Higher-order boundary regularity estimates for nonlocal parabolic equations

Blow-up Estimates for Higher-order Semilinear Parabolic Equations

Boundary Value Problems for Higher Order Parabolic Equations

We consider a constant coefficient parabolic equation of order 2m and establish the existence of solutions to the initial-Dirichlet problem in cylindrical domains. The lateral data is taken from spaces of Whitney arrays which essentially require that the normal derivatives up to order m−1 lie in L2 with respect to surface measure. In addition, a regularity result for the solution is obtained if...

Nonlocal higher order evolution equations

In this paper we study the asymptotic behavior of solutions to the nonlocal operator ut(x, t) = (−1) n−1 (J ∗ Id − 1)n (u(x, t)), x ∈ R which is the nonlocal analogous to the higher order local evolution equation vt = (−1)(∆)v. We prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. Moreover, we prove that the solutions of the nonlocal problem conver...

Boundary Regularity Estimates for Nonlocal Elliptic Equations in C and C Domains

We establish sharp boundary regularity estimates in C and C domains for nonlocal problems of the form Lu = f in Ω, u = 0 in Ω. Here, L is a nonlocal elliptic operator of order 2s, with s ∈ (0, 1). First, in C domains we show that all solutions u are C up to the boundary and that u/d ∈ C(Ω), where d is the distance to ∂Ω. In C domains, solutions are in general not comparable to d, and we prove a...

Regularity estimates for parabolic integro- differential equations and applications

We review some regularity results for integro-differential equations, focusing on Hölder estimates for equations with rough kernels and their applications. We show that if we take advantage of the integral form of the equation, we can obtain simpler proofs than for second order equations. For the equations considered here, the Harnack inequality may not hold. Mathematics Subject Classification ...