Properties of the Support of Solutions of a Class of 2-Dimensional Nonlinear Evolution Equations.

Abstract

In this work we consider equations of the form ∂tu + P(D)u + u^{t}∂xu = 0, where P(D) is a two-dimensional differential operator, and l ∈ N. We prove that if u is a sufficiently smooth solution of the equation, such that supp u(0), supp u(T) ⊂ [−B, B] × [−B, B] for some B > 0, then there exists R_0 > 0 such that supp u(t) ⊂ [-R_0,R_0]×[-R_0,R_0] for every t ∈ [0, T].

Keywords

• Nonlinear evolution equations
• weighted Sobolev spaces
• Carleman estimates