# Parabolic problems

1. Mar 1, 2005

### Feynman

Let the parabolic problem:
$$\displaystyle\left\{ \newline\begin{array}{ccc} \newline\frac{\partial u}{\partial t}-\Delta u= f_{0}& \Omega_{T}=\Omega\times ]0,T[\\ \newline\frac{\partial{u}}{\partial n}=f_{1}& \Gamma_{T}=\partial\Omega\times ]0,T[ \\ \newline u(.,0)=u_{0}& \Omega \end{array}\right.$$

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

2. Apr 7, 2005

### Feynman

so? i need help