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Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

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.

Galerkin time-stepping methods for nonlinear parabolic equations

Georgios Akrivis, Charalambos Makridakis (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

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.

Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system

Yanjiang Li, Zhongqing Yu, Yumei Huang (2024)

Czechoslovak Mathematical Journal

The self-consistent chemotaxis-fluid system n t + u · n = Δ n - · ( n c ) + · ( n φ ) , x Ω , t > 0 , c t + u · c = Δ c - n c , x Ω , t > 0 , u t + κ ( u · ) u + P = Δ u - n φ + n c , x Ω , t > 0 , · u = 0 , x Ω , t > 0 , is considered under no-flux boundary conditions for n , c and the Dirichlet boundary condition for u on a bounded smooth domain Ω N ( N = 2 , 3 ...

Global controllability properties for the semilinear heat equation with superlinear term.

A. Y. Khapalov (1999)

Revista Matemática Complutense

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...

Global existence and regularity of solutions for complex Ginzburg-Landau equations

Stéphane Descombes, Mohand Moussaoui (2000)

Bollettino dell'Unione Matematica Italiana

Si considerano equazioni di Ginzburg-Landau complesse del tipo u t - α Δ u + P u 2 u = 0 in R N dove P è polinomio di grado K a coefficienti complessi e α è un numero complesso con parte reale positiva α . 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 α < C α , dove C dipende da K e N .

Global existence and stability of some semilinear problems

Mokhtar Kirane, Nasser-eddine Tatar (2000)

Archivum Mathematicum

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.

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