Let us compile a list of counterexamples in PDE, similar in spirit to the books Counterexamples in topology and Counterexamples in analysis. Eventually I plan to type up the examples with their detailed derivations.
Please give one example per answer, preferably with clear descriptions and pointers to literature.
Scheffer has shown that there is a nontrivial weak solution $u(x,t)\in L^2(\mathbb R^2\times\mathbb R)$ to the incompressible Euler equations in 2D
$$\begin{cases} \frac{\partial u}{\partial t}+\nabla\cdot(u\otimes u) +\nabla p=0, \\[5pt] \nabla\cdot u=0 . \end{cases}$$ such that $u(x,t)\equiv 0$ for $|x|^2+|t|^2>1$. In other words, the solution is identically zero for $t<-1$, then "something happens" and the solution becomes non-zero, and for all $t>1$ the solution vanishes again. In the real world, this would look like if the water suddenly started to move in a cup that stands firmly on a table.
See V. Scheffer, "An inviscid flow with compact support in space-time", Journal of Geometric Analysis, vol. 3 (1993), pp. 343-401.
In a paper of 2001, N. Filonov has constructed a second order uniformly elliptic operator in divergence form on $\mathbb R^n$ (where $n\ge3$) with Hölder continuous coefficients and compactly supported eigenfunctions. (It is known that this can't happen for Lipschitz continuous coefficients since then one has the unique continuation property.)
Lewy's Example gives a linear PDE with analytic coefficients where local solvability fails.
http://en.wikipedia.org/wiki/Lewy%27s_example
Fritz John's PDE book has a detailed discussion.
In his classic 1935 work Tikhonov showed that the Cauchy problem for the heat equation with 0 initial data has nonzero solutions. He also identified uniqueness classes, of course.
Here are three nontrivial examples with uniformly elliptic equations.
1 Nadirashvilli and Valduts example of a solution to a fully nonlinear uniformly elliptic PDE with constant coefficients which is not $C^2$.
Nadirashvili, N. and Vlăduţ, S. Nonclassical solutions of fully nonlinear elliptic equations. Geom. Funct. Anal. 17 (2007), no. 4, 1283–1296.
The original example was in dimension 12. In a later work (joint with Vladimir Tkachev) they brought the dimension down to 5. The example is known to be impossible in dimension 2. Dimensions 3 and 4 are still open.
2 Safonov's example of a uniformly elliptic equation in 3D whose solution cannot be Lipschitz continuous.
Safonov, M. V. Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients. (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; translation in Math. USSR-Sb. 60 (1988), no. 1, 269–281
Such example is known to be impossible in dimension 2.
3. Plis' example of a uniformly elliptic equation with Holder coefficients for which the unique continuation property does not hold.
Pliś, A. On non-uniqueness in Cauchy problem for an elliptic second order differential equation. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 1963 95–100. 35.42
In particular, he proves that there exists a non zero solution to some uniformly elliptic PDE in 3D with Holder coefficients which is identically zero outside of a ball.
It is related to the example that Hendrik Vogt suggested. Again, in dimension 2 it would not be possible.
In the paper "One cannot hear the shape of a drum", Carolyn Gordon, David L. Webb and Scott Wolpert give an example of two simply connected regions which are isospectral but not isometric. The article is available here. It was in answer to the article "Can you hear the shape of a drum by Mark Kac. Also see this article for more information about this problem.
:-)
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Commented
Jun 26, 2011 at 12:13
It seems that non-uniqueness is the main source of counter-example, at least in the above answers. So, one more:
Consider the heat equation for harmonic maps: $$ u_t-\Delta u+|\nabla u|^2u=0,\qquad |u(x,t)|\equiv1,\qquad(1) $$ with prescribed boundary data $u=g$. A steady solution is a $\phi$ (a harmonic map) such that $$-\Delta \phi+|\nabla u|^2\phi=0,\qquad |\phi(x,t)|\equiv1$$ and $\phi=g$ on the boundary. It is a critical point of the functional $$I[z]:=\int_\Omega|\nabla z|^2dx$$ under the constraints that $|z|\equiv1$ in $\Omega$ and $z=g$ on the boundary.
One may choose $g$ such that there exists a harmonic map $\phi$ that does not minimize locally $I[z]$. In this case, the Cauchy problem for (1), with initial data $\phi$, has two solutions. One is $\phi$, and the other one is time-dependent, with $I[u(t)]$ non-constant (it decays).
This result was due to Bethuel, Coron, Ghidaglia, and Soyeur. See also the work of Coron and later Bertsch, Dal Passo, and van der Hout.
in 1955 Ennio De Giorgi constructed an example of parabolic-type linear equation, whose Cauchy problem has non-unique solution. An English translation of this paper appears in De Giorgi's collected works.
To be more specific, he constructs 4 smooth functions $a(x,t)$, $b(x,t)$, $c(x,t)$, and $u(x,t)$ defined on the strip $\mathbb{R}\times[0,1]$, such that
$$ \frac{\partial^8u}{\partial t^8} = a\frac{\partial^4u}{\partial x^4} + b\frac{\partial^2u}{\partial x^2}+cu, $$
on the whole strip,
$$ \frac{\partial^nu}{\partial t^n} = 0, $$
identically on the line $t=0$ for $n=0,\ldots,7$, and $u$ not identically zero.
There is an example of non-uniqueness of solutions of the Cauchy problem for the heat equation in a class of functions with possible rapid growth at infinity. The example is constructed using the theory of quasi-analytic classes. See, for example, Section 1.9 in A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ, 1964.
Lewy's example and most other allied examples of non-solvable operators have only complex coefficients. Leaving open the question of unsolvable operators with real coefficient. F.Treves in 1962 has constructed a fourth order operator iterating Lewy operator.
If $P$ is Lewy operator then $PP\bar{P}\bar{P}$ is a real coefficient fourth order operator which is not local solvable.
A particularly simple example is Norton's dome, with height given as a function of radial distance on the surface of the dome by
$h = \frac{2}{3g}r^{3/2}$
where $g$ is the gravitational constant near the surface of the earth. The dome has a curvature singularity at the apex. And, if we model a mass at the ($r=0$) apex of this dome with zero velocity, we find that Newton's equation does not have a unique solution; the mass can "fall" at any arbitrary time $t$ for no reason at all.
Norton's paper about the dome: http://philsci-archive.pitt.edu/2943/
A helpful reply: http://philsci-archive.pitt.edu/3195/
In this paper of Caicedo and Castro https://www.aimsciences.org/journals/displayArticles.jsp?paperID=4083 they prove that for the seminilinear wave equation $$\square u + \lambda u + h(u) = c\sin(x+t)$$ subject to double periodic conditions $u(x,t)=u(x,t+2\pi)=u(x+2\pi)$ there is no continuous solutions for |c| large enough. Here $h$ could be any continuous function with compact support and $-\lambda\notin\sigma(\square)$. It is not very hard to prove the existence of weak solutions. The thing here is that the data $c\sin(x+t)$ is smooth (actually analytic) but there is no regularity at all.