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.
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 .