PDE classification and well-posedness

The order is the highest derivative order appearing. A linear equation is linear in the unknown and all its derivatives, with coefficients depending only on the independent variables. A semilinear equation allows nonlinear lower-order terms while retaining a linear principal part with prescribed coefficients. A quasilinear equation remains linear in its highest derivatives, but their coefficients may depend on the unknown or its lower derivatives. Fully nonlinear equations depend nonlinearly on highest derivatives. These labels describe mathematical structure rather than the visual complexity of the formula.

For a scalar second-order equation in x and y, write the principal part as $A u_{xx}+2B u_{xy}+C u_{yy}$. The symmetric coefficient matrix determines the local type. A definite matrix is elliptic; opposite signs of its eigenvalues give hyperbolic type; rank-one degeneracy gives the familiar parabolic case under additional structural assumptions. For variable coefficients, type can change across the domain. For quasilinear equations, evaluating the principal coefficients may require knowing the solution and its first derivatives. This two-variable test does not classify every PDE system or every higher-order equation.

Hadamard well-posedness has three parts: an admissible solution exists for each admissible datum, it is unique within the stated solution class, and changing the data slightly changes the solution slightly in the chosen topology. A theorem therefore specifies spaces for both inputs and outputs, and often a time interval. Continuous dependence might hold in an energy norm while a stronger derivative norm requires additional regularity. Local existence on a data-dependent interval is different from a global theorem for all positive time.

An estimate is often the practical expression of this promise. For source-free heat flow with the same homogeneous Dirichlet or homogeneous Neumann conditions, or periodic conditions, the L2 difference between solutions cannot increase as time advances. A wave estimate naturally compares both initial displacement gradients and initial velocities. Those are different measurements because the equations propagate different information. Boundary compatibility matters too: a continuous solution on a closed space-time cylinder cannot simultaneously have endpoint temperature zero and an initial profile assigning that endpoint temperature one. Weaker solutions can accommodate some mismatches, but their regularity at the corner must then be described honestly.

On a periodic interval, a heat mode sin(nx) is multiplied by exp(−κn²t). Forward evolution suppresses fine-scale differences; reconstructing the initial state multiplies the same difference by exp(κn²t). Thus backward heat reconstruction is not continuously dependent in the same L2 topology, even though one can algebraically invert every individual nonzero mode. Choosing increasingly high frequencies constructs small final perturbations whose reconstructed initial states remain large. This is an instability argument, not simply a warning about floating-point arithmetic.

Use the spectral lab to increase mode count N and then increase coefficient decay p. Observe how added short wavelengths can be reduced by smaller coefficients. The displayed family is exactly a finite sum of sin(nx−t)/nᵖ; playback changes phase and does not apply heat damping. It illustrates the distinction between coefficient size and derivative size, since differentiating a mode introduces factors of n. A well-posedness claim needs the appropriate norm and evolution operator; the animation is a spatial-scale analogy rather than an experiment proving stability.