Blowup in diffusion equations: A survey

A survey of the blowup problem in diffusion up to 90s to the equation

The topics listed here can be used for further research for equations of other types.

Existence of solution in different forms

1. Weak solution
2. Very weak solution
3. Variation of parameter with the fundamental solution, the solution is a fixed point 
4. Continuation theorem. In another word, we have the local existence of solution in appropriate settings. If the solution blowup in some sense (usually one or more norms of the solution becomes infinite), is the conditions for the local existence violated? More precisely, which norm should blowup?

Blowup criteria

1. Subsolution and supersolution: If a subsolution blows up at some time, then the solution should blow up earlier.
2. Monotonicity of the solution. If



then the solution is nondecreasing.

3. Fourier coefficients method (multiplication by the first eigenfunction)
4. Cancavity methods

Fujita's phenomena: every nontrivial (nonnegative) solution blows up in finite time

Qualitative results

1. Geometry: easy to blowup in larger domain and large data; the effect of domain on the blowup time if started with zero iniitial data and positive constant f
2. Location of blowup points, Hausdorff dimension of the blowup sets
3. Asymptotic blowup profile
4. Continuation after blowup

Numerical analysis of blowup

1. Time-stepping and the blowup time
2. Time-stepping criteria
3. Adaptive mesh