Thread: Quasilinear parabolic equations with nonlocal boundary conditions

1. Quasilinear parabolic equations with nonlocal boundary conditions

In the lamma4.4 and 4.5 and 4.6 the required convergence is strong but in the proof she indicate just the weak convergence so why?
what does it mean $L^q(0,T;\ H^{-r}(\Omega))$ or how can I read it mathematicaly?

2. Re: Quasilinear parabolic equations with nonlocal boundary conditions

If you notice the convergence is strong with respect to a functional: For example, in 4.4 it's asserted that $\displaystyle (|u_m|^{p-2}u_m,v)\to (|u|^{p-2}u,v)$ for $\displaystyle v$ in a suitable space, and this is the definition of weak convergence. The case of the other lemmas are analogous by Riesz-representation for $\displaystyle L^p$ or by deifnition of $\displaystyle H^{-s}$.

As for $\displaystyle L^q(0,T;\ H^{-r}(\Omega))$ it's the space of strongly measurable functions $\displaystyle u:[0,T) \to H^{-r}(\Omega)$ such that $\displaystyle \int_0^T \| u\|^q_{H^{-r}} <\infty$.

I suggest you take a look at Evans' "Partial Differential Equations" before continuing.

3. Re: Quasilinear parabolic equations with nonlocal boundary conditions

http://ejde.math.txstate.edu/Volumes/2011/18/chen.pdf

what's the differences between generalized solution and week solution?
what's the differences between nonlocal boundary conditions and local boundary conditions?
what's the differences between $L^q(0,T;\ H^{-r}(\Omega))$ and $L^q(0,T)$?

I want to know how the absolute value (1.1) make the problem change (what will happen if we delete the absolut value)?

4. Re: Quasilinear parabolic equations with nonlocal boundary conditions

1) I thought they were interchangeable, but if I had to guess I would say the generalized solution admits arbitrary distributions while weak solution refers to $\displaystyle H^{s}$ solutions for some exponent $\displaystyle s\geq 1$ (or any suitable Sobolev space).

2) A local problem, basically, is one in which the solution depends, pointwise, only on the values 'near' each point. Your boundary conditions are nonlocal since at each point, the value depends on the value of the solution in the whole domain.

3)Notice that functions in the first space have values on the (infinite dimensional Banach) space $\displaystyle H^{-s}$ while the other has functions with codomain $\displaystyle \mathbb{R}$. This causes some technical difficulties (see the appendix in Evans' book).

4) You need that absolute value, how would you define $\displaystyle (-1)^{\frac{5}{4}}$ without appealing to complex numbers, branch cuts, etc? Basically it's there to make the term behave like $\displaystyle |u|^{p-1}$ but without the restrictions on both the values it can take and the differentiability at a possible steep zero.