I am studying this topic and there are general examples of PDEs having a unique solution, proved by the energy method. However, I am wondering if this is not true for every PDE — under what conditions is the solution not unique?

I know $\Delta u = f(x)$ in $\Omega$ with $u = \phi$ on boundary of $\Omega$ has a unique solution. But what about something like $\Delta u - u^3 = f$ with similar other data?


Examples abound even in the world of ODEs.

Consider first the boundary value problem for $x\colon [0,1] \to \mathbb{R}$: $$\ddot{x}+\lambda x = 0;\quad x(0)=x(1)=0.$$ Of course this admits the trivial solution $x=0$, but when $\lambda = k^2 \pi^2$ for $k\in \mathbb{N}$ this also admits the non-trivial solution $x(t)=\sin(k\pi t)$.

Next consider the initial-value problem $x\colon [0,T) \to \mathbb{R}$: $$ \dot{x}(t) = f(x(t)); \quad x(0)=x_0 .$$

By the Picard theorem, this equation admits a unique solution for some $T>0$ if $f \colon \mathbb{R} \to \mathbb{R}$ is locally Lipschitz. But this fails if $f$ is non-Lipschitz. Indeed, take $f(x)= x^{\frac{1}{3}}$ and $x_0=0$. Then there is the trivial solution $x=0$, but also the non-trivial solution $x(t)=\sqrt{\frac{8}{27}} t^{\frac{3}{2}} $.

One lesson you can take from this is that it depends a lot on the boundary conditions. For boundary-value problems for elliptic PDE one can cook up similar eigenvalue-type problems exhibiting non-uniqueness (have a look into $-\Delta u = \lambda u$ for $u \colon \Omega \to \mathbb{R}$ with $u|_{\partial \Omega} =0$ when $\Omega \subset \mathbb{R}^n$ is an open bounded subset). For the above initial-value problem one has uniqueness provided solutions are sufficiently smooth, and this is a rule which holds by and large for many evolutionary equations. There are some exceptions which should be known about (look into infinite energy solutions to the heat equation), but it makes a reasonably good rule of thumb that solutions to evolutionary equations are uniquely determined by the initial data provided the solutions are sufficiently regular. To see an example of failure of uniqueness (in the PDE world) within the class of weak solutions to an initial-value problem, I suggest you look at the well known shock-wave solutions to Burger's equation.

In answer to your question: "how to tell?". For someone starting out in PDE, the short answer is that there is no general way to tell. You have to treat the PDEs that arise on a case by case basis. But I hope this answer goes some way towards convincing you that it is not a trivial matter :)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.