We consider the problem \begin{equation}\label{1} \dfrac{\partial u}{\partial t}-\Delta u + \omega F(u)= f(x,t), \ x \in \mathbb{R}^n, t > 0, \end{equation} with the initial condition \begin{equation}\label{2} u(x,0)=u_0. \end{equation} with $1$ periodic in $x$ boundary conditions.
where $f$ is $1$-periodic function $\mathbb{R}^n \times \mathbb{R}_+$, $f \in C^{2,1}(\mathbb{R}^n \times \overline{\mathbb{R}}_+)$ and $F$ is non linear function, lipschitzian increasing with $F(0)=0$ and $w>0$.
We search a solution $1$ periodic in $x$: $u \in H^1_{\#}(\mathbb{R}^n)$.
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Our aim is to prouve using the Galerkin methode that the problem admit a unique solution $u \in H^1_{\#}(\mathbb{R}^n)$.
I try this:
First, we have the following theorem:
\textbf{Theorem}
Let $T > 0$ and $u_0 \in L^2(\mathbb{R}^n)$. We suppose that $f \in L^2((0,T),L_{\#}^2(\mathbb{R}^n)$. Then there exists a unique solution $u$ such as \begin{equation}\label{3} \displaystyle\int_0^T \langle \partial_t u,v\rangle_{(H^1_{\#})',H^1} \mathrm{d}s +\displaystyle\int_0^T (\displaystyle\int_{(0.,1)^n} \nabla u \cdot \nabla v \mathrm{d}x) \mathrm{d}s + \omega \displaystyle\int_0^T (\displaystyle\int_{(0,1)^n} F(u) v \mathrm{d}x) \mathrm{d}s= \end{equation} $$ = \displaystyle\int_0^T (\displaystyle\int_{(0,1)^n} f v \mathrm{d}x) \mathrm{d}s, \ \forall v \in L^2((0,T),H^1_{\#}(\mathbb{R}^n)). $$ \begin{equation}\label{4} u(x,0)=u_0. \end{equation}
\textbf{end Theorem}
\textbf{Step $1$}
We introduce an sequence $w_1,\ldots,w_m$ of functions with the following properties: $$ \begin{cases} w_i\in H^1_{\#}(\mathbb{R}^n),\\ \forall i, w_1,\ldots,w_m \mbox{ are lineary independant },\\ \mbox{finite linear combinations of } w_i \mbox{ are dense in } H^1_{\#}(\mathbb{R}^n). \end{cases} $$ Such this sequence exists since $H^1_{\#}(\mathbb{R}^n)$ is an separable space.
We search an approximation solution into the form \begin{equation}\label{5} u_m(x,t)= \sum_{i=1}^m \alpha_{im}(t) w_i(x), \end{equation} where $\alpha_{im}$ will be determined by the conditions: \begin{equation}\label{6} \displaystyle\int_0^T \langle \partial_tu_m,w_j\rangle_{(H^1_{\#})',H^1_{\#}} \mathrm{d}s + a(u_m,w_j)+ \displaystyle\int_0^T (\displaystyle\int_{(0,1)^n)} F(u_m) w_j \mathrm{d}x) ds\\ = \displaystyle\int_0^T(\displaystyle\int_{(0,1)^n} f w_j \mathrm{d}x) \mathrm{d}s, \ \forall 1 \leq j \leq m, \end{equation} where \begin{equation}\label{7} a(u,v)= \sum_{i=1}^n \displaystyle\int_{(0,1)^n} \dfrac{\partial u}{\partial x_i} \dfrac{\partial v}{\partial x_i} \mathrm{d} x. \end{equation} We complete the system of non linear ordinary differential equations with the initial conditions: \begin{equation}\label{8} u_m(0)=u_{0m}, u_{0m}= \sum_{i=1}^m \alpha_{im} w_i \to u_0 \mbox{ in } H^1_{\#}(\mathbb{R}^n) \mbox{ when } m \to +\infty. \end{equation}
My question is how we finish this step 1? please.