Long Time Behavior Of Fractional Porous Medium Equations

In this paper,we are mainly concerned with the long-time dynamic for solu-tion of the following porous medium equation with fractional diffusion operators:where(-?)?/2 fractional operator,? ?(0,2),m ? 1,? is a bounded domain in RN(N ? 1)with sufficiently regular boundary(?)?,u0 is the initial value,h ? L?(?)is the time-independent external force and g(s)is the nonlinearity term.In Chapter 3,we mainly consider m>1 and the equation with the spectral fractional Laplacian.This Chapter has three parts.In the first part we study the existence of solution for the equation with the nonlinear term which is polynomial growth.We get that for every u0 ? Lm+1(?),there exists a weak solution u(x,t)such that um(x,t)? L2(0,T;H0?/2(?))Because of the nonlinearity of main oper-ator(Au =(-?)?/2(|u|m-1u)),the Faedo-Galerkin method is difficult to work.We discretize the time,consider the difference equation,and then approximate the solution by the solution of the difference equation to overcome the difficult.The nonlinearity of g leads to the invalidity of the method used to prove the existence of solution in[89].Actually,the nonlinear term causes the contraction of the operator(I + ?A)-1 is lost.Hence,the Crandall-Liggett theorem is not work,while this theorem is critical method to prove the existence of solution in[89].We apply compactness theorem to work out the problem.In the second part,we investigate the well-posedness of the equation with the nonlinear term which has not an upper growth restriction.We prove that the existence of the solution for the equation with initial value in Lm+1(?)and the approximate solution generate a continuous semigroup in L1(?)(see theorem 3.16).Firstly,we prove the existence of the solution with initial value in Lm+1(?).On the one hand,we also discretize the time and consider the difference equation.However,because of the nonlinear term without an upper growth restriction,common methods(such as the fixed point methods and the calculus of variations)is difficult to prove the iteration is work,which is actually the solvability of an elliptic equation.In order to overcome the difficult,we apply the Faedo-Galerkin method for the elliptic equation.We first prove the existence of solution for equations in Rn(see lemma 3.4),and then get our approximate solution of elliptic equation.However,even the equations are finite-dimensional,it is difficult to prove the existence of the solutions,because of the nonlinear term.We use the theory of maximal monotone operators to solve this problem.On the other hand,we can not get the weak limit of g(u?),because of the nonlinear term without an upper growth restriction.Hence,we apply the theory of Orlicz space to overcome the problem.Secondly,in order to prove the approximate solution generate a continuous semigroup in L1(?),we first prove the existence of the approximate solution in L1(?).Because the equation is degenerate,we add a perturbation term to get the solution of the equation with initial value u0 ? Cc?(?)and then we get the L1-L? estimate for the approximate solution.At last,we approximate the non-smooth initial value in L1(?)by sufficiently smooth initial value in Cc?(?),and then we get the uniqueness.the last part,we investigate the long-time behavior of the equation.We get the existence of global attractor by prove the system has a compact absorb set(see theorem 3.18).Then,we consider the dimension of global attractor by the theory of Z2-index.We prove that the dimension of the global attractor is infinity(see theorem 3.21).In Chapter 4,we concern with m = 1 and the equation with the restricted fractional Laplacian.We prove that the well-posedness of weak solutions(see theorem 4.2),the existence of(L02(?),L02(?)-global attractor(see theorm 4.5)and the existence of(L02(?),H?/2(?)-global attractor(see theorm 4.8).There exists two difficulties for the absence of an upper growth restriction of g.Firstly,We prove the existence of weak solutions by the Galerkin method.However,it is impossible to estimate the boundedness of g(um)(per Galerkin sequence um)to determine its weak limit.In order to overcome this difficulty,we apply the weak compactness theorem in an Orlicz space.Secondly,in order to prove the uniqueness,we act the truncation function ?k(w)of w to the difference of two equations instead of w.However,there exists a new problem:how to estimate the main operator.we use the method of ?-harmonic extension to overcome the problem([25,89]).Furthermore,we obtain the(L02(?),L02(?)-global attractor.Because of we can not to prove the continuity of semigroup in H?/2(?),in order to overcome this difficulty,inspired by the idea of the norm-to-weak continuous semigroup proposed in[77],we obtain the existence of a(L02(?),H0?(?))-global attractor.Due to to the shortage of the regularity of solutions in more regular phase space.To overcome this difficulty,we prove the asymptotic compactness in H?/2(?)of the semigroup by virtue of asymptotic a priori estimates.Combining the theory of the norm-to-weak continuous semigroup,we prove the existence of a global attractor in H?/2(?).