Mathematics / advanced
Parabolic equations and diffusion
Understand smoothing and long-time behavior through heat kernels, energy methods, comparison, and nonlinear diffusion.
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Inside this module
Module 01 advanced
Parabolic equations and diffusion
Understand smoothing and long-time behavior through heat kernels, energy methods, comparison, and nonlinear diffusion.
- 01
Heat kernels and semigroup smoothing
Derive diffusion's averaging structure while distinguishing whole-space kernels, periodic modes, and initial traces.
29 min - 02
Weak solutions and Galerkin approximation
Construct an energy-class solution without assuming the differentiability one hopes to prove.
31 min - 03
Energy dissipation and uniqueness
Use the correct boundary terms and coercive norm to prove stability and distinguish decay from conservation.
29 min - 04
Parabolic maximum principles
Control interior values from the past and boundary while respecting forcing and reaction signs.
30 min - 05
Duhamel's principle and forcing
Accumulate source contributions through the correct heat semigroup and distinguish mild from classical solutions.
30 min - 06
Parabolic regularity and scaling
Match space-time scales, quantify positive-time derivative control, and identify the regularity that coefficients can support.
32 min - 07
Reaction–diffusion and traveling fronts
Reduce a Fisher–KPP wave to an ODE and distinguish an exact front, a linear tail test, and a spreading theorem.
33 min - 08
The porous-medium equation and finite propagation
Track a degenerate diffusion front with a mass-preserving self-similar profile and the appropriate weak interpretation.
33 min - 09
Entropy methods and convergence to equilibrium
Choose a nonnegative functional, calculate its dissipation, and obtain a rate only from a valid functional inequality.
32 min - 10
Blow-up criteria and continuation
State a local solution framework, identify its continuation norm, and prove finite-time growth through comparison.
34 min