# Why should we give special attention to at most polynomially growing solutions of PDEs?

The equation $$\begin{gather} \frac{\partial u}{\partial t} (t,x) = \frac{1}{2} \text{Trace}[\sigma(x) \sigma(x) (\text{Hessian}_x u)(x,t)] + \langle \mu (x) , (\nabla_x u) (t,x) \rangle, \\ u(0,x) = \varphi(x), \end{gather}$$ is called Kolmogorov partial differential equation.

I know about a result which states, that this equation has a unique, at most polynomially growing viscosity solution under some assumptions including continuity for $$\varphi$$ and local Lipschitz continuity for $$\sigma, \mu$$.

I have seen several solution-existence theorems of this type associated with PDEs and I am confused by the emphasis placed on "at most polynomial growth" of solutions seemingly required for uniqueness in several cases. I am fairly new to the theory of partial differential equations and I wonder why we only seem to care about solutions which grow at most polynomially.

What about other solutions which might exist and do not fulfill this growth requirement?

Are at most polynomially growing solutions the only ones which are interesting in applications?

I see no reason to single out at most polynomially growing solutions and give them special attention if there might be many other solutions; it seems like a rather arbitrary property to me. So why are we satisfied with existence theorems like the one about Kolmogorov PDEs above?

If someone could shed some light on what makes solutions of this type particularly interesting, that would be fantastic!

Kind regards,

Joker

It is well-known (see Evans Ch.2) that solutions of parabolic equations are not unique without some growth conditions at $$\infty$$. For the linear heat equation, it is required that $$|u(x,t)|\leq e^{A|x|^2}$$, so polynomial and exponential growth is allowed. So it is not unusual to restrict to, say, polynomial growth solutions, since this would imply existence and uniqueness. Functions with polynomial growth constitute a very wide class of functions in the context of solving PDEs. Most of the time, one has to place far stronger growth conditions, say bounded, or linear growth, to get uniqueness of viscosity solutions (in the nonlinear setting).
ADDITION: When the growth bounds are violated, solutions are not unique. For the heat equation, there are infinitely many solutions that violate the growth condition $$|u(x,t)|\leq e^{A|x|^2}$$. These solutions are so large as $$x\to \infty$$ that heat flows very quickly from $$\infty$$ in towards the origin and the solution exhibits finite time blow up. So the solutions are poorly behaved and do not represent the physical phenomenon being modeled (e.g., heat flow), so are normally discarded as being "non-physical".