Research On Well-posedness Of Time Fractional Wave Equations

In this thesis,we mainly study the well-posedness of time fractional wave equation,which is applied to simulate the anomalous diffusion phenomena,a signalling problem closely related to seismology,the memory and hereditary properties of various materials and processes etc.This thesis includes five main chapters:In Chapter 1,we introduce some preliminaries which will be useful throughout this thesis.Based on the fact that the time fractional wave equation can be transformed into an abstract fractional evolution equation in a specific domain,in the first section of Chapter 2,by using the operator theory of cosine family,we give two integral representations of the solutions for fractional evolution equations,and then the properties of the related solution operators are considered.Because the time fractional wave equations can simulate the propagation of mechanical waves with random effects in viscoelastic media,in the second section,we study a class of time fractional stochastic wave equations.First,we transform this type of equation into an abstract form of fractional evolution equation combining with the related results of the previous section,moreover,by applying stochastic analysis tools and fixed point theory,we establish the existence and approximate controllability of solutions for the present equation.From the point that time fractional wave equations with control term can be regarded as a fractional evolution inclusion problem,in the third section,we consider a class of nonlocal fractional evolution inclusion problems,by applying the theory of measure of noncompactness,we get the existence of solution for the problem and the compactness of solution set.As an application,we apply this problem to a controllability problem of time fractional wave equation.Considering the property that time fractional wave equation with damped term can be used for describing the interaction between vector electric field and electromagnetic characteristics of materials,in Chapter3,we mainly study the well-posedness of time fractional damped wave equation.Firstly,by applying the method of eigenvalue expansion,we give a representation of formal solution for the problem and a definition of mild solution.Moreover,we establish the well-posedness and regularity results of linear problem,and by applying the compact method and fixed point technique,we also obtain the existence,continuation and blow-up selection for the semilinear problem.In Chapter 4,we study a time fractional wave equation in the whole space,by means of harmonic analysis and the properties of MittagLeffler functions,we first define a new mild solution.Furthermore,we consider the related properties of the solution operators and the global well-posedness of the linear problem,by applying the properties of the solution operators,we obtain local well-posed results and related blow-up selection in different spaces for the nonlinear problem.In Chapter 5,we study the existence and regularuzation of solution for a backward problem of a class of nonlinear time fractional wave equations.By using the methods of eigenvalue expansion and operator theory,we first get a definition of mild solution,we further consider the properties of these solution operators and the existence and uniqueness of solution of the present problem.Since the problem is severely ill-posed,by means of the general regularuzation method,we also consider the convergence rate of the regularized solutions.