Explosion et existence globale pour quelques problèmes paraboliques
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Ce livre �tudie plusieurs probl�mes paraboliques non-lin�aires. Premi�rement, on consid�re l''�quation de Burgers dans un domaine born� r�el. On �tudie les propri�t�s des solutions de cette �quation lorsqu''on impose des conditions dynamiques sur le bord et lorsque la donn�e initiale est positive. On s''int�resse � l''ordre de croissance et aux points d''explosion des solutions r�guli�res via une �tude du profil de la solution. Ensuite, on �tudie les solutions stationnaires de l''�quation de Burgers param�tr�e. A l''aide du plan des phases, on d�montre l''existence de solutions stationnaires sous diff�rentes conditions au bord (Dirichlet et Neumann). En faisant varier le param�tre lambda, on provoque une bifurcation dans le plan des phases, impliquant de profonds changement au niveau de l''existence des solutions. On s''int�resse aussi � l''explosion des solutions non-stationnaires de l''�quation de Burgers param�tr�e dans un domaine r�el non-born�. Finalement, on �tudie le ph�nom�ne de Fujita. On montre que le ph�nom�ne de Fujita, connu dans le cas des conditions de Dirichlet et de Neumann, reste vrai sous les conditions dynamiques et les conditions de Robin.
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