Parabolic equations are the place where PDE technique becomes concrete fast: they are time‑directed, they smooth rough data, and they reward careful bookkeeping. If you can prove the right statement for the heat equation and its close relatives, you have learned a proof pattern that reappears across nonlinear diffusion, fluid models with dissipation, and many coupled systems.
This guide is not a list of theorems to memorize. It is a map of proof moves that actually get used. The goal is to show how a typical parabolic argument is assembled from a small toolkit: a priori estimates, compactness, weak formulations, and a stability step that identifies the limit as the intended solution.
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A clean model problem
Work on a bounded domain $\Omega\subset\mathbb{R}^d$ with smooth boundary, and consider the inhomogeneous heat equation with homogeneous Dirichlet boundary condition:
A standard target is: given $u_0\in L^2(\Omega)$ and $f\in L^2(0,T;H^{-1}(\Omega))$, there exists a unique weak solution
with $\partial_t u\in L^2(0,T;H^{-1}(\Omega))$, and the solution satisfies an energy estimate.
This is the parabolic “hello world.” Almost every parabolic existence proof is a variation on how to produce (and use) an estimate of this type.
Step zero: decide the level of solution and the test space
Your first proof choice is not about cleverness. It is about the correct notion of solution.
A classical solution requires $u$ \to be twice differentiable in space and once in time, and boundary conditions are pointwise. For many data sets that is unrealistic. A weak solution is designed so that you can talk about $\Delta u$ as a distribution and interpret the equation through integration by parts.
For the model problem, the weak formulation is:
Find $u$ with the regularity above such that for almost every $t\in(0,T)$ and for every test function $\varphi\in H_0^1(\Omega)$,
This choice is not cosmetic. It tells you which estimates are available, what “multiply the PDE by $u$” means, and what compactness tools will apply.
The central parabolic proof move: energy estimates
The engine is an inequality that bounds the size of $u$ by data. For the heat equation, the basic energy identity comes from testing with $\varphi=u(t)$. Formally,
The left pairing satisfies $\langle \partial_t u, u\rangle = \frac12\frac{d}{dt}\|u\|_{L^2}^2$ (this becomes rigorous once you know $u\in C([0,T];L^2)$). Then
You now pay for forcing with Cauchy–Schwarz and Young’s inequality in the dual pairing:
Plugging in yields
Integrate over $[0,t]$ \to get the estimate
This inequality is the backbone. It gives uniform bounds in precisely the spaces that define the weak solution.
What you should notice about the structure
Energy estimates are a proof template, not a trick.
- The PDE has a coercive operator (here $-\Delta$ with Dirichlet boundary), and testing with $u$ produces a nonnegative dissipation term.
- The forcing appears on the \right, and a duality estimate plus Young moves part of it back into dissipation.
- You get time‑integrated control over a spatial derivative. That is what parabolic smoothing is made of.
When you move to nonlinear diffusion, you change the test function and the coercivity term, but the logic is the same.
Existence: build approximations that inherit the estimate
The energy estimate is useful only if you can produce approximations that satisfy it uniformly. Two standard choices are:
- Galerkin approximation (finite-dimensional ODE system in time)
- Implicit time discretization (Rothe method)
Both are workhorses because they respect the structure of the energy identity.
Galerkin in one page
Let $\{w_k\}$ be an orthonormal basis of $L^2(\Omega)$ consisting of eigenfunctions of the Dirichlet Laplacian: $-\Delta w_k = \lambda_k w_k$ with $w_k\in H_0^1\cap H^2$. For each $n$, seek an approximate solution
Impose the weak formulation only against the span of $\{w_1,\dots,w_n\}$. This yields an ODE system:
with initial condition $a_j^{(n)}(0)=\langle u_0,w_j\rangle$.
This ODE system has a unique absolutely continuous solution, so $u_n$ exists.
Now test the Galerkin system with $u_n$ itself (meaning take the linear combination of the equations weighted by $a_j^{(n)}$). Because the eigenfunctions diagonalize the Laplacian, you recover the same energy estimate, uniformly in $n$:
That uniformity is the whole point.
Compactness: extract a convergent subsequence
From the estimate you learn:
- $u_n$ is bounded in $L^2(0,T;H_0^1)$.
- $u_n$ is bounded in $L^\infty(0,T;L^2)$.
To identify a limit, you need some compactness in space–time. A standard route uses the Aubin–Lions lemma, which requires a time derivative bound. You get it from the equation:
interpreted in $H^{-1}$. The Laplacian maps $H_0^1\to H^{-1}$ continuously, and $f\in L^2(0,T;H^{-1})$, so
which is uniformly bounded.
Now apply Aubin–Lions with the compact embedding $H_0^1(\Omega)\hookrightarrow L^2(\Omega)\hookrightarrow H^{-1}(\Omega)$. You conclude (after passing \to a subsequence):
- $u_n\to u$ strongly in $L^2(0,T;L^2)$
- $u_n\rightharpoonup u$ weakly in $L^2(0,T;H_0^1)$
- $\partial_t u_n\rightharpoonup \partial_t u$ weakly in $L^2(0,T;H^{-1})$
This is a typical parabolic convergence package: one strong convergence to pass nonlinearities, one weak convergence to pass coercive terms, and a weak time derivative control to justify time continuity.
Identification: pass to the limit in the weak formulation
The Galerkin equations say: for each fixed test function $\varphi\in H_0^1$ that lies in some finite span (or after a density argument for arbitrary $\varphi$), we have
Use weak convergence of $\partial_t u_n$ and $\nabla u_n$ \to pass to limits. You obtain the same identity for $u$. That is existence.
At this point a common mistake is to assume the initial condition is automatic. It is not. You need a result that gives a representative of $u$ continuous into $L^2$ and identifies $u(0)=u_0$. Standard parabolic theory supplies this: if $u\in L^2(0,T;H_0^1)$ and $\partial_t u\in L^2(0,T;H^{-1})$, then $u\in C([0,T];L^2)$. You then compare with the Galerkin initial data (or use a weak formulation with time test functions) \to show the initial value matches.
Uniqueness: difference + energy
Uniqueness uses the same energy pattern with no new ideas. If $u,v$ are two weak solutions with the same data, let $w=u-v$. Then
Test with $w$ \to get
Hence $\|w(t)\|_{L^2}^2$ is nonincreasing and starts at $0$, so it stays $0$. That gives $u=v$.
This is a key parabolic lesson: once you have coercivity, uniqueness is often cheaper than existence.
Maximum principles and comparison: the other parabolic backbone
Energy is not the only structural estimate. For scalar parabolic equations, comparison is often the shortest route to qualitative information.
For the heat equation with $f=0$ and Dirichlet data, the maximum principle says that $\sup_{\Omega} u(t)$ is controlled by boundary and initial values. In the weak setting, you usually implement this by testing with $(u-k)_+$, the positive part above a level $k$, and deriving a differential inequality for $\|(u-k)_+\|_{L^2}$. The core move is:
- Choose a convex function $\eta$ and test with $\eta'(u)$
- Use that $\eta''\ge 0$ \to keep the diffusion term nonnegative
The maximum principle is the gateway to De Giorgi–Moser type iteration, Harnack inequalities, and regularity for rough coefficients, but even at the elementary level it provides robust bounds without differentiating solutions.
Regularity: differentiate the equation only after you can pay for it
A proof strategy pitfall in PDE is differentiating too early. Parabolic regularity is real, but it is not free.
The typical safe progression is:
- Prove existence of a weak solution with minimal assumptions
- Derive improved estimates (energy at higher levels, or test with $-\Delta u$ when allowed)
- Conclude $u$ belongs \to a better space, then interpret the equation more strongly
For example, if $f\in L^2(0,T;L^2)$ and $u_0\in H_0^1$, you can test the equation with $-\Delta u$ \to get
hence
This is a second‑tier estimate and it upgrades $u$ \to $L^2(0,T;H^2\cap H_0^1)$ when the domain is smooth enough. The point is not the specific space; it is the sequence of payments and upgrades.
The parabolic proof template you can reuse
When you leave the heat equation, you keep the same skeleton:
- Choose a notion of solution aligned with your data and coefficients
- Identify a coercive quantity and test the PDE to get an a priori estimate
- Build approximations that respect that estimate
- Use compactness (often Aubin–Lions) \to take limits
- Use stability to identify the limit as a solution
- Use an energy inequality on differences for uniqueness
- Upgrade regularity only after you have estimates that justify the extra operations
This is why starting with parabolic equations is so instructive: the toolkit is small, and the structure is visible. Once you can execute this proof reliably, the rest of parabolic theory feels like systematic variations rather than unrelated tricks.
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