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

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 Problem*aka*mixed problem*): 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.

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.