Quantization And Convergence Of Evolution Equations With Exponential Growth On A Closed Riemann Surface

Flow method is an effective method in partial differential equations and geometric analysis.It plays a positive role in solving the existence of solutions to elliptic equations,especially to the critical cases.In this dissertation,we are concerned in two kinds of nonlinear evolution equations:evolution equation with critical exponential growth and mean field type flow at a critical case on a closed Riemann surface.For the former,we analysis the concentration behavior of positive solutions,and particularly derive an energy identity for such a solution.For the latter,we study the convergence of the flow at a critical case.In particular,this gives a sufficient condition of the existence of solutions to the mean field equation.Gradient estimate is an important method in geometric analysis.One can use gradient estimates to derive Harnack inequalities,Liouville’s theorem and so on.In the last of this dissertation,we discuss gradient estimates and Liouville’s theorem of f-Laplacian nonlinear equations.Our main results areⅠ.Quantization for an evolution equation with critical exponential growth on a closed Riemann surfaceTheorem 0.1 Let(M,g)be a closed Riemann surface.Given any smooth initial function u0≥0,there exists a unique smooth solution u≥0 of the equationwith the invariance of the Moser-Trudinger energy for all t>0.For a suitable sequencee tk → ∞,the functions u(tk)→ cweakly in H1(M),Mwheree u∞ is a solution to the problemm-△gu∞+τu∞=λ∞u∞eu∞2 for some constant λ∞ ≥ 0.Moreover,either u(tk)→ u∞ strongly in H1(M),(M),λ∞ ＞ 0and 0<u∞ ∈ H1(M),or there exist N ∈N and points x(ⅰ)∈ M different from each other,I(ⅰ)∈N,1,≤i≤N,such that as κ→∞,we have(|▽gu(tk)|2+τu2(tk))dυg(?)(|▽gu∞|2+τu∞2)dυg+ ∑i=1 N 4πI(l)δx(l)akly in the sense of measures.Ⅱ.The convergence of the mean field type flow at a critical caseTheorem 0.2 Let(M,g)be a closed Riemann surface and Q be a smooth function on M with ∫M Qdυg=8π.Suppose that υ0(x)∈ C2+α(M),υ(t)is the solution of Suppose that A(x)2 log h(x)achieves its maximum at x0 and Q(x0)>2K(x0),where K is the Gauss curvature of M.Then there exists an initial data υ0 ∈ C2+α(M)such that υ(t)converges to a smooth function υ∞ in H2(M)and υ∞ satisfyies△υ∞-Q+8π eυ∞/∫Meυ∞dυg=0 In particular,we reprove the result of Ding-Jost-Li-Wang[1].Ⅲ.Gradient estimates of nonlinear equations for f-LaplacianTheorem 0.3 Let(M,g,e-f dυg)be an n-dimensional complete noncompact sm ooth metric measure space with Ricfm ≥-(n+m+1)H for H ≥ 0.Suppose that h(x)∈ C2(M)and △fh ≥ 0.If u(x)is a positive solution of△fu + huα=0 for 1<α<n+m/n+m+2(n+m≥4),then As a corollary,we obtain the following Liouville theorem.Corollary 0.4 Let(M,g,e,e-f dυg)be an n-dimensional complete noncompact smooth metric measure space with Ricfm 0.Suppose that h(x)∈C2(M),△fh ≥0 and there exists a point x0 ∈ M such that h(x0)≥ 0.If 1<α<n+m/n+m-2(n+m≥4),then equation △fu + huα=0 doesnsn’t have a nonconstant positive solution on M.Theorem 0.5 Let(M,g,e-f dυg)be an n-dimensional complete noncompact sm ooth metric measure space.Assume the m-Bakry-Emery Ricci tensor of M is bounded from below by-(n+m-1)H,and H 0.Suppose h(x,t)is a function defined onM ×[0,∞)which is C2 in the x-variable and C in the t-variable.If u(x,t)is a positive solution of(△f-(?)/(?)t)u(x,t)+h(x,t)uα(x,t)=0 on M ×[0,∞).For 0<α<1,assume that h ≥0,△fh≥ 0.Then|▽u|2/u2 + 1/αhuα-1-1/α ut/u ≤ m+n1/2α2t +(m+n)(n+m-1)H/2α(1-α)For α≥1,assume that-M1≤h≤M3,△fh≥-K1,and |▽h|≤K2.Thenwhere M2=sup{uα-1(x,t)|(x,t)∈M ×[0,∞} and c4>0,s>α are constants.