A Class Of Fully Nonlinear Parabolic Equations And Its Application In Image Restoration

In this paper we study a class of fully nonlinear parabolic equations with sourceswhere m≥1, 0≤u₀∈C₀(R^N),Φ(x,s) andΨ(x,s) are nonlinear functions withx∈R^N, s∈[0,∞). They come from a variety of diffusion phenomena appearing widelyin nature, such as the resistive diffusion of a force-free magnetic field,dynamics of bio-logical groups, curve shortening ?ow, spread of infectious disease, elasticity with damp-ing, Bellman-Dirichlet type problem and so on. The main aim of this paper is to study thequalitative theory of this class of equations and their application in image restoration.In Chapter 2, we firstly study the classical solvability forΦ(x,u) > 0. It shouldbe noticed that the existence of classical solutions is hard to be obtained in general sincethat for m = 0, the equation is degenerate or singular at the points where ?u(x,t) = 0.As far as we know, there are few papers concerned with the classical solutions of thiskind of equation, especially for multi-dimensional case. But it is possible to happen. Ourmethod is based on topological degree, which allows us to transform the problem intothe classical solvability of some kind of linearized equations, which can be solved byusing the Rothe method, together with some technique in the regularity theory about thegeneralized porous medium equation. In addition, the comparison principle also plays animportant role for studying this problem, the proof of which based on the time evolutionof the extreme of solutions is simple but interesting. Furthermore, we study the localexistence of continuous solutions for degenerate caseΦ(x,u) = uq(q≥0) and singularcaseΦ(x,u) = uq(q < 0) respectively. The desired generalized solutions will be obtainedas the limit of some subsequence of classical solutions of some regularized problem. So,we need to establish some uniform estimates on those solutions. The key one is the Ho¨ldernorm estimates, which are obtained based on some functional framework. Moreover,by the monotonicity of the operator and the compactness of Lp(0,T; B), together withsome domain decomposition techniques, we obtain the strong convergence of Laplacianof those classical solutions and then obtain the local existence of continuous solutions bya limiting process. In Chapter 3, based on the local existence and comparison principle of solu-tions obtained in Chapter 2, we study the long time behavior of solutions for the caseΦ(x,u) = uq(q∈R) andΨ(x,u) = up(p > 0). It was shown that there exist two crit-ical exponents of p: the global existence exponent p0 and the critical blow-up exponentpc. Noticing the fully nonlinearity of the diffusion operator makes problematic applyingmany effective methods from the classical theory of weak solutions, we introduce nonlin-ear local capacity and investigate the blow up profile for p > q +m by capacity estimates.Furthermore, by introducing sub- and super-solutions, we investigate the blow-up andglobal existence profile for the case 0 < p≤q + m and the global existence profile forp > pc. Different from the former research on the critical exponents of p, we find thereexists a critical exponent for q: q0 = m, such that if q > m, the interval of p in which nonontrivial global solution exists vanishes.In Chapter 4, we apply a class of fully nonlinear parabolic equations in imagerestoration, which remedy the defect of TV model on"ramps". Firstly we study a classof adaptive denoising modelsBased on nonlinear semigroup theory, we study the well-posedness of this model. Thepropagation direction of horizontal curve in image depends on the sign of ?u. The nu-merical experiments illustrate its effectiveness in protecting boundaries and ramps withno staircase effect. Next we applyin multiplicative denoising for ramps. The diffusion speed is proportional to the noiseintensity and the diffusion stops in the region with no noise.