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.

A related question:

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.

A related question:

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.

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.

A related question: