$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$

1.2. Initial and Boundary Value Problems

1. Problems for PDEs
2. Notion of 'well-posedness'

Problems for PDEs

We know that solutions of ODEs typically depend on one or several constants. For PDEs situation is more complicated. Consider simplest equations \begin{gather} u_x =0, \label{eq-1.2.1}\\ v_{xy}=0 \label{eq-1.2.2} \end{gather} with $u=u(x,y)$ and $v=v(x,y)$. Equation (\ref{eq-1.2.1}) could be treaded as an ODE with respect to $x$ and its solution is a constant but this is not a genuine constant as it is constant only with respect to $x$ and can depend on other variables; so $u(x,y)=\phi(y)$.

Then for solution of (\ref{eq-1.2.2}) we have $v_y= \phi(y)$ where $\phi$ is an arbitrary function of one variable and it could be considered as ODE with respect to $y$; then $(v-g(y))_y=0$ where $g(y)=\int \phi(y)\,dy$, and therefore $v-g(y)=f(x)\implies v(x,y)=f(x)+g(y)$ where $f,g$ are arbitrary functions of one variable.

Considering these equations again but assuming that $u=u(x,y,z)$, $v=v(x,y,z)$ we arrive to $u=\phi(y,z)$ and $v=f(x,z)+g(y,z)$ where $f,g$ are arbitrary functions of two variables.

Solutions to PDEs typically depend not on several arbitrary constants but on one or several arbitrary functions of $n-1$ variables. For more complicated equations this dependance could be much more complicated and implicit. To select a right solutions we need to use some extra conditions.

The sets of such conditions are called Problems. Typical problems are

• IVP (initial value problem): one of variables is interpreted as time $t$ and conditions are imposed at some moment; f.e. $u|_{t=t_0}=u_0$;
• BVP (boundary value problem) conditions are imposed on the boundary of the spatial domain $\Omega$: f.e. $u|_{\partial\Omega}=\phi$ where $\partial\Omega$ is a boundary of $Omega$;
• IVBP (initial-boundary value problems aka mixed problems): one of variables is interpreted as time $t$ and some conditions are imposed at some moment but other conditions are imposed on the boundary of the spatial domain.

Remark 1. In the course of ODEs students usually consider IVP only. F.e. for the second-order equation like \begin{equation*}u_{xx}+ a_1 u_{x}+a_2 u=f(x)\end{equation*} such problem is $u|_{x=x_0}=u_0$, $u_x|_{x=x_0}=u_1$. However one could consider BVPs like \begin{gather*}(\alpha_1 u_x+\beta_1 u)|_{x=x_1}=\phi_1,\\ (\alpha_2 u_x+\beta_2 u)|_{x=x_2}=\phi_2 \end{gather*} where solutions are sought on the interval $[x_1,x_2]$. Such are covered in advanced chapters of some of ODE textbooks (but not covered by a typical ODE class). We will need to cover such problems later in this Textbook.

Notion of "well-posedness"

We want that our PDE (or the system of PDEs) together with all these conditions satisfied the following requirements:

• Solutions must exist for all right-hand expressions (in equations and conditions);
• Solution must be unique;
• Solution must depend on this right-hand expressions continuously.

Such problems are called well-posed. PDEs are usually studied together with the problems which are well-posed for these PDEs. Different types of PDEs "admit" different problems.

Sometimes however one needs to consider ill-posed problems.

$\Leftarrow$  $\Uparrow$  $\Rightarrow$