If $u_m \rightharpoonup u$, how to show using monotonicity that $f(u_m) \rightharpoonup f(u)$?

Question

Let $$u_m \rightharpoonup u \quad \text{(weakly) in $L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$}.$$ We are given $f:\mathbb R \to \mathbb R$, a Lipschitz continuous invertible map which is monotone $$(f(x)-f(y))(x-y) \geq 0\quad\text{for all $x, y$}$$ and we have $$f(u_m) \rightharpoonup f(v) \quad \text{in $L^2(0,T;H^1(\Omega))$}$$ for some $v \in L^2(0,T;H^1(\Omega))$ (assume that for $u \in L^2(0,T;H^1)$, $f(u)$ and $f^{-1}(u)$ are in $L^2(0,T;H^{1}(\Omega))$.)

Is it possible to show that indeed $v=u$, i.e. $f(u_m) \rightharpoonup f(u)$? I can't seem to do it by using the monotonicity method.

Answer 1

Do you really mean weak convergence in $L^\infty$ or weak-$*$ convergence? If it is the latter, this does not seem correct. Forget about the spatial dependence and consider functions depending only on t. Now let $u_m(t)=1+\sin(mt)$, and let $f$ be some monotone function which agrees with $f(x)=x+x^3$ on the interval $[0,2]$. Then $u_m$ converges to 1, but $f(u_m)$ converges to 7/2.

Answer 2

Check Proposition 2.1(i) in the book Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Verlag, 2010, by Viorel Barbu.