Shrinkage Of The Moment Of The Set Of Parabolic Equations

This paper discusses the instantaneous shrinking property of the support for solutions of semilinear heat equations and degenerate parabolic equations. Under the condition0 ≤ u_x) → 0(|x| →∞),we prove that the solution of the following problem has the ISS property:where m ≥ 1, 0 < p < 1.We also study some properties, such as existence, uniqueness and the ISS phenomenon of the following parabolic variational inequalities:Comparison principles are established for these equations, and by these principles, we investigate the relationship between the absorption term and the initial data which implies the occurence of the the instantaneous shrinking phenomena. Also the sharpness of these conditions are showed in a proper way.