Example of linear parabolic PDE that blows up Does anyone have an example of a linear parabolic PDE that blows up in finite time in a Sobolev space setting? How does one show blow up for that particular example?
The one in Evans unfortunately is nonlinear. I want to study linear ones first. Thanks.
 A: There are many.  Take for instance, a Cauchy problem for the heat equation
$$
\begin{cases}
u_t=u_{xx}\,\,,\quad x\in\mathbb{R},\; t>0,\\
u|_{t=0}=e^{x^2},\; x\in\mathbb{R},
\end{cases}
$$
that does possess a unique classical solution 
$$
u(x,t)=\frac{e^{\frac{x^2}{1-4t}}}{\sqrt{1-4t}}
$$
on $\,Q=\mathbb{R}\times [0,1/4)\,$.  Nowadays, checking the uniquenes of this solution is seen as nearly a routine exercise.  Though academically, it is well to remember that uniqueness in this Cauchy problem is guaranteed by the classical uniqueness theorem discovered by A. Tychonov: 
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=6410&option_lang=eng   If need be, this classical blowup solution may  well be treated as a weak solution in some weighted Sobolev space.
With little or no change, this example can be modified to fit initial boundary value problems for the heat equation on the half-axis $\mathbb{R}_{+}=\{x\in\mathbb{R}\,\colon\, x>0\}$. Indeed, the initial boundary value problem
$$
\begin{cases}
u_t=u_{xx}\,\,,\quad x>0,\; t>0,\\
u|_{x=0}=0,\quad t\geqslant 0,\\
u|_{t=0}=xe^{x^2},\; x\geqslant 0,
\end{cases}
$$
with the first boundary condition does possess a unique classical solution 
$$
u(x,t)=\frac{xe^{\frac{x^2}{1-4t}}}{(1-4t)^{3/2}}
$$
on $\,Q_{+}\!=\mathbb{R}_{+}\!\times [0,1/4)\,$. While for the initial boundary value problem
$$
\begin{cases}
u_t=u_{xx}\,\,,\quad x>0,\; t>0,\\
u_x|_{x=0}=0,\quad t\geqslant 0,\\
u|_{t=0}=e^{x^2},\; x\geqslant 0,
\end{cases}
$$
with the second boundary condition, its unique classical solution on $\,Q_{+}\!=\mathbb{R}_{+}\!\times [0,1/4)\,$ coincides with that of the Cauchy problem: 
$$
u(x,t)=\frac{e^{\frac{x^2}{1-4t}}}{\sqrt{1-4t}}\,.
$$
As to the third boundary condition,
constructing a somewhat similar blowup example for the heat equation might prove a good question on math.stackexchange.com .
