Many equations in analysis describe how a quantity changes along one variable: time for an oscillator, arc length for a curve, or a parameter for an [ordinary differential equation](/page/Ordinary%20Differential%20Equation). A partial differential equation begins when the unknown depends on several variables at once. The derivative in one direction can no longer tell the whole story, because spatial variation, temporal change, boundary behaviour, and geometry all interact in the same equation.

The simplest warning comes from heat flow. If $u(t,x)$ denotes temperature, then knowing only the time derivative $\partial_t u$ does not determine the evolution, because heat also reacts to spatial curvature through $\Delta u$. A steep linear temperature profile and a curved one may have the same value and the same first spatial derivative at a point, but only the curved profile has local heat imbalance. The PDE records that imbalance.

[example: Heat Flow Needs Spatial Curvature] Think of $u_1(x)=0$ and $u_2(x)=x^2$ as instantaneous spatial temperature profiles on $\mathbb{R}$, not as full time-dependent solutions of the [heat equation](/page/Heat%20Equation). At $x=0$, their values agree: \begin{align*} u_1(0)=0 \end{align*} and \begin{align*} u_2(0)=0^2=0. \end{align*} Their first derivatives also agree, since \begin{align*} u_1'(x)=0 \end{align*} and \begin{align*} u_2'(x)=2x, \end{align*} so \begin{align*} u_1'(0)=0 \end{align*} and \begin{align*} u_2'(0)=2\cdot 0=0. \end{align*} On $\mathbb{R}$, the Laplacian is the [second derivative](/page/Second%20Derivative). For $u_1$, differentiating $u_1'(x)=0$ gives \begin{align*} \Delta u_1(x)=u_1''(x)=0. \end{align*} For $u_2$, differentiating $u_2'(x)=2x$ gives \begin{align*} \Delta u_2(x)=u_2''(x)=2. \end{align*} In particular, \begin{align*} \Delta u_1(0)=0 \end{align*} while \begin{align*} \Delta u_2(0)=2. \end{align*} If $U_1$ and $U_2$ solve the heat equation $\partial_t U_i=\Delta U_i$ with initial data $U_i(0,x)=u_i(x)$, then evaluating the equation at $(0,0)$ gives \begin{align*} \partial_t U_1(0,0)=\Delta U_1(0,0)=\Delta u_1(0)=0 \end{align*} and \begin{align*} \partial_t U_2(0,0)=\Delta U_2(0,0)=\Delta u_2(0)=2. \end{align*} The second profile starts changing at the origin while the first does not, even though their value and slope at the origin are the same. The heat equation is therefore sensitive to second spatial variation, not only to slope. [/example]

This chapter introduces the basic language for partial differential equations, explains why boundary and initial data are part of the problem rather than decoration, and organizes the first major families of equations by the features that control their analysis. The subject connects naturally with [Cambridge IA Differential Equations](/page/Cambridge%20IA%20Differential%20Equations), [Cambridge II Analysis of Functions](/page/Cambridge%20II%20Analysis%20of%20Functions), and the geometric viewpoint developed in [Cambridge III Differential Geometry](/page/Cambridge%20III%20Differential%20Geometry).

## Definition

The central object is not just an equation, but an equation imposed on an unknown function over a domain. The domain specifies where derivatives are taken, the target specifies what kind of quantity is being solved for, and the differential operator specifies which local measurements of the function enter the law.

Before writing a general PDE in coordinates, we fix the notation used to record all derivatives up to a chosen order. A multi-index is a vector $\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{N}_0^n$, with length $|\alpha|=\alpha_1+\cdots+\alpha_n$, and

\begin{align*} \partial^\alpha u=\frac{\partial^{|\alpha|}u}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}. \end{align*}

The space $C^k(U;\mathbb{R}^m)$ consists of maps $U\to\mathbb{R}^m$ whose partial derivatives through order $k$ are continuous. When a formula uses variables indexed by all $\alpha$ with $|\alpha|\leq k$, those variables represent the finite jet of values and derivatives of $u$ up to order $k$ at a point.

[definition: Partial Differential Equation] Let $U$ be an open subset of $\mathbb{R}^n$, let $m,q \in \mathbb{N}$, and let $k \in \mathbb{N}$. For each multi-index $\alpha$ with $0\leq |\alpha|\leq k$, let $z_\alpha \in \mathbb{R}^m$, and let \begin{align*} F: U \times \prod_{0\leq |\alpha|\leq k}\mathbb{R}^m &\to \mathbb{R}^q \end{align*} be a specified function. For $u\in C^k(U;\mathbb{R}^m)$, the induced differential operator has type \begin{align*} P:C^k(U;\mathbb{R}^m)&\to C^0(U;\mathbb{R}^q). \end{align*} It is defined by \begin{align*} P(u)(x)&=F\left(x,\{\partial^\alpha u(x):0 \leq |\alpha| \leq k\}\right). \end{align*} A partial differential equation of order at most $k$ for an unknown function $u: U \to \mathbb{R}^m$ is an equation of the form \begin{align*} F\left(x,\{\partial^\alpha u(x):0 \leq |\alpha| \leq k\}\right)=0 \end{align*} for all $x \in U$. [/definition]

The notation hides a large amount of structure. The same displayed form may describe a scalar equation, a coupled system, a stationary law, or an evolution equation. Even so, the definition isolates the local character of the subject: a PDE constrains a function by comparing its values and its partial derivatives at each point.

## Notions of Solution

The strongest first interpretation asks for all derivatives in the displayed equation to exist pointwise. This is the natural starting point when the coefficients and data are smooth, and it is the version closest to multivariable calculus.

[definition: Classical Solution] Let $U$ be an open subset of $\mathbb{R}^n$, let $m,q,k\in\mathbb{N}$, and let \begin{align*} F: U\times\prod_{0\leq |\alpha|\leq k}\mathbb{R}^m&\to\mathbb{R}^q \end{align*} define a partial differential equation of order at most $k$ for an unknown map $u:U\to\mathbb{R}^m$. A classical solution of this equation is a function $u\in C^k(U;\mathbb{R}^m)$ such that \begin{align*} F\left(x,\{\partial^\alpha u(x):0\leq |\alpha|\leq k\}\right)&=0 \end{align*} for every $x\in U$. [/definition]

Classical solutions are valuable when they exist, but many natural equations create corners, shocks, or functions whose derivatives exist only after [integration by parts](/theorems/210). This pressure leads to weak formulations, where the equation is tested against smooth compactly supported functions instead of being read point by point.

[definition: Weak Solution] Let $U$ be an open subset of $\mathbb{R}^n$, let $m,q\in\mathbb{N}$, let $X$ be a function space of maps $U\to\mathbb{R}^m$, and let $\mathcal{T}\subset C_c^\infty(U;\mathbb{R}^q)$ be a specified test-function space. Let \begin{align*} \mathcal{A}:X\times \mathcal{T}&\to\mathbb{R} \end{align*} be the residual functional associated to the chosen weak formulation of the equation. A weak solution in $X$ for this weak formulation is a function $u\in X$ such that \begin{align*} \mathcal{A}(u,\varphi)=0 \end{align*} for every $\varphi\in\mathcal{T}$. [/definition]

The residual functional is part of the formulation, not a canonical object attached to the words "partial differential equation" alone. It is usually built by multiplying the differential expression by a [test function](/page/Test%20Function), integrating over the domain, and applying the chosen integration-by-parts identity. For a scalar equation one often takes $\mathcal{T}=C_c^\infty(U)$; for a system with $q$ equations, vector-valued tests in $C_c^\infty(U;\mathbb{R}^q)$ record each residual component. For example, a divergence-form equation is integrated by parts, placing derivatives on the test function. This is the bridge from PDE to Sobolev spaces and functional analysis.

[example: Weak Form of Poisson's Equation] Let $U$ be a bounded open subset of $\mathbb{R}^n$, let $f\in L^2(U)$, and first suppose $u\in C^2(U)$ satisfies $-\Delta u=f$ pointwise. For a test function $\varphi\in C_c^\infty(U)$, multiplying the equation by $\varphi$ and integrating gives \begin{align*} \int_U (-\Delta u)\varphi\,d\mathcal{L}^n=\int_U f\varphi\,d\mathcal{L}^n. \end{align*} Since $\Delta u=\sum_{i=1}^n \partial_{x_i}\partial_{x_i}u$, the left-hand side is \begin{align*} \int_U (-\Delta u)\varphi\,d\mathcal{L}^n=-\sum_{i=1}^n\int_U (\partial_{x_i}\partial_{x_i}u)\varphi\,d\mathcal{L}^n. \end{align*} Because $\varphi$ has compact support in $U$, one-dimensional [integration by parts](/theorems/2098) in each coordinate has no boundary term, so \begin{align*} -\int_U (\partial_{x_i}\partial_{x_i}u)\varphi\,d\mathcal{L}^n=\int_U (\partial_{x_i}u)(\partial_{x_i}\varphi)\,d\mathcal{L}^n. \end{align*} Summing over $i$ gives \begin{align*} \int_U (-\Delta u)\varphi\,d\mathcal{L}^n=\sum_{i=1}^n\int_U (\partial_{x_i}u)(\partial_{x_i}\varphi)\,d\mathcal{L}^n. \end{align*} Since $\nabla u\cdot\nabla\varphi=\sum_{i=1}^n(\partial_{x_i}u)(\partial_{x_i}\varphi)$, the identity becomes \begin{align*} \int_U \nabla u\cdot\nabla\varphi\,d\mathcal{L}^n=\int_U f\varphi\,d\mathcal{L}^n. \end{align*} The weak formulation therefore declares $u$ to solve $-\Delta u=f$ when this last identity holds for every $\varphi\in C_c^\infty(U)$; only the first weak derivatives of $u$ appear, even though the classical equation contains second derivatives. [/example]

## Data and Well-Posed Problems

### Boundary Data