My question comes from Taylor's PDE textbook, volume III.
Consider a semilinear hyperbolic system, $u_t=Lu+g(u)$, $u(0)=f$, where $Lu=\sum_j A_j \partial_{x_j}u$, $g(0)=0, \ |g'(u)| \le C$, take $M=\mathbb{T}^n$.
- Let $u_\epsilon$ be a solution to an approximating equation:
$$u_{\epsilon_t} = J_{\epsilon}LJ_\epsilon u_\epsilon +J_\epsilon g(J_\epsilon u_\epsilon), \ u_\epsilon(0)=f$$
Show that:
$$d/dt\| u_\epsilon \|_{L^2}^2 \le C \| u_\epsilon \|_{L^2}^2 , \ d/dt \| \nabla u_\epsilon \|_{L^2}^2 \le C \| \nabla u_\epsilon \|_{L^2}^2$$
Deduce that there exists a solution to the approximating equation for any $\epsilon>0$ which is defined for all $t\in \mathbb{R}$ and for all compact $I \subset \mathbb{R}$: $u_\epsilon$ bounded in $L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$.
The above question I solved prior to question 13.
13) Deduce that passing that passing to a subsequence $u_{\epsilon_k}$ we have a limit point $u\in L^{\infty}_{loc}(\mathbb{R},H^1(M))\cap Lip_{loc}(\mathbb{R}, L^2(M))$. s.t $u_{\epsilon_k} \rightarrow u \in C(I,L^2(M))$ in norm for all compact $I\subset \mathbb{R}$; hence, $g(J_{\epsilon_k}u_{\epsilon_k})\rightarrow g(u)$ in $C(\mathbb{R}, L^2(M))$, and $u$ solves the original semilinear hyperbolic pde. Examine the issue of uniqueness.
My problem is in exercise 13, I thought that the first part is a simple conclusion from problem 12 of Arzela-Ascoli theorem, but I am not sure it that's much easy.
As for the uniqueness part, I believe we need to use here Gronwall's inequality on the norm, we get for if two solutions of the semilinear hyperbolic pde, $u1,u2$, then by denoting $w=u1-u2$, and $h(w)=g(u1)-g(u2): $$\| w(t) \|_{L^2}^2 = (w(0),w(t)) + \int_0^t (Lw(s)+h(w(s)), w(t))ds$$
I need somehow to estimate $(Lw(s),w(t))$ from above, which I don't see how. I see how to esitamte $(h(w(s)),w(t)$, by Cauchy-Schwarz we have $(h(w(s)),w(t))\le \|h(w(s))\|_{L^2} \| w(t)\|_{L^2}$, $\| h(w(s)) \|_{L^2} = \| w(s)\int_0^1 h'(rw)dr \|_{L^2} \le \| w(s)\|_{L^2} \| \int_0^1 h'(rw)dr\|_{L^{\infty}} \le 2C \| w(s) \|_{L^2} \sup_{v\in \mathbb{R}^n} |h'(v)| \le 2C\| w(s) \|_{L^2}$, since $|h'(v)|=|g'(u2+v)-g'(u2)| \le C+C =2C$.
I appreciate your help here. Thanks!!!
P.S I forgot to mention that $J_\epsilon u = j_\epsilon * u$, where $j_\epsilon$ is Friedrichs mollifier.