Let $\Omega$ be a bounded domain. I write $L^2$ instead of $L^2(\Omega)$ etc. Let $u \in L^2(0,T;H^1)$ with weak derivative $u’ \in L^2(0,T;H^{-1})$. Consider $$f(u(x,t)) = \begin{cases} -1 &: u(x,t) \in (-\infty, -1)\\ u &: u(x,t) \in (-1, 1)\\ 1 &: u(x,t) \in (1, \infty). \end{cases}$$ We have $f(u) \in L^2(0,T;L^2)$. Does it make sense […]

My quesion involves the weak time derivative. In the book: ‘Partial Differential Equations’ by Evans the time derivative $u’$ of a function $u: [0,T] \rightarrow H^1_0(U)$ is defined by an element $u_t \in L^2(0,T;H^{-1}(U))$ such that $$ \forall \phi \in C_0^{\infty}([0,T]): \int_{[0,T]} u \phi’ dt = -\int_{[0,T]} u’ \phi dt $$ Where $u’\phi$ and $u […]

Suppose I have a function $u\in L^{2}([0,T]\times\Omega)$ for some bounded domain $\Omega$ in $\mathbb{R}^{2}$ or $\mathbb{R}^{3}$. I managed to prove that this implies $u\in L^{2}(0,T; L^{2}(\Omega))$. Now I want to show that opposite is not true. This is where I struggle. According to Fubini’s theorem, it is sufficient to give example of a function $u\in […]

If $\Omega$ is a bounded $C^1$ domain, is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$? Are they the same? I know this is true when instead of $(0,\infty)$ we have a bounded interval.

What does weak convergence in $L^{2}(0,T;H^{-1}(\Omega))$ means? $\Omega$ is open, bounded, has boundary smooth and etc…

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