equation.

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...

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.