Just spotted that Yemon has already mentioned this one in the comments :)

Unless I'm misunderstanding something, isn't (3) just (2) with $u_0 = u_l 1_{(-\infty,0)} + u_r 1_{(0,\infty)}$? ($1_B$ being the characteristic function of a set $B$). As things stand, you seem to be asking "how does the solution to an evolution PDE depend on its initial condition?". If this is what you're asking, I'd suggest that you're unlikely to get much help here, but/since there are lots of books which cover this: maybe try Hormander's "Lectures on Nonlinear Hyperbolic Differential Equations" or something similar.

Re. topology: what topology do you give to $C(\Omega,\Omega)$ (compact open presumably)? What relative topology does this give you for $\Omega$ when you regard it as a subspace?

$a = ab$ for every $b$

If the question is along the lines of: "What does $S:=C(\Omega,\Omega)$ tell us about $\Omega$?" ($\Omega$ being some nonempty topological space), I think the subset of $S$ consisting of maps with one-point images tells you quite a lot.

I think the word `Hessian' might be enough to help you out here ;)

I believe that the operator spoken of in the second part is the `Hessian' (the integral disappears because it's part of the inner product). I think this is usually defined by requiring that $f''(u)\phi \psi = (\phi,(\mathrm{Hess}f)(u)\psi)$ for all $\phi$ and $\psi$, where $(.,.)$ is the inner product for the space in which you're working.

I think you might want to define $E_n$ as $C^1([0,1]^n)/\mathbb{C}$ (i.e. consider everything modulo the constant functions and leave out the sup-norm). With this definition, I agree that $E_1\approx C([0,1])$. I think I'm right to say that for $n>1$, $E_n$ is the space of all closed 1-forms on $[0,1]^n$ with continuous coefficients (`closed' being interpreted in the distributional sense). Just a thought!

Which results are available to you depends more on (a) the properties of your kernel $K$ and (b) the function space in which you're looking for solutions $f$. Could you provide some context about the setting in which you're working? If you're not sure, maybe provide a few details about what you know about your kernel and maybe we can work the rest out from that. I'd imagine that whether your kernel is defined over a square as opposed to a rectangle makes no difference, but cannot say this with confidence without more details about your kernel or the setting in which you intend to work.

The $C^{2+\alpha,\beta}(Q_T)$ in my last comment should have been a $C^{2+\alpha,1+\beta}(Q_T)$, and what I'm really interested in is whether you agree that, for $u$ in this space, $u$ and its first order $x$ and $t$ derivatives are Lipschitz continuous with respect to the metric given by $d((x,t),(x',t')) = |x-x'|^\alpha + |t-t'|^\beta$. I suspect that'll suffice, but will check and post an answer once I'm clearer on your definitions.

Hi again. I'm not sure I'm quite clear on the definitions of your spaces. Could you perhaps confirm (or give an indication to the contrary) that all the functions in your first space $C^{2+\alpha,\beta}(Q_T)$ have $u, \dfrac{\partial u}{\partial x}, \dfrac{\partial u}{\partial t}\in C^{\alpha,\beta}(\bar Q_T)$? I think this is likely to be all you need (provided $\Omega$ is an interval or similar) to get a compact embedding here, but I'm reluctant to work on an interpolation inequality until I'm clear on what spaces you're considering. Feel free to refer me to a reference if you like :)

I think you should be ok if $\Omega \subset \mathbb{R}$ is a finite union of bounded open intervals (I think you may want some sort of boundary smoothness for compactness of this sort of embedding in general). I'm not sure you'll find the statement you want explicitly, but you might be able to derive it from suitable interpolation estimates. Might be worth looking what google turns up on interpolation inequalities in parabolic Holder spaces.

Do you mean to say that $\Omega$ is a bounded interval here? If so, isn't it the case that all the domains you're considering here are bounded rectangles (where you'll almost certainly have a compact embedding)? Unless I'm missing something, I think you should be able to get what in that setting using something like the interpolation inequality trick described here: books.google.co.uk/…

I'm not an expert in these matters, but I think the situation is this: while the 'differential version' is {\em really} just a compact way of writing the integral equation, it can (as the respondent in the second link says) "serve as a guide to deducing one integral equation from another". The 'differentials' one encounters in connection with the Ito integral (and in measure theory) are not the same as the `real' ones encountered in differential geometry.

Something I should have said before: to avoid 'silly' cases, you probably want to insist that your $p_i$s are all distinct (which obviously requires $n>1$) and your $M_i$s are (at least) not all zero. Could you possibly provide some background? The answer to your question as currently stated is just `no' as currently stated if you're interested in a unique minimiser. One other case to consider; what happens if all your $M_i$ are isometries and your $p_i$ all lie on a sphere?