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.

• Section 7.1.4 of Evans gives a pretty good reason why you don't expect to be able to find such equations easily. Usually blow-up for parabolic PDEs uses strongly the nonlinearity to defeat the maximum principle and the dissipation; therefore I don't see how studying "linear ones" will help your understanding in general. – Willie Wong Feb 4 '14 at 14:43
• (That said, you can always force the solution to blow-up by introducing a singularity in the coefficients, but I suspect that's not what you are looking for.) – Willie Wong Feb 4 '14 at 14:43

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:
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 .
• @Lion: A Cauchy problem for the heat equation requires only one initial condition, and nothing else, since real axis $\mathbb{R}$ has no boundary. Meanwhile, it is a good idea to modify this example to fit initial boundary value problems. I will edit my answer to include such examples. – mkl314 Feb 6 '14 at 12:19