Existence Of Solution For A Fourth-order Parabolic Equation

In this thesis,we study the initial-boundary value problem for a four-order parabolic equation ut+Δ2u =γdiv(|u|m(?)u),x ∈Ω,t ∈(0,T),(?)u/(?)r =(?)Δu/(?)n= 0,X∈(?)Ω,t ∈(0,T),u(x,0)= u0(x),X∈Ω,where Ω(?)RN(n≥ 2)is a bounded domain,with smooth boundary(?)Ω and λ>0 is a constant.Using Kato’s method,we establish the existence,uniqueness and regularity of the solu-tion to the model,in suitable spaces,namely C0([0,T];Lp(Ω))where p = n(m+1)/2,with m>0,n ∈ N,and n ≥ 2.We also show the global existence solution to the nonlinear parabolic equation for small initial data.Our main tools are Lp-Lq estimates,regularization property of linear part of e-tΔ2 and successive approximations.