The fundamental solution of a family of anisotropic fourth-order parabolic equations: small-time asymptotics and related Gaussian estimates

We study the fundamental solutions of a class of uniformly elliptic fourth-order parabolic equations in two dimensions, whose principal symbols are not necessarily strongly convex. For constant-coefficient operators, we derive precise small-time asymptotic formulas along certain special directions. These formulas are obtained by the method of steepest descent and reveal the dependence of the asymptotic behavior on the coefficients of the symbol. We subsequently consider operators with bounded measurable coefficients, subject to Dirichlet boundary conditions, and establish Gaussian bounds for their fundamental solutions by means of Davies' exponential perturbation method. The exponential constant in these bounds is shown to be sharp, as a consequence of the constant-coefficient asymptotics.