Uniqueness for fractional parabolic and elliptic equations with drift

{contraction and Uniqueness for Quasilinear Elliptic{parabolic Equations

We prove L1{contraction principle and uniqueness of solutions for quasilinear elliptic{parabolic equations of the form @t[b(u)] div[a(ru; b(u))] + f(b(u)) = 0 in (0; T ) ; where b is monotone nondecreasing and continuous. We only assume that u is a weak solution of nite energy (see [1]). In particular, we do not suppose that the distributional derivative @t[b(u)] is a bounded Borel measure or a...

Nonlinear Elliptic and Parabolic Equations with Fractional Diffusion

Backward Uniqueness for Parabolic Equations

It is shown that a function u satisfying |∂t + u| M (|u| + |∇u|), |u(x, t)| MeM|x| in (R \ BR) × [0, T ] and u(x, 0) = 0 for x ∈ R \ BR must vanish identically in R \ BR × [0, T ].

Elliptic and Parabolic Equations

Elliptic equations: 1. Harmonic functions 2. Perron’s method 3. Potential theory 4. Existence results; the method of suband supersolutions 5. Classical maximum principles for elliptic equations 6. More regularity, Schauder’s theory for general elliptic operators 7. The weak solution approach in one space dimension 8. Eigenfunctions for the Sturm-Liouville problem 9. Generalization to more dimen...

Quantitative uniqueness estimates for second order elliptic equations with unbounded drift

In this paper we derive quantitative uniqueness estimates at infinity for solutions to an elliptic equation with unbounded drift in the plane. More precisely, let u be a real solution to ∆u + W · ∇u = 0 in R2, where W is real vector and ‖W‖Lp(R2) ≤ K for 2 ≤ p <∞. Assume that ‖u‖L∞(R2) ≤ C0 and satisfies certain a priori assumption at 0. Then u satisfies the following asymptotic estimates at R ...