All Questions

Let $f\in\mathcal{S}(\mathbb{R}^n)$, $a\in(-1,1)$, how i can prove that the following problem: $$\text{div}(t^a\nabla u)=0,\quad\text{in }\mathbb{R}^n\times(0,\infty),$$ $$ u(x,0)=f(x),\quad\forall x\...

Here we're integrating "Banach-valued" functions $u: \Omega \rightarrow C_0(\mathbb{R}^n))$ , and by $u \in L^1(\Omega; C_0(\mathbb{R}^n))$ I mean that $$\int_{x \in \Omega} \| u(x) \|_{\...

Let $\Omega$ be a bounded and regular open subset $\Omega$ of $\mathbb{R}^N$ and $u:[0,\infty)\times \Omega\to \mathbb{R}$ be a smooth function (for example a smooth solution to a PDE). Thus the ...

I am working to establish an estimate in $X^{s,b}$ spaces to prove local well-posedness of a certain equation, and I need to consider some sub-cases. In particular, I wish to show that the following ...

Say that $f:\Omega\times\mathbb{R}\to\mathbb{R}$, where $\Omega\subset\mathbb{R}^N$ is an open set, is a function such that: $f(x,\cdot)\in L^1_{\text{loc}}(\mathbb{R})$ for a.a. $x\in\Omega$ $f(\...