The Applications of Microlocal Analysis in æ-Evolution Equations
Xiaojun Lu · 2010
σ-evolution equations generalize the classical models of wave equations, Klein-Gordon equations, Petrowsky type equations 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 σ-evolution equations with time-dependent coefficients and long time behavior for a special type of σ-evolution equation with structural damping and viscoelasticity from the perspective of higherorder energy decay rates. As an elementary introduction to the phenomenon of loss of regularity, we will consider weakly hyperbolic equations with finite/infinite degenerating coefficients. With the theory of Confluent Hypergeometric Functions, one can represent the solutions explicitly and obtain the precise loss by considering asymptotic behaviors of the special functions. For σ-evolution models with singularity near the origin, the joint influence from the principal σ-Laplacian operator, degenerating part and oscillating part is of prime concern in the discussion of regularity behavior of the solutions. We will apply the sophisticated techniques from 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 case. Furthermore, in order to demonstrate the optimality of the estimates, 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. For σ-evolution models with structural damping and viscoelasticity, three types of damping phenomena will be considered carefully, namely, constant dissipation, increasing structural dissipation and decreasing structural dissipation, which represent the distribution resistance of ideal conductor, superconductor and semiconductor in the electromagnetic field respectively. The joint effect from the principal σ-Laplacian operator, dissipation types and the regularity of the initial Cauchy data