On a nonlinear parabolic equation with fractional Laplacian and integral conditions

Abstract

The paper deals with non-classical initial-boundary value problems for parabolic equations with a fractional Laplacian. We study the existence and uniqueness of a mild solution to our problem. The continuous dependence of the solution on the given data is shown and the ill-posedness of the mild solution at t = 0 is also considered. In order to avoid such ill-posedness, we construct a regularized solution using the Fourier truncation method. An $L^p$ error estimate and the convergence rate between the regularized solution and the exact solution are obtained.

Keywords:

• Parabolic equation
• fractional Laplacian
• ill-posed problem
• regularization

2010 Mathematics Subject Classifications:

• 35R11
• 35B65
• 26A33

Disclosure statement

No potential conflict of interest was reported by the author(s).