On Hyperbolic/Parabolic σ-Evolution Equations

σ-evolution equations generalize the classical models of wave equations, heat equations, Schrodinger equations, Klein-Gordon equations, Petrowsky type equa-tions and Non-Kowaleskian type equations, etc. They are useful models for the description of anomalous dynamic behaviors, such as charge carrier transport in amorphous semiconductors, nuclear magnetic resonance diffusometry in percolative and porous media, transport on fractal geometries, diffusion of a scalar tracer in an array of convection rolls, dynamics of a bead in a polymeric network, transport in viscoelastic materials, etc.The main aim of this monograph is to study the regularity behavior of solutions for strong/weak hyperbolicσ-evolution equations with time-dependent coefficients, higher-order energy decay rates for a special type of hyperbolicσ-evolution equa-tions with structural damping and viscoelasticity, and non-null controllability theory for a type of parabolicσ-evolution equations.In Chapter 3, as an elementary introduction to the phenomenon of loss of regu-larity and difference of regularity of initial Cauchy data, we mainly consider Cauchy problem of weakly hyperbolic equations with finite/infinite degenerating coefficients. With the theory of Confluent Hypergeometric Functions, one can represent the so-lutions explicitly and obtain the precise estimates by considering asymptotic behav-iors of the special functions. This chapter also demonstrates the optimality of the regularity discussion in Chapter 5.In Chapter 4 and Chapter 5, for hyperbolic a-evolution models with singu-larity near the origin, the joint influence from the principalσ-Laplacian operator, degenerating part and oscillating part is of prime concern in the discussion of regu-larity behavior of the solutions. We apply the sophisticated techniques from modern micro-local analysis to explore the upper bound of loss of regularity and give the formula in precisely calculating the critical point which separates the loss case from the no loss one. Furthermore, in order to demonstrate the optimality of the esti- mates, delicate counter-examples with periodic coefficients will be constructed to show the lower bound of loss of regularity by the application of Floquet theory and instability arguments.In Chapter 6, we mainly consider a special type of hyperbolicσ-evolution mod-els with structural damping and viscoelasticity. Three types of damping phenom-ena will be considered carefully, namely, constant dissipation, increasing structural dissipation and decreasing structural dissipation, which represent the distribution resistance of ideal conductor, semiconductor and superconductor in the electromag-netic field respectively. The joint effect from the principalσ-Laplacian operator, dissipation types and the regularity of the initial Cauchy data will be explored by the modern micro-local analysis. In particular, there is a sort of compensation phe-nomenon between the regularity of initial Cauchy data and the high-order energy decay rates.In Chapter 7, we mainly consider the non-null controllability theory for a type of parabolicσ-evolution equations comprised of self-adjoint positive definite operators with compact resolvent on the smooth manifold with boundary. Applying the theory of pseudo-differential operators on the smooth manifold, we transform the exact controllability into a dual problem. Afterwards, one reduces the dual problem into a Moment problem, and typical initial data leading to non-null controllability will be constructed accordingly.In this monograph, all theσ-Laplacian models are discussed on RN, actu-ally, the conclusions can be applied successfully to Laplace-Beltrami operators on a smooth compact manifold without boundary, especially on the torus TN. For instance, many of our counter-examples are constructed on the torus due to the lack of finite propagation speed for general hyperbolic a-evolution equations whenσ≠1.