Consider the following nonlinear partial differential equations, the "semilinear heat equation" : $$\partial _t u -\Delta u + u^p=0,$$ where $p \geqslant 2$ is an integer.
A standard way for obtaining some weak solutions is to approximate this equation, construct strong solutions, prove some energy bounds and then pass to the limit. Usually the nonlinear term is the more difficult and one often prove some strong compactness result which I want to avoid.
To be precise, let $(f_n)$ be a sequence of real-valued functions, and assume that this sequence is bounded in the space $L^2(\mathbb{R}^3)$. Then up to an extraction we can assume that $f_n \rightharpoonup f$ an $L^2$ function. Obviously this does not implies $f_n^p \rightharpoonup f^p$. In order to prove statements as $\int f_n^p \phi \rightarrow \int f^p\phi$, one often prove that $f_n \rightarrow f$ in $L^p$.
I find it quite unnecessary, so here is my question :
Let $(f_n)_{n \geqslant 0}$ be a bounded sequence of measurable functions over a space $L^2$ so that without loss of generality $f_n \rightharpoonup f$. What are the requirements to ensure that $f_n^p \rightharpoonup f^p$ in $L^2$-weak ?
I am looking for theorems that give some "minimal requirements", in the spirit of the Vitali's convergence theorem that gives a characteristion of sequences of functions $f_n \to f$ a.e for the convergence in $L^p$, but instead I want weak limits.
Thank you,
Such nonlinear functions are not weak continuous. Only $p=1$ works. Just take oscillating sequences $f_n=sign(\sin(\pi n x))+1$ on $(0,1)$, then $f\rightharpoonup 1$ in every $L^p(0,1)$, $p<\infty$. But $f_n^p\rightharpoonup 2^{p-1}$.
I am not aware of any assuptions that do not bring some compactness or strong convergence of $(f_n)$ like $f_n$ could be monotonically increasing sequence, or bounded in some $W^{1,p}$.
If you are dealing with nonlinear monotone equations, there is the so-called Minty trick, which cleverly avoid passing to the weak limit in nonlinear terms.