  In this paper, we address stability of parabolic linear Partial Differential
Equations (PDEs). We consider PDEs with two spatial variables and spatially
dependent polynomial coefficients. We parameterize a class of Lyapunov
functionals and their time derivatives by polynomials and express stability as
optimization over polynomials. We use Sum-of-Squares and Positivstellensatz
results to numerically search for a solution to the optimization over
polynomials. We also show that our algorithm can be used to estimate the rate
of decay of the solution to PDE in the L2 norm. Finally, we validate the
technique by applying our conditions to the 2D biological KISS PDE model of
population growth and an additional example.
