The Studies Of The Solution For A Class Of Quasilinear Elliptic Equations

In this paper, we study the quasilinear elliptic equations with dynamical boundary conditions as follows u=0, on Γ0×(0,T)(1.2) u(x,0)=u0(x), ut{x,0)=u1(x),on Γ上(1.4) where p <m+1<1+2,2<p <n, p <1+2<(n-1)p/n/p and u0(x), u1(x) are the initial functions which we have known.The existence of global solutions, as well as the energy decay estimate of global solutions and the blow-up of the local solution for system(1.1)-(1.4) are discussed by the theory of potential well in this paper.In chapter two, the potential well theory is introduced and the related theorems are given.In chapter three, the existence theorem of global weak solutions of system (1.1)-(1.4) is obtained with the help of potential well theory and Galerkin method.Theorem1Let u0(x)∈V0,u1(x)∈L2(Γ1)∩Lm+1(Γ1). If u0(x)∈W and E(0)<d, then there exist weak solutions for system (1.1)-(1.4) such that u∈L∞(0,T;V0), ut∈L∞(0,T;L2(Γ1)∩Lm+1(Γ1)),and u∈W,0<t<+∞In chapter four, by M.Nakao’s difference inequality, a decay estimate of global solution for system (1.1)-(1.4) is derived.Theorem2Let u0∈W,u1∈L2(Γ1), if E(0)<d, then we can get decay estimate as follows In the last chapter, the sufficient condition of the nonexistence for the local solution isobtained by the concavity method and the theory of potential well.Theorem3Let u (x,t)is a local solution for system (1.1)-(1.4) for t [t0,Tmax), ifthere exist a t0[0,Tmax)satisfyingu (t0,x) Weand E (t0) d,then the solution blows upin finite time.