Building examples in PDE is not about hunting for exotic formulas. It is about controlling which features of the equation are active. A good example isolates a mechanism: propagation, smoothing, boundary influence, loss of regularity, blow‑up, finite‑speed effects, dispersion, or the failure of an estimate outside its hypotheses.
This recipe is meant to be used. It gives a workflow for constructing solutions and counterexamples that teach you something structural, not just computational.
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Start by declaring the mechanism you want to expose
Every good PDE example has a headline mechanism. Common targets include:
- Smoothing and dissipation (parabolic)
- Rigidity and maximum principles (elliptic)
- Finite propagation speed and characteristics (hyperbolic)
- Boundary layers and incompatibilities (bounded domains)
- Instability or blow‑up in nonlinear models
- Failure of uniqueness at low regularity (weak formulations)
Pick one. Then select the simplest equation class that contains it.
Choose the PDE type by matching the mechanism
You do not need the most general equation. You need the minimal model.
Parabolic: diffusion and smoothing
Use the heat equation $u_t-\Delta u=0$ or semilinear heat $u_t-\Delta u = u^p$. These expose:
- Immediate smoothing for $t>0$
- Energy decay and maximum principles
- Blow‑up vs global existence depending on dimension and exponent
Elliptic: spatial constraints and boundary control
Use Laplace $-\Delta u=0$ or Poisson $-\Delta u=f$. These expose:
- Interior regularity and boundary influence
- Maximum principles and comparison
- Singularities from rough data or geometry
Hyperbolic: wave propagation
Use the wave equation $u_{tt}-\Delta u=0$ or transport $u_t + b\cdot \nabla u =0$. These expose:
- Characteristics
- Finite speed of influence
- Formation of discontinuities in nonlinear conservation laws
Once you commit \to a type, you can choose symmetry and scaling to manufacture an explicit solution.
Exploit symmetries first: translation, scaling, rotation
Symmetries are the easiest way to produce examples because they reduce PDE to simpler forms.
Translation invariance gives plane waves and traveling profiles
For equations with constant coefficients on $\mathbb{R}^d$, try solutions of the form $u(x,t)=g(x\cdot \xi – ct)$ or $u(x,t)=g(x-ct\xi)$. In hyperbolic equations, this aligns with characteristics.
For the transport equation $u_t + c\,u_x=0$ on $\mathbb{R}$, the general solution is
which is the cleanest “propagation without smoothing” example you can have. It teaches that norms like $\|u\|_{L^p}$ are preserved, while derivatives can be as rough as the initial data.
Rotational symmetry reduces to radial equations
If you want a singularity at the origin, radial symmetry is often the fastest route. For Laplace’s equation in $\mathbb{R}^d$, radial harmonic functions are of the form
This single formula yields many classic examples:
- A harmonic function with a nonremovable singularity at $0$
- A function in $H^1_{\text{loc}}$ but not in $H^1$ globally (depending on dimension)
- Boundary value problems where the boundary data forces a singular interior behavior
Scaling reveals criticality and the right norms
For many PDE, scaling tells you which spaces are natural. For the heat equation, scaling is
For the semilinear heat equation $u_t-\Delta u=u^p$, scaling becomes
This is not a theoretical luxury. It tells you how to pick initial data that stresses the equation:
- Choose data concentrated at a point to test blow‑up
- Choose data spread out to test decay
- Choose borderline integrability to test whether estimates need strict assumptions
Scaling is the example‑builder’s compass.
Use separation of variables on bounded domains
On bounded $\Omega$, separation of variables turns PDE into spectral expansions. This is the primary tool for examples where boundary conditions matter.
Heat equation on an interval: explicit decay rates
Take $\Omega=(0,\pi)$ with Dirichlet boundary. The eigenfunctions are $\sin(nx)$, eigenvalues $n^2$. If $u_0(x)=\sin(nx)$, then
This yields a simple family of examples:
- High frequency components decay faster
- Any initial datum decomposes into modes, and each mode decays at a rate tied to its frequency
- Estimates like $\|\nabla u\|$ gain time‑integrated control because the higher modes are heavily damped
You can tune $n$ \to show sharpness of constants in inequalities, or to test numerical schemes (higher modes are the first to be mishandled).
Wave equation on an interval: persistence and resonance
For $u_{tt}-u_{xx}=0$ with Dirichlet boundary on $(0,\pi)$, the same eigenfunctions yield
This contrasts sharply with the heat equation:
- There is no decay in amplitude without damping
- Energy is conserved
- High frequency means rapid oscillations, not rapid decay
The pair of examples is a strong teaching tool because it isolates “dissipation” as the difference, not algebraic complexity.
Construct counterexamples by breaking one hypothesis at a time
A counterexample is most useful when it violates exactly one assumption in a theorem. The recipe is:
- Identify the theorem’s hypotheses and the claimed conclusion
- Decide which hypothesis you will violate (and keep the rest)
- Construct a family that approaches the boundary of that hypothesis
- Show the conclusion fails or the constant blows up
Example: loss of maximum principle when the sign condition is broken
For elliptic equations, the maximum principle often requires a sign on the zero‑order term. Consider
and recall that if $c\ge 0$ under suitable conditions, maximum principles hold. If $c$ takes negative values, you can create interior maxima that violate comparison.
A concrete way to build this is to choose $c\equiv -\lambda$ constant and take $u$ as an eigenfunction for $-\Delta$ with eigenvalue $\lambda$. Then
holds, and eigenfunctions change sign and have interior extrema. This shows why the sign condition is not negotiable.
Example: nonuniqueness at low regularity for transport
For transport $u_t + b(x)\cdot\nabla u=0$, uniqueness of weak solutions can fail when $b$ is too rough. A classroom‑level version is to show that if you allow nonsmooth coefficients, characteristic curves may not be unique, and the transported field becomes ambiguous. You can keep this example honest by:
- Writing the characteristic ODE $\dot X = b(X)$
- Choosing a vector field with nonunique integral curves (classically, Hölder but not Lipschitz at the origin)
- Defining weak solutions by pushing forward initial data along different characteristic selections
Even if you do not write the full construction, the mechanism is clear: weak formulations alone do not restore uniqueness when the flow map is not well defined.
Manufacture nonlinear phenomena with self-similar forms
Nonlinear PDE often admit self-similar solutions that reveal blow‑up, spreading, or profile selection.
Blow-up sketch for semilinear heat
Consider $u_t-\Delta u = u^p$ with $p>1$. Seek a self-similar form
where $\alpha=\frac{1}{p-1}$.
Plugging in yields an elliptic equation for $U$ with a confining drift term. Existence of such profiles is delicate, but the structure itself is already an example‑building tool:
- The scaling exponent predicts the blow‑up rate
- The similarity variable predicts how the blow‑up region shrinks
- You can test numerics against the predicted rate even without a closed form profile
This is how PDE examples often work in practice: the example is a family with a predicted scaling law and a clear mechanism, not a single explicit formula.
Traveling waves for reaction-diffusion
For $u_t – u_{xx} = f(u)$ on $\mathbb{R}$, traveling waves $u(x,t)=U(x-ct)$ reduce the PDE to an ODE:
Phase‑plane analysis then builds examples of fronts and pulses. These examples teach:
- Speed selection and stability
- How diffusion interacts with reaction terms
- Why boundary conditions at infinity matter
A checklist you can apply to build a PDE example quickly
When you want a new example, run this checklist and fill in the blanks.
- Mechanism: smoothing, propagation, boundary layer, blow‑up, nonuniqueness, dispersion
- PDE type: elliptic, parabolic, hyperbolic, mixed
- Domain: whole space, half-space, bounded domain, manifold
- Symmetry: radial, traveling, separable, periodic
- Scaling: what is invariant, what is critical
- Solution ansatz: separation of variables, Fourier modes, fundamental solution, similarity form
- Stress test: which theorem hypothesis are you pushing against
One worked example that mixes several techniques: boundary layers in a singular perturbation
Consider
with small $\varepsilon>0$. This ODE is a PDE example in disguise: it is the one‑dimensional prototype for convection–diffusion.
Solve explicitly:
Impose boundary conditions:
Then
So
As $\varepsilon\to 0$, $u(x)\to 0$ for $x<1$ but $u(1)=1$. The transition occurs in a layer of width $\varepsilon$ near $x=1$.
This example is powerful because it is explicit and structural:
- The diffusion term is tiny, yet it is the only term that enforces the boundary condition at the inflow/outflow \end
- Estimates in $H^1$ blow up like $\varepsilon^{-1/2}$, showing why uniform bounds may fail
- Numerical schemes that do not resolve the layer will show artificial oscillations
That is the core of good example construction: a simple formula that explains a mechanism and predicts where proofs will be delicate.
What “research thorough” looks like in examples
Being thorough does not mean adding features. It means extracting consequences.
After building an example, push it through the standard questions:
- Which norms stay bounded, which blow up, and at what rate
- What the example implies about sharpness of constants
- Whether the example survives perturbation (is it stable)
- How the example changes with dimension or domain geometry
- What it suggests about numerical resolution requirements
If you do this, you will build examples that are not just demonstrations, but tools for proof design.
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