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\title{Partial Differential Equations}
\author{Homework 5}
\date{due April 22, 2014}
\maketitle


Let $U \subset \R^n$ be open, bounded, and with $C^2$ boundary, and
set $U_T = U \times (0,T]$.

Consider the semilinear parabolic equation 
\begin{gather*}
  u' - \Delta u = f(u) \qquad \text{in } U_T \,, \\
  u = 0 \qquad \text{on } \partial U \times [0,T] \,, \\
  u = g \qquad \text{on } U \times \{ t=0 \} \,,
\end{gather*}
where 
$u \colon \overline U_T \to \R$, $g \in L^2(U)$, and, for simplicity,
assume that 
\[
  f(u) = u^{p-1}
\]
for some $p \geq 2$.  

Modify the discussion of weak solutions for linear parabolic equations
from Evans, Section~7.1 to the case above.  Write out the argument
completely, where you can be brief on those parts that need no
modification and should be careful and detailed when considering the
effect of the nonlinearity.

Some things to consider:
\begin{itemize}
\item You need to carefully define a notion of weak solution so that
you can make sense of the nonlinear right hand side.  In particular,
you might need to require the solution to lie in certain space-time
$L^p$ spaces.  
\item In general, you cannot hope to get a solution for arbitrary
$T$.  Expect your \emph{a priori} estimates on the sequence of
approximate solutions to blow up in a finite time which depends on the
initial data $g$.  What you still need to show is that the blow up
time is independent of the approximation parameter $m$.

\item You should expect to find a restriction on $p$ which makes the
estimates possible.  

\item A most important tool is the Gagliardo--Nirenberg--Sobolev
inequality.  In the form given by Evans, Section~5.6, Theorem~1, you
will need to use it with the $L^2$ norm on $Du$ on the right hand
side, and you will need to combine it with a H\"older inequality to
get an estimate of the form
\[
  \lVert u \rVert_{L^q} \leq C \, \lVert Du \rVert_{L^2}^\theta \, 
  \lVert u \rVert_{L^2}^{1-\theta} \,.
\]
What is the relation between $n$, $q$, and $\theta$?  Does this
estimate hold for $u \in H_0^1$?

\item Note that if a sequence is weakly converging in $L^p$ and
strongly converging in $L^q$ for, then it will converge strongly in
every intermediate space $L^r$ with $q \leq r < p$. (Why?)  You will
need to use this for spaces $L^p$ in space and time when passing to
the limit in the weak formulation.
\end{itemize}



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