The Cauchy Problem For A Class Of Metaparabolic Equations

This paper is concerned with the Cauchy problem of the nonlinear metaparabolic equation whereΝ(u)=λ|u|σu,α>0,λ∈R.In the weighted Lebesgue space Lp,α,we deal with the study of the existence and uniqueness and the Long-time asymptotic behaviour of solutions of the Cauchy problem.In the case when the initial date are small our approach is based on a detailed study of the Green's function and estimate of Symbol and the use of the contraction-mapping method,we establish the existence and uniqueness of the global solutions and the solution has the long-time asymptotics u(t,x)=AG0(t,x)+O(t-(?)-γ), (?)t→∞, as t→∞uniformly with respect to x∈Rn,where 0<γ<min((?),(?)σ-1),G0(t,x)= t-(?)G(t-(?)(·))and A is a constant.In the other case,we can remove the smallness condition on the initial dataυ0(x), and letλ<0,σ>0(n=1,2,3,4).We can prove the existence and uniqueness of the global solutions.Moreover,letλ<0,σ>(?),we can have the above long-time asymptotic behaviour. In order to prove the conclusion,we need to use the four inportant lemmas and the Sovolve's embedding theorem,Fourier splitting method,contraction-mapping method and so on.