Application of Liouville theorems for parabolic system with non linear gradient terms

Abstract

In this thesis, Liouville theorem was applied for parabolic sys tems with nolinear gradient terms. The analytical study design was used to achieve the stated objectives, different mathematical procedures were followed by estmating small positive constant, showed that existance of integers and at the end estimated the en ergy. The obtained result were liouville theoremfor scaling invari ant nonlinear parabolic equations and systems with no nontrivial whole solution ensure optimal universal estimates of solution to related initial and initial boundary value situations. In the case of non-negative solutions and the system, the converse is true. The parabolic Liouville theorem is true. whenever the correspond ing elliptic Liouville theorem for the system is true. We showed that the same result holds without the positivity assumptions on G and F. In particular, cover the primary conclution in Bartsch et al. (2010) in the scalar case. Furthermore, also prove a parabolic Liouville theorem for solutions in R n + ×R fulfilling homogeneous Dirichlet boundary conditions on ∂R n + ×R, as this theorem is re quired to prove universal estimates of related system solutions in Ω×(0,T), where Ω ∈ R n is a smooth domain. Finally, we prove universal estimates for specific parabolic systems using our Liouville theorem.