Marcelo R. Ebert

Theory of damped wave models with integrable and decaying in time speed of propagation

We study the Cauchy problem for damped wave equations with a time-dependent propagation speed and dissipation. The model of interest is utt − a(t)2Δu + b(t)u t = 0,u(0,x) = u0(x),ut(0,x) = u1(x). We assume a ∈ L1(ℝ+). Then we propose a classification of dissipation terms in non-effective and effective. In each case we derive estimates for kinetic and elastic type energies by developing a suitable WKB analysis. Moreover, we show optimality of results by the aid of scale-invariant models. Finally, we explain by an example that in some estimates a loss of regularity appears.

Global Existence of Small Data Solutions to the Semilinear Fractional Wave Equation

In this paper, we find the critical exponent for the global existence of small data solutions to the semilinear fractional wave equation in low space dimension.

Partial Differential Equations in Models

We begin with a discussion of various demands on mathematical modeling. We explain how to model technical processes as convection, diffusion, waves, or hydrodynamics. For this reason we introduce partial differential equations as Laplace equations heat equations wave equations or Schrodinger equations that play a central role in applications. These models are treated in later chapters.

Basics for Partial Differential Equations

This chapter is devoted to mathematical prerequisites, including a detailed discussion of classification of partial differential equations and systems of partial differential equations, as wells classification of domains in which a process takes place, of notions of solutions and additional conditions as initial or boundary conditions to the solutions.

Linear Hyperbolic Systems

This chapter is devoted to aspects of linear hyperbolic systems. We have in mind mainly two classes of systems, symmetric hyperbolic and strictly hyperbolic ones. First we discuss these classes of systems with constant coefficients. Fourier analysis coupled with function-theoretical methods imply well-posedness results for different classes of solutions. Then, we treat such systems with variable coefficients. On the one hand we apply the method of characteristics introduced in Chap. 6 to derive a local existence result in time and space variables. On the other hand we discuss the issue of energy estimates and well-posedness for both cases of symmetric hyperbolic and strictly hyperbolic systems.

Phase Space Analysis for the Heat Equation

This chapter explains in an elementary way via the Cauchy problem for the heat equation without with and mass term how phase space analysis and interpolation techniques can be used to prove L p − L q estimates on and away from the conjugate line \(\frac {1}{p}+\frac {1}{q}=1\), p ∈ [1, ∞]. Here we distinguish between L p − L q estimates for low regular and for large regular data.

Heat Equation—Properties of Solutions—Starting Point of Parabolic Theory

There exists comprehensive literature on the theory of parabolic partial differential equations. One of the simplest parabolic partial differential equation is the heat equation. By means of this equation we explain qualitative properties of solutions as maximum-minimum principle, non-reversibility in time, infinite speed of propagation and smoothing effect. Moreover, we explain connections to thermal potential theory. Thermal potentials prepare the way for integral equations for densities in single- or double-layer potentials as solutions to mixed problems.

Wave Equation—Properties of Solutions—Starting Point of Hyperbolic Theory

There exists comprehensive literature on the theory of hyperbolic partial differential equations. One of the simplest hyperbolic partial differential equations is the free wave equation. First, we introduce d’Alembert’s representation in 1d and derive usual properties of solutions as finite speed of propagation of perturbations, existence of a domain of dependence, existence of forward or backward wave fronts and propagation of singularities. There a long way to get representation of solutions in higher dimensions, too. The emphasis is on two and three spatial dimensions in the form of Kirchhoff’s representation in three dimensions and by using the method of descent in two dimensions, too. Representations in higher-dimensional cases are only sketched. Some comments on hyperbolic potential theory and the theory of mixed problems complete this chapter.

Laplace Equation—Properties of Solutions—Starting Point of Elliptic Theory

There exists comprehensive literature on the theory of elliptic partial differential equations. One of the simplest elliptic partial differential equations is the Laplace equation. By means of this equation we explain usual properties of solutions. Here we have in mind maximum-minimum principle or regularity properties of classical solutions. On the other hand we explain properties as hypoellipticity or local solvability, too. Both properties are valid even for larger classes than elliptic equations. Moreover, a boundary integral representation for solutions of the Laplace equation shows the connection to potential theory. Boundary value problems of potential theory of first, second and third kind are introduced and relations to the theory of integral equations are described.

Method of Characteristics

The method of characteristics is applied in studying general quasilinear partial differential equations of first order sich as, for example, convection or transport equations. It is shown how the notion of characteristics allows for reducing the considerations to those for nonlinear systems of ordinary differential equations. An application to the continuity equation describing mass conservation completes this chapter.