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Parabolic problems

  1. Mar 1, 2005 #1
    Let the parabolic problem:
    \newline\frac{\partial u}{\partial t}-\Delta u= f_{0}& \Omega_{T}=\Omega\times ]0,T[\\
    \newline\frac{\partial{u}}{\partial n}=f_{1}& \Gamma_{T}=\partial\Omega\times ]0,T[ \\
    \newline u(.,0)=u_{0}& \Omega \end{array}\right.[/tex]

    Then the weak formulation of this problem is :[tex]
    \newline Trouver & u\in L^{2}(0,T;H^{1}(\Omega))\cap C(0,T;L^{2}(\Omega))\\
    \newline\int_{0}^{T}[a(u(t),v)\phi(t)-(u(t),v)\phi^{,}(t)]dt=(u_{0},v)\phi(0) +\int_{0}^{T}(f_{0}(t),v)\phi(t) dt\\
    \newline + \int_{0}^{T}<f_{1}(t),v>_{H^{-\frac{1}{2}},H^{\frac{1}{2}}}\phi(t)dt&\forall\phi\inD([0,T[)et\forallv\in H^{1}(\Omega)\end{array}\right.
    and \displaystyle (h,g)= \int_{Omega} h(x)g(x) dx and $\displaystyle a(u,v)=(\nabla u,\nabla v).[/tex]
    So how prof that this weak problem have a solution? and u verify .[tex](1)_{1} and (1)_{3}.[/tex]
    u verify .[tex](1)_{3}.[/tex]?
  2. jcsd
  3. Apr 7, 2005 #2
    so? i need help
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