Thank you a lot for your help.

So we don't have any compact imbedding???

I can compute $\gamma_*$ and I restricted $p < {\gamma _*} = \frac{{2\left( {\gamma + 1} \right)}}{\gamma }$.

We have $W_\gamma ^{1,2}\left( {0,1} \right) \hookrightarrow L_\gamma ^p\left( {0,1} \right)$ is continuous provided by $p \in \left( {2,{\gamma _*}} \right)$.

Thank you for your help. We consider a bounded sequence $\left\{ {{u_n}} \right\} \subset W_\gamma ^{1,2}$. We have ${H^1}\left( {\varepsilon ,1} \right) \hookrightarrow \hookrightarrow C\left( {\left[ {\varepsilon ,1} \right]} \right)$. If we use diagonal argument, we can find $u \in C\left( {\left( {0,1} \right]} \right)$ such that ${\left\| {{u_{{n_k}}} - u} \right\|_{C\left( {\left[ {\varepsilon ,1} \right]} \right)}} \to 0$. But I cant prove that $u \in L_\gamma ^p\left( {0,1} \right)$ and ${\left\| {{u_{{n_k}}} - u} \right\|_{L_\gamma ^p}} \to 0$. Can you help me?

At first we dont have $u\left( t \right) \in {L^2}\left( \Omega \right)$. So we cant define $\left\| {u\left( t \right)} \right\|$ and we dont have $$\left\| {u\left( t \right)} \right\| \leqslant {\left\| \phi \right\|_{{L^\infty }\left( {0,T;{L^2}} \right)}} + {\left\| k \right\|_{{L^\infty }\left( {0,T} \right)}}\int_0^t {\left\| {u\left( s \right)} \right\|ds} .$$

It's not homework problem. I consider the nonlinear wave equation with boundary conditions $${u_x}\left( {0,t} \right) - u\left( {0,t} \right) = {u_x}\left( {1,t} \right) + u\left( {1,t} \right) = 0,$$ and inital conditions $$u\left( {x,0} \right) = {u_0}\left( x \right),\,{u_t}\left( {x,0} \right) = {u_1}\left( x \right).$$ I proved the existence of weak solution with ${u_0} \in V$. Now i want to prove the existence of weak solution with ${u_0} \in {H^1}$ by by a density argument.