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.
